Download SELECTED TOPICS IN QUANTUM MECHANICS Pietro Menotti

1
SELECTED TOPICS IN QUANTUM MECHANICS
Pietro Menotti
Dipartimento di Fisica, Università di Pisa, Italy
2
INDEX
Foreword
Chapter I:
Remarks on the black-body radiation
5
1. Introduction
2. Wien’s law and the number of photons
3. Energy fluctuations in the black body
References
Chapter II:
The superposition principle
11
1. Observables, spectrum, eigenstates and probability
2. The superposition principle
3. Standard expansion of a vector
4. Observable with bounded but otherwise arbitrary spectrum
5. Continuum spectrum in the Dirac formalism
References
Chapter III: The correspondence principle
19
References
Chapter IV: The time evolution
23
1. Introduction
2. Composition properties of U
3. Construction of the solution to the time evolution equation
4. Expansion in T -ordered integrals
References
Chapter V:
Properties of the hamiltonian operator
29
1. Introduction
2. Some properties of the spectrum
3. Lower boundedness of the spectrum
Chapter VI: Description of composite systems
33
Chapter VII: The density operator
37
3
1. Introduction
2. The time evolution of the density operator
3. Subsystems of composite systems
Chapter VIII: Measure in quantum mechanics
41
References
Chapter IX: Symmetry transformations
45
1. Space translations
2. Symmetry transformations
3. Rotations
4. The euclidean group
References
Chapter X:
The spin
55
1. Realization of half-integer angular momenta
Chapter XI: Identical particles
59
1. Introduction
2. Two particle system
3. Three particle system
Chapter XII: The formal theory of scattering
73
1. Introduction
2. The Lippmann-Schwinger equation
3. The adiabatic theorem
4. “In” and “out” states
5. The Moller wave function operators and the S matrix
6. The unitarity of the S matrix and the optical theorem
7. Example on partial waves
References
Chapter XIII: Scattering from a central potential
1. Introduction
2. Jost functions
3. Physical interpretation of the singularities
87
4
4. Levinson theorem
5. Construction of special potentials
6. Discussion of the singularities of the S matrix
References
Chapter XIV: Functional formulation of quantum mechanics
71
1. Introduction
2. Trotter formula
3. Appendix: Proof of Trotter formula
References
Chapter XV: The violation of Bell inequalities and the non separability
of quantum mechanics
1. Introduction
2. Clauser-Holt-Horn-Shimony inequality
3. Violation of Bell (CHHS) inequality in quantum mechanics
References
81
5
Foreword
The suggested books for the course on Quantum Mechanics were
P.A.M. Dirac The Principles of Quantum Mechanics Oxford University Press
L.D. Landau and E.M. Lifshitz Quantum Mechanics Pergamon Press
M. Born Atomic physics London, Blackie and Son Limited, Glasgow
Reference to other books were also frequently made and these are found at the end of
each chapter of these notes.
6
Chapter 1
Remarks on the black-body radiation
1.1
Introduction
The energy density of the electromagnetic field is given by
u=
ǫ0 2
(E + c2 B 2 ) (MKS);
2
u=
1
(E 2 + B 2 ) (Gauss) .
8π
It is a quadratic form in the fields Ej , Bj . As such the time average of the energy possesses
a spectral decomposition.
Given f (t), −T /2 < t < T /2 let us write
1
f˜(ωn ) = √
T
1 X iωn t ˜
e f (ωn );
f (t) = √
T n
Z
T
2
f (t)e−iωn t dt
− T2
ωn = 2πn/T . f˜(−ω) = f˜∗ (ω) due to the reality of f . The time average value is
1
T
Z
Z
1
1 ∞
1X ˜
2
2
2
˜
˜
|f (ωn )| →
dω|f (ω)| =
|f (t)| dt =
dω|f(ω)|
T
2π
π
−T /2
0
n
Z
T /2
2
Thus we can write
u(T ) =
Z
∞
uν (ν, T )dν .
0
Experimentally equilibrium in a black body is reached in very short times, of the order
of L/c being L the linear dimension of the black body.
Kirchhoff’s theorems assure the universality of uν (ν, T ), i.e. the independence form the
material of the walls, volume, shape of the cavity and from the position of the detected
radiation.
The proof relies on the II principle of thermodynamics.
7
8
CHAPTER 1. REMARKS ON THE BLACK-BODY RADIATION
Stefan-Boltzmann law
u(T ) = K1 T 4
The derivation is based on
1) II principle of thermodynamics Q/T = dS;
2) Electromagnetic nature of the radiation: in fact from the expression of the momentum
density
P = ǫ0 (E ∧ B) (MKS); P =
1
(E ∧ B) (Gauss)
4πc
we have for a plane wave
u
c
and for an isotropic radiation it follows for the pressure p = u/3. Moreover by integration
|P | =
one obtains for the entropy
4
S = K1 V T 3 + const.
3
The radiance of the black body is given by
(1.1)
power = σT 4 ∗ area
where σ, is the Stefan constant related to K1 by the kinematic relation σ = cK1 /4.
Experimentally σ = 5.67 10−8 J/m2 s K 4 .
Example: The temperature of the sun is T = 6000 K. The apparent radius of the sun is
1/4 of degree = 0.0043. It follows that the power of the sun radiation on the earth is
w = σ (0.0043)2 (6000)4 K 4 = 1358 watt/m2 sec .
Wien’s law
ν
uν (ν, T ) = ν 3 f ( )
T
(1.2)
is derived from
1) Second principle of thermodynamics;
2) Doppler effect.
First part: If one compresses slowly the black radiation in a vessel with reflecting walls
the radiation stays black; only its temperature changes according to (1.1).
Second part: Render this compatible with the Doppler effect
v
ν ′ = ν(1 + 2 cos θ);
c
(v/c << 1) .
Result: the radiation has the functional form (1.2).
1.1. INTRODUCTION
9
1) By integration of Wien’s law, Stefan-Boltzmann law follows .
2) From eq.(1.2) Wien displacement law follows
λmax T = const;
experimentally
const = 2.9 10−3 m K
Example: The temperature of the sun is 6000K
6000 K 4800 10−10 m = 2.9 10−3 m K
4800 Angstrom = 4800 10−10 m (we are in the visible).
Universality allows us to replace the walls with a set of harmonic oscillators. In such a
setting one can compute the
average emitted power
w̄e =
2e2 2 2e2 ω 2
Ēν
|ẍ| =
3c3
3mc3
(Gauss);
e2 =
qe2
(MKS)
4πǫ0
and the average absorbed power
w̄a =
πe2
uν (ν, T ) .
3m
From the energy balance we have
uν (ν, T ) =
8πν 2
Ēν
c3
(1.3)
All the problem is to compute Ēν .
Replacing Ēν with the value given by the classical equipartition theorem 2 12 kT one
obtains the Rayleigh-Jeans law
8πν 2
uν (ν, T ) = 3 kT
c
experimentally well verified at low frequencies but in total disagreement at high frequencies (the ultraviolet catastrophe).
The number of normal modes of the electromagnetic field in a cavity is
4πν 2
8πν 2
dνV
=
2
dνV ≡ Nw .
c3
c3
(1.4)
As the electromagnetic field is equivalent to and ensemble of harmonic oscillators, we can
obtain the same result applying the classical equipartition principle to these oscillators
obtaining again, and much more quickly the Rayleigh-Jeans law.
10
CHAPTER 1. REMARKS ON THE BLACK-BODY RADIATION
Planck hypothesis: the harmonic oscillator has energy level quantized by integers
0, ǫν , 2ǫν , 3ǫν . . . . The partition function gives
X
Z=
e−βǫν =
n
Ēν = −
1
1 − e−βǫν
∂ log Z
ǫν
.
= βǫν
∂β
e −1
Substituting in (1.3) and imposing Wien law we have Planck’s law
uν (ν) =
8πν 2 hν
c3 (ehν/kT − 1)
from which
Stefan constant σ =
2π 5 k 4
;
15c2 h3
displacement constant λM T =
h c
.
k 4.965
With the experimental values of σ and of Wien displacement constant one obtains k =
1.38 10−23 J/K, h = 6.67 10−34 Js.
Why the spacing of the energy levels of the harmonic oscillator is assumed to be constant?
Let us examine the partition function for large T (small β).
Z ∞
X
−βEn
Z=
e
→
e−βE ρ(E)dE
0
n
where ρ(E) is the level density. Suppose that for E > EM the distribution be ρ(E) = c1 E α
with α > −1. For small β we have
Z EM
Z
−α−1
Z=
ρ(E)dE + c1 β
0
0
from which it follows
Ē =
∞
xα e−x dx ≈ c1 β −α−1 Γ(α + 1)
α+1
= (α + 1)kT .
β
If we do not want to violate the Rayleigh-Jeans law, which works fine at small frequencies,
we must choose α = 0 (constant asymptotic spacing). The simplest assumption is to
extend this constant spacing to all energies.
If α < −1 we have
Z
0
∞
ρ(E)dE < ∞
i.e. a finite number N of levels. In this case for T → ∞ we have
Ē = −
E1 + · · · + EN
∂ log Z
→
∂β
N
1.2. WIEN’S LAW AND THE NUMBER OF PHOTONS
11
i.e. the average of the energies. Again we violate Rayleigh-Jeans law.
Example: The best black body is provided by the cosmic background radiation.
The decoupling of the radiation from matter occurred about at 400 000 Yr when the
universe had a volume 1000−3 times the present volume. The present temperature is
T ≈ 2.725 K.
A temperature of decoupling of 3000K is consistent with V2 /V1 = T13 /T23 = 10003 due to
an adiabatic expansion i.e. conservation of entropy, see eq.(1.1).
Keep in mind that the cosmic background radiation has to be corrected by the motion of
the earth of about 600 Km/sec.
1.2
Wien’s law and the number of photons
If we accept that the black body radiation is a collection of photons, the number of
photons is given by
dn =
i.e.
1 ν 2 ν
dν
( ) f ( )V T 3
h T
T
T
V T3
N=
h
Z
∞
x2 f (x)dx
0
which under an adiabatic transformation V T 3 = const, see eq.(1.1), is left unchanged.
The Doppler effect
v
δν = 2 ν cos θ
c
with v << c speed of the moving mirror, is consistent with a momentum of the photon
p=
hν
c
1.3
and an energy of the photon E = hν. In fact the work of the mirror is
Z
Z
L = Fx vdt = v Fx dt = v∆px = v 2 p cos θ = ∆E .
Energy fluctuations in the black body
The average value of the square of the energy fluctuations is given by
−
∂ Ē
= (E 2 ) − (Ē)2 = (∆E)2 = kT 2 CV .
∂β
Starting from Planck’law
Ē =
8πν 2 hν
∆νV ≡ hνNp
c3 (ehν/kT − 1)
12
CHAPTER 1. REMARKS ON THE BLACK-BODY RADIATION
we obtain
(∆E)2 = hν Ē +
where Nw is given in eq.(1.4).
Ē 2
c3
2
2
Ē
=
(hν)
N
+
p
8πν 2 ∆νV
Nw
The first term is the typical particle fluctuation with Np independent systems (∆E)2 =
ε0 E ≈ ε20 Np . The second part does not contain h and as such is a classical wave contri-
bution to the energy fluctuations. In fact the number of classical waves in the volume V ,
and in the frequency interval ∆ν is given by
Nw =
8πν 2 ∆νV
c3
and if we write
Ē = ǫ(ν)Nw
we expect a fluctuation
∆E = ǫ(ν)
i.e.
p
Nw
(∆E)2 = ǫ(ν)2 Nw = Ē/Nw .
The presence in the fluctuation of the energy of two terms, one particle-like and the other
wave-like is the first evidence of the dual nature of matter (in the present case of light).
References
[1] M. Born “Atomic physics” London, Blackie and Son Limited, Glasgow
140423
Chapter 2
The superposition principle
2.1
Observables,
spectrum,
eigenstates
and
probability
1. Two states of the same physical system, produced by two different preparing apparatus
will be considered the same state if they give rise to the same statistical distributions of
results when measuring any measurable quantity.
2. We assume that it is meaningful to consider linear superposition of two or more states
with complex coefficients; in different words that the space of states is a vector space on
the complex numbers. This is supported by the success of the wave description of matter.
3. Given a measurable quantity we shall call spectrum of such a quantity the set of all
possible result of the measurement of such a quantity on all states of the system. Such a
spectrum is assumed to be composed of real numbers; complex spectra would correspond
to the “simultaneous measurement” of two real numbers, an occurrence we want to exclude
as we know that the measurement of whatever quantity can alter in an unpredictable way
the state of the system. For simplicity sake we shall start with the discrete spectrum.
We shall call eigenstate of a measurable quantity a state such that performing on it the
measure of such a quantity one obtains with certainty a given value (of the spectrum). If
the set of the eigenstates of a certain measurable quantity is a complete set in the space
of the states we shall call such measurable quantity an observable.
4. Probabilistic interpretation of the state vectors.
Generalizing Born probabilistic interpretation of the wave function we postulate that the
(i)
probability PA (ψ) to find the i-th value of the spectrum measuring the observable A on
13
14
CHAPTER 2. THE SUPERPOSITION PRINCIPLE
the state ψ is given by
(i)
(i)
PA (ψ) = (ψ, KA ψ) .
From this it follows that ψ e eiα ψ represent the same state.
If we consider the vector ψ ′ = ρψ we have that
(i)
(i)
PA (ψ) = (ψ ′ ,
KA ′
ψ)
|ρ|2
for any A and whatever i.
Thus we can say that the set of vectors ρψ, ρ 6= 0 (ray in Hilbert space) describe the same
state. It is useful to work with normalized vectors.
To the null vector we cannot associate a state because we cannot compute from it any
probability.
Notice that if ψ1 is eigenstate to the value a1 e ψ2 is eigenstate to the value a2 different
from a1 the two states are independent. This because if the two vectors were proportional
they would give the same probabilities and thus they would be experimentally indistinguishable, while being eigenstates to different eigenvalues they are distinguishable. More
generally the vector ψ = c1 ψ1 + c2 ψ2 never will be the null vector for c1 e c2 both different
from zero and ψ1 e ψ2 representing different states.
2.2
The superposition principle
We shall assume the following “superposition principle”:
Given a state ψ, combination of eigenstates ψk of the observable A of spectrum {ak }, the
possible results of the measure of A on ψ are the ak relative to the ψk which appear in
the expansion of ψ.
It follows that if ψ1 e ψ1′ are vectors representing eigenstates of A to the same values a1
we have that measuring A on ψ = c1 ψ1 + c′1 ψ1′ we shall find with certainty the value a1
and thus ψ is an eigenstate of A to the value a1 . Thus we can speak of eigensubspaces of
an observable A relative to a given value of the spectrum.
Being the set of the eigenstates of A complete, every state vector can be written as
ψ=
X
ci ψi
(2.1)
i
with ψi eigenstates of A all relative to different values of the spectrum.
This because we can collect all vectors which represent eigenstates relative to the same
P
eigenvalue in a single eigenstate. Notice also that a combination ψ = i ci ψi with ci not
2.2. THE SUPERPOSITION PRINCIPLE
15
all equal to zero cannot be the null vector; otherwise we would have
ψ1 = −
1 X
ci ψi
c1 i>1
which is contradictory as, from the r.h.s. measuring A on ψ1 we should find only values
different from a1 . Thus the states ψi are linearly independent.
(i)
We pose now the problem to express PA (ψ) in term of the coefficients of the expansion
(2.1) with normalized ψi .
To this end we keep the ψi fixed and we vary properly the ci .
According to our axiom we have
(i)
(i)
PA (ψ) = (ψ, KA ψ)
and thus
(1)
PA (ψ) =
X
(1)
c∗l Alm cm .
lm
(1)
Choosing cl equal to 1 and the others zero we derive for l = 1 A11 = 1 and for l 6= 1
(1)
All = 0.
Choosing the vector obtained by normalizing φ = εψ1 + ψ2 i.e.
ψ=
and imposing
N 2 = (φ, φ)
ε∗ (1)
ε∗ ε
(1) ε
+
A12 + A21 2 ≥ 0;
2
2
N
N
N
(1)
PA (ψ) =
(1)
ε
1
ψ1 + ψ2 ;
N
N
∀ε
(1)
we have A12 = A21 = 0. Considering the vector obtained by normalizing φ = b2 ψ2 + b3 ψ3
with b2 e b3 not both equal to zero and imposing
(1)
(1)
PA (ψ) =
(1)
(1)
b∗2 A23 b3 + b∗3 A32 b2
=0
N2
(1)
we have A23 = A32 = 0 .
Thus we reached the conclusion
(i)
PA (ψ) = c∗i ci .
As a corollary we have
X
i
c∗i ci =
X
(i)
PA (ψ) = 1
i
as with certainty we obtain some result. From this the orthogonality property follows.
P
Let us consider in fact φ = j bj ψj , with bj not all vanishing. We know that such vector
16
CHAPTER 2. THE SUPERPOSITION PRINCIPLE
is not the null vector. If we denote by N its norm the normalized vector is ψ =
with cj =
1
b.
N j
Thus we have
X
X
X
b∗k bk .
c∗k ck =
b∗l (ψl , ψm )bm = N 2
N2 =
j
cj ψj
k
k
lm
P
This identity in the bi tells us that (ψl , ψk ) = δlk . Thus we reach the fundamental
result: Eigenvectors of the same observable relative to different values of the spectrum
are orthogonal.
2.3
Standard expansion of a vector
In each eigensubspace relative to the value ak of the spectrum of A one chooses a complete
(i)
orthonormal basis ψk . Every vector can be written in the form
X (i) (i) X
ψ=
ck ψk =
ck ψk
ki
k
with
X
(i)
(i)
ck ψk = ck ψk
from which
i
It follows that
(i)
PA (ψ) =
X
i
2.4
|ck |2 =
X
i
(i)
|ck |2 .
(i)
|ck |2 .
Observable with bounded but otherwise arbitrary
spectrum
We refer in this section to an observable with bounded but otherwise arbitrary spectrum
(point, continuous, mixed ... ). The treatment will be rigorous.
We divide the spectrum in a finite number of intervals [ai−1 , ai ), i = 1, . . . n.
Axiom 1: ProbA [ai−1 , ai ) = (ψ, KA [ai−1 , ai )ψ)
Axiom 2: If measuring A on the state φj we obtain with certainty a value in the interval
[akj −1 , akj ), the only values we can obtain by measuring A on the state
X
φj
j
are contained in the union of the intervals [akj −1 , akj ).
Axiom 3 (completeness): Every vector can be obtained as the sum of vectors
ψ=
n
X
i=1
ψ[ai−1 , ai )
2.4. OBSERVABLE WITH BOUNDED BUT OTHERWISE ARBITRARY SPECTRUM17
with the property that measuring A on ψ[ai−1 , ai ) (when ψ[ai−1 , ai ) 6= 0) we are sure to
find a value in [ai−1 , ai ).
Consequence 1: The non zero vectors ψ[ai−1 , ai ) are independent. Otherwise for some k
we could write ψ[ak−1 , ak ) as a combination of others. Then from the l.h.s. we would be
sure to measure a value in the interval [ak−1 , ak ) while from the r.h.s. we would be sure
to find a value outside [ak−1 , ak ).
Consequence 2 (uniqueness): If
ψ=
X
φj
(kj all different)
j
then φj = ψ[akj −1 , akj ) otherwise for some j we could write φj − ψ[akj −1 , akj ) 6= 0 as a
combination of vectors for which the measure gives results outside [akj −1 , akj ). Thus the
decomposition
ψ=
n
X
ψ[ai−1 , ai )
i=1
is unique, the operation ψ → ψ[ai−1 , ai ) is linear and the decomposition of ψ[ai−1 , ai ) is
just itself, i.e. the operation ψ → ψ[ai−1 , ai ) ≡ Pi ψ is a projector Pi = Pi2 . Repeating the
reasoning used in the discrete case we obtain
a) ProbA [ai−1 , ai ) = (ψ[ai−1 , ai ), ψ[ai−1 , ai ))
b) (ψ[aj−1 , aj ), ψ[ai−1 , ai )) = 0 for i 6= j
i.e. Pi are orthogonal projectors. In fact
0 = (Pi ψ, (1 − Pi )ψ) = (ψ, Pi+ (1 − Pi )ψ)
Pi+ = Pi+ Pi i.e. Pi = Pi+ , Pi Pj = 0 per i 6= j.
Thus the average value of the measures of A is
ā =
lim
max ∆i →0
X
a′i (ψ[ai−1 , ai ), ψ[ai−1 , ai )) =
i
lim
max ∆i →0
X
a′i (ψ, Pi ψ)
i
with ai−1 ≤ a′i ≤ ai .
Such a limit exists because
X
(ψ[ai−1 , ai ), ψ[ai−1 , ai )) = 1
i
(Compare two elements with one which belongs to a refinement of the two sets of intervals).
We construct the operator  by
Âψ ≡
lim
max ∆i →0
X
i
a′i ψ[ai−1 , ai ) .
18
CHAPTER 2. THE SUPERPOSITION PRINCIPLE
Such limit exists: Compare two element with one which belongs to the refinement of the
two sets of intervals and notice that due to orthogonality we have
X
||
ci ψ[ai−1 , ai )|| ≤ max(|ci |) ||ψ|| .
(2.2)
We can define Pi = E(ai ) − E(ai−1 ) with E(a) = P [−∞, a) and thus
Z
X
′
 = lim
ai (E(ai ) − E(ai−1 )) ≡ a dE(a)
(2.3)
i
max ∆i →0
i
where due to (2.2) the convergence is uniform. Â is an hermitean operator (bounded
self-adjoint) as uniform limit of hermitean operators and we have
ā = (ψ, Âψ).
The probability is given by
ProbA [aj−1 , aj ) = (ψ, (E(aj ) − E(aj−1 ))ψ) .
Every self-adjoint operator (also unbounded) can be written in the form (2.3) [1].
On the whole it has to be noticed that the real chracterization of an observable is its
spectral family. In fact as far as the spectrum is concerned if we consider instead of A,
tanh A, the spectrum of such operator, discrete or not, is contained in the interval [−1, 1].
The bounded operator tanh A is perfectly defined from the spectral decomposition of A.
This does not mean that we can use always observables represented by bounded operators.
E.g. Stone theorem relates the group of evolution operators U(t) with the hamiltonian
H which is its generator and in most cases such generator is an unbounded (self-adjoint)
operator (see Chapter IV and V). Working with tanh H would be completely unpractical.
2.5
Continuum spectrum in the Dirac formalism
The “transcription dictionary” between the above described spectral-family formalism
and the generalized eigenvectors formalism, in the simplest case is
Z b
P [a, b) =
da′ ψa′ ◦ ψa′
a−0
with (ψa′ , ψa ) = δ(a − a′ ).
In this paragraph we give without any pretense of rigor the treatment of an observable
with a non degenerate continuous spectrum in the formalism of Dirac i.e. employing a
vector space larger than an Hilbert space.
2.5. CONTINUUM SPECTRUM IN THE DIRAC FORMALISM
19
The rigorous treatment of such a space (the Gelfand triplet) is a very elaborate subject
[2]. On the whole, for a rigorous treatment the spectral-family treatment is simpler while
Dirac’s formalism is more elegant and more inspiring.
Given a physically measurable quantity Q with continuous spectrum such quantity will
be called observable if
1) For every point q of the spectrum a vector |qi is associated such that every vector of
the Hilbert space (or better a proper dense subset of vectors of the Hilbert space) can be
written as a continuous superposition of vectors |qi
Z
|ϕi = c(q)|qidq .
(2.4)
2) If a vector |ϕI i can be written as an integral on the interval I
Z
|ϕI i = c(q)|qidq
I
the only values of the spectrum of Q which can be obtained measuring Q on |ϕI i are
contained in the interval I.
Considering now finite linear combinations of vectors |ϕI1 i, |ϕI2 i etc. relative to intervals
with zero intersection and reasoning exactly as in the case of the discrete spectrum we
obtain the following results
a) given the normalized vector |ϕi i.e. hϕ|ϕi = 1
Z
|ϕi = c(q)|qidq
the probability to find, measuring Q on |ϕi, values contained in the interval I is given by
hϕI |ϕI i with
|ϕI i =
Z
c(q)|qidq
I
b) Vector |ϕIj i relative to intervals with zero intersection are orthogonal.
As for I1 ∩ I2 = ∅ we have
hϕI2 |ϕI1 i =
Z Z
I2
I1
c∗2 (q)hq ′ |qic1 (q)dq ′ dq = 0
for arbitrary c1 (q) e c2 (q), we have that hq ′ |qi = 0 for q ′ 6= q.
If we assume that hq ′ |qi is a function we reach the paradox that the norm of every vector
(2.4) is zero, as the integrand is different from zero only on the zero measure set q ′ = q.
This shows the necessity to widen the Hilbert space, if we insist in dealing with eigenstates
of an observable with continuum spectrum.
20
CHAPTER 2. THE SUPERPOSITION PRINCIPLE
The way out is to admit that hq ′|qi be a distribution, in particular
hq ′ |qi = k(q)δ(q ′ − q) .
The function k(q) must be positive otherwise we violate the positivity of the norm in
p
Hilbert space. Thus we can normalize the vectors |qi dividing by k(q) obtaining the
standard normalization.
hq ′ |qi = δ(q ′ − q) .
(2.5)
Using the standard normalization we have for the probability to find a value of the spectrum in the interval I is
PI = hϕI |ϕI i =
Z
c∗ (q)c(q)dq
I
from which it follows that the probability density is given by
P (q) = c∗ (q)c(q) .
Exploiting (2.5) we have
c(q) = hq|ϕi
and thus
Z
|ϕi =
|qihq|ϕidq
from which it follows the completeness relation
Z
|qihq|dq = 1 .
In completely analogous manner as done in the case of the discrete spectrum we can
associate the observable Q the operator Q̂ given by
Z
Q̂ = q|qihq| dq .
We have
hϕ|Q̂|ϕi =
For any polynomial P we have
P(Q̂) =
Z
Z
qP (q)dq = q̄ .
P(q)|qihq| dq
which is extended to any function f defined on the spectrum of Q by
Z
f (Q̂) = f (q)|qihq| dq .
2.5. CONTINUUM SPECTRUM IN THE DIRAC FORMALISM
21
References
[1] F. Riesz and B. Nagy Functional Analysis Dover Publications, New York
[2] I.M. Gelfand and N. Ya. Vilenkin Generalized functions Vol 5, Academic press, New
York, London
140427
22
CHAPTER 2. THE SUPERPOSITION PRINCIPLE
Chapter 3
The correspondence principle
We look in the space of operators for an operation which has the same algebraic properties
of the classical Poisson brackets, i.e. a binary operation (A, B) which induces on the space
of operators a Lie algebra [1,2].
Expanding
(Ai Aj , Bi Bj )
in two different orders and using Leibniz rule one obtains [1]
[Ai , Bi ](Aj , Bj ) = (Ai , Bi )[Aj , Bj ]
for whatever choice of the operators Ak Bk . Defined
X
[L] =
ai [Ai , Bi ]
i
(L) =
X
ai (Ai , Bi )
i
one obtains
[L](Aj , Bj ) = (L)[Aj , Bj ]
[L](L) = (L)[L] .
We admit that there exists an [L] which possesses the inverse [L]−1 .
It follows that
(Aj , Bj ) = [Aj , Bj ](L)[L]−1
(Aj , Bj ) = [L]−1 (L)[Aj , Bj ] .
But [L]−1 (L) = (L)[L]−1 and thus (L)[L]−1 commutes with all commutators and thus
with the algebra generated by all commutators. A set M of operators is said irreducible
if there exists no proper subspace of H invariant under the action of M.
23
24
CHAPTER 3. THE CORRESPONDENCE PRINCIPLE
Let M be an algebra of bounded operators such that if A ∈ M also A+ ∈ M
The following three statements are equivalent [3]
1. M is irreducible.
2. The commutant of M (i.e. M′ ) is cI.
3. Every vector ψ 6= 0 is cyclic i.e. linear span(Mψ) = H .
Thus if the algebra M generated by commutators is irreducible we have M′ = {λI}, and
thus
(Aj , Bj ) = c[Aj , Bj ]
and c is a universal constant. If we want to respect the same dimensional relations as in
the classical Poisson brackets, c must have the dimension of an action. Moreover imposing
that (A, B) with A e B hermitean be hermitean, we have that c has to be pure imaginary.
Let us set
1
[Aj , Bj ]
i~
where ~ is a constant with the dimensions of an action, to be determined experimentally.
(Aj , Bj ) =
Note: The statement (3) tells us that in case of irreducibility we can replace the vectors
with operators; to superpose two states Aψ and Bψ is equivalent to consider αA + βB
and thus the vector space of states is actually the linear space of operators.
Example 1.
Let us consider the Hilbert space of the L2 functions f (q) (space of the the wave functions
describing a single spinless particle). Consider on such a space the operators q̂k given by
the multiplication by qk and the operators t̂j given by the action of −i ∂q∂j . We have
[q̂k , t̂j ] = iδkj .
Thus we have one commutator [L] (actually more than one) which admits inverse [L]−1
and (L)[L]−1 will commute with the whole algebra generated by commutators. We have
[q̂kn , t̂j ] = inq̂ n−1 δkj
m−1
[q̂k , t̂m
δkj
j ] = imt̂
i.e. the algebra of commutators contains all powers on q̂k and all powers of p̂j . Consider
now an operator F which commutes with all q̂k . We have with
hq′ |F |qi = f (q′ , q)
0 = hq′ |[q̂k , F ]|qi = (qk′ − qk )f (q′ , q)
25
i.e.
hq′ |F |qi = δ(q′ − q)f (q) = δ(q′ − q)f (q′ ) .
If now F commutes with t̂j we have
0 = hq′ |[t̂j , F ]|qi = −iδ(q′ − q)
∂f (q′ )
∂qj′
from which we have f (q) = const. and thus we proved statement (2). Thus the only binary
operation on operators on the considered Hilbert space which possesses the algebraic
properties of Poisson brackets is the commutator. In particular the only translation of
the Poisson brackets to operators on the Hilbert space L2 of the functions of three variables
is
[q̂j , q̂k ] = 0,
[p̂j , p̂k ] = 0,
[q̂j , p̂k ] = i~ δjk ,
(j, k =, 1, 2, 3)
(3.1)
where ~ is a constant to be determined experimentally e.g. by diffraction of electrons
on crystals (there are also more precise ways of identifying ~). Following the formalism employed above one can easily prove von Neumann theorem: All realization of the
commutation relations (3.1) are unitary equivalent. In fact after writing
p̂k = −i~
∂
+ R̂k
∂qk
the last of eq.(3.1) tells us
hq′ |R̂k |qi = δ(q′ − q)rk (q)
i.e.
p̂k = −i~
∂
+ rk (q) .
∂qk
The second of eq.(3.1) tells us that
∂rk (q)
∂rj (q)
=
∂qj
∂qk
i.e. for a simply connected space
rk (q) =
∂
Λ(q)
∂qk
Λ(q) real. Perform now the unitary transformation on the base |qi
|qi = eiΛ/~ |qii .
We have
Λ
hhq|p̂k |ψi = ei ~ − i~
= −i~
Λ
∂Λ ∂Λ −i Λ
∂
∂
hq|ψi = ei ~ − i~
e ~ hhq|ψi
+
+
∂qk ∂qk
∂qk ∂qk
∂
hhq|ψi .
∂qk
26
CHAPTER 3. THE CORRESPONDENCE PRINCIPLE
For a rigorous proof of von Neumann theorem starting from Weyl algebra
eipa/~ eiqb e−ipa/~ = eiqb eiab
which has the advantage of dealing only with bounded operators, see [4].
Example 2.
Let us consider the two-dimensional Hilbert space (which describes spin 1/2) with the
operators
1
s1 =
2
0 1
1 0
!
;
1
s2 =
2
0 −i
i
0
!
;
1
s3 =
2
1
0
0 −1
!
[sj , sk ] = iεjkmsm
and it holds s2k = I/4. The operators s1 , s2 , s3 plus the identity form a complete set of
matrices and thus we have that the algebra generated by the commutators is irreducible.
It follows that the unique operation which respects the properties of the Poisson brackets
is the commutator.
References
[1] P.A.M. Dirac, “The principles of quantum mechanics”, Chapt.4, Clarendon Press,
Oxford.
[2] J. Grabowski e G. Marmo, “Binary operations in classical and quantum mechanics”
Banach Center Publications, vol. 59, Polish Academy of Sciences.
[3] W. Thirring, “Quantum mechanics of atoms and molecules”, Chapt.2 par. 2.3, Springer
Verlag.
[4] W. Thirring, “Quantum mechanics of atoms and molecules”, Chapt.3 par. 3.1, Springer
Verlag.
140430
Chapter 4
The time evolution
4.1
Introduction
For the time evolution operator U(t2 , t1 ) we shall require:
1) Linearity, which we consider a fundamental feature of quantum mechanics.
2) Im U = H, i.e. we can produce at time t2 whatever state vector provided we start at
time t1 with a proper vector.
3) Isometry of U(t2 , t1 ). This is not an optional requirement if we want to maintain
linearity. In fact if we try to normalize the transformation (φ1 and φ2 orthogonal, |a| =
6 1)
Uφ1 = φ1 ;
Uφ2 = aφ2 ;
U(φ1 + cφ2 ) = φ1 + caφ2
by going over to U ′
U ′ φ1 = eiα1 φ1 ;
U ′ φ2 = eiα2 φ2 ;
U ′ (φ1 + cφ2 ) = b(φ1 + caφ2 )
we would have from the linearity of U ′
U ′ (φ1 + cφ2 ) = eiα1 φ1 + ceiα2 φ2 6= const(φ1 + caφ2 )
which is contradictory.
From 2) and 3) it follows that U(t2 , t1 ) is unitary. In fact given ζ ∈ H, an η exists such
that
ζ = Uη .
Then
U + ζ = η,
UU + ζ = ζ
and thus U has an inverse which equals U + .
U(t2 , t1 )U + (t2 , t1 ) = I
27
28
4.2
CHAPTER 4. THE TIME EVOLUTION
Composition property of U
We shall denote the inverse of U(t2 , t1 ) by U(t2 , t1 )−1 ≡ U(t1 , t2 ).
Case of invariance under time translation.
U(t1 + τ, t1 ) = U(t2 + τ, t2 )
from which
U(t2 , t1 ) = U(t2 − t1 , 0) ≡ V (t2 − t1 ) = V (−t2 + t1 )−1
V (t1 )V (t2 ) = V (t2 )V (t1 ) = V (t1 + t2 ) .
Thus we have a one parameter group of unitary transformations. We shall require weak
continuity in t.
We have now the fundamental
Stone theorem: If a one parameter abelian group of unitary transformations is weakly
continuous it can be written as the exponential of a self-adjoint operator.
U(t) = exp(−iAt)
and we have
dU(t)
|t=0 = −iA
dt
in the sense that for all and only all ψ ∈ D(A) we have
(4.1)
U(ǫ) − U(0)
ψ = −iAψ.
ǫ→0
ǫ
lim
Vice-versa given a self-adjoint operator A we have that U(t) = exp(−iAt) is a one parameter unitary group weakly continuous and eq.(4.1) holds.
We have
dU(t)
= −iAU(t) = −iU(t)A.
dt
In fact
U(ǫ) − U(0)
U(t + ǫ) − U(t)
ψ = lim
U(t)ψ =
ǫ→0
ǫ→0
ǫ
ǫ
U(ǫ) − U(0)
ψ = −iU(t)Aψ
= U(t) lim
ǫ→0
ǫ
which shows that if ψ ∈ D(A) we have also U(t)ψ ∈ D(A) and viceversa.
lim
There exists an other important correspondence between self-adjoint operators and unitary operators given by the Cayley transformation
U = (A − iI)(A + iI)−1
4.2. COMPOSITION PROPERTY OF U
29
and it inverse
A = i(I + U)(I − U)−1 .
U is any unitary transformation with 1 not a characteristic value. Thus the evolution
equation becomes
dψ
= Hψ
dt
with H self-ajoint operator with the dimensions of energy.
i~
In absence of invariance under time translation we have, provided U(t, t1 ) admits a
bounded derivative
dU(t, t1 )
= −iA(t, t1 )U(t, t1 )
dt
But as
dU(t, t1 )
= −iA(t, t2 )U(t, t2 )U(t2 , t1 ) = −iA(t, t2 )U(t, t1 )
dt
we have that A depends only on t.
dU(t, t1 )
= −iA(t)U(t, t1 ).
dt
It is easily seen that the unitarity of U imposes A(t) = A+ (t). If A(t) = A after constructing U(t) = exp(−iA(t − t1 ) one shows that U + (t)U(t, t1 ) = const i.e. U(t, t1 ) =
exp(−iA(t − t1 )). The general time-dependent case in which A(t) is not bounded is rather
difficult to handle as the domain itself of A(t) may change in time [2].
The reason why A(t) is so important is that the description of physics local in time is
simple while in general the solutions are complicated.
We must now physically identify the operator A(t).
The Heisenberg picture is the most suitable one:
(φ(t), B(t)ψ(t)) = (φ(t0 ), BH (t)ψ(t0 ))
with
BH (t) = U + (t, t0 )B(t)U(t, t0 )
and
∂BH (t)
1
dBH (t)
=
+ [BH (t), HH (t)]
dt
∂t
i~
∂BH (t)
which also defines ∂t . As the commutation relations are invariant under unitary
transformation we have also
[qHj (t), qHl (t)] = 0,
[pHj (t), pHl (t)] = 0,
1
[qHj (t), pHl (t)] = δjl
i~
and comparing with classical mechanics in which
∂B(t)
dB(t)
=
+ [B(t), H(t)]P B
dt
∂t
we have that HH (t) has to be identified with the Hamiltonian.
30
4.3
CHAPTER 4. THE TIME EVOLUTION
Construction of the solution to the time evolution equation
In the case in which H is independent of time the problem is solved by Stone theorem.
If H(t) is constant on intervals we have
i
U(tn , t0 ) = Πj e− ~ Hj (tj −tj−1 )
If H(t) is properly approximated by an H constant over intervals we have
U(tn , t0 ) =
4.4
lim
i
i
max ∆j →0
Πj e− ~ Hj (tj −tj−1 ) ≡ Πe− ~ H(t)dt .
Expansion in T -ordered integrals
∂U(t2 , t1 )
= H(t2 )U(t2 , t1 ).
∂t2
To give an idea of the expansion in T -ordered integrals we refer to the case in which H(t)
i~
is norm continuous and ||H(t)|| < c and its derivative exists in uniform sense. This occurs
in a number of interesting cases. The most important one occurs in perturbation theory.
In that case the Hamiltonian is replaced by Ṽ (t) = ei
H0 t
~
V (t)e−i
H0 t
~
. If V (t) is bounded
(and hermitean as it must be) also Ṽ (t) is bounded and hermitean.
One sets U = 1 + K(t2 , t1 ) and we write the Volterra equation
Z
Z
i t2
i t2
′
′
H(t )dt −
H(t′ )K(t′ , t1 )dt′ .
K(t2 , t1 ) = −
~ t1
~ t1
(4.2)
Iterating we have Dyson series
Z ′
Z
Z
i 2 t2 ′ t ′′
i t2
′
′
H(t )dt + −
dt
dt H(t′ )H(t′′ ) + . . .
U(t2 , t1 ) = 1 −
~ t1
~
t1
t1
Such a series converges uniformly in norm, as also the series of its derivatives. It follows
that such a series solves the starting equation.
Uniqueness
For a solution of eq.(4.2) we have
d(U + U)
=0
dt2
i.e. U + U = 1. It follows that U is isometric and thus has norm 1. Thus K(t2 , t1 ) has
norm less or equal to 2. If there are two solutions the difference W must obey
Z t2
H(t′ )W (t′ , t1 )dt′
W (t2 , t1 ) = −i
t1
4.4. EXPANSION IN T -ORDERED INTEGRALS
which implies
||W (t2 , t1 )|| ≤ c
Iterating we have
31
t2
Z
t1
||W (t, t1 )||dt ≤ 4c(t2 − t1 )
||W (t2, t1 )|| ≤ 4(c(t2 − t1 ))n /n!
i.e. W = 0.
Unitarity:
Reshuffling Dyson series we have that it holds
∂U(t2 , t1 )
= U(t2 , t1 )H(t1 ) .
∂t1
−i~
(4.3)
In fact e.g.
i
K2 (t2 , t1 ) ≡ (− )2
~
and
From (4.3) it follows
Z
t2
′
dt
t1
Z
t′
t1
i
H(t )H(t )dt = (− )2
~
′
′′
dK2 (t2 , t1 )
i
= −(− )2
dt1
~
′′
Z
t2
Z
t2
′′
dt
t1
Z
t2
H(t′ )H(t′′ )dt′
t′′
H(t′ )dt′ H(t1 ) .
t1
d(U(t2 , t1 )U + (t2 , t1 ))
=0
dt1
and as for t1 = t2 , U(t2 , t2 ) = 1 we have UU + = 1.
References
[1] H. Thirring: Quantum mechanics of atoms and molecules, Springer, Chapter 3.3
[2] J.D. Dollard and C.N. Friedman: Product integration with applications to differential
equations, Addison Wesley; pag. 112
140609
32
CHAPTER 4. THE TIME EVOLUTION
Chapter 5
Properties of the hamiltonian
operator
5.1
Introduction
Often the hamiltonian operator is given in the form of a differential operator. The formal
expression of a differential operator in general is not sufficient to define the operator on an
Hilbert space insofar it is necessary to give its definition domain. This because due to the
Hellinger-Toeplitz theorem, an operator defined on the whole Hilbert space and hermitean
is necessarily bounded. On the other hand differential operators are unbounded and thus
if self-adjoint cannot be defined on the whole Hilbert space. In general the problem of
the definition of a self-adjoint operator (or of the self-adjoint extension of an operator) is
rather subtle. Let us consider for concreteness a very useful case i.e. the operator
H =−
~2 2
∇ + V (r)
2m
with V (r) which goes to zero for r → ∞ and
|V (r)| < const. r −3/2+ε
for r → 0. Let us assume as initial domain of definition of H the set of infinitely differ-
entiable functions with compact support D0 (H) = C0∞ ; these are obviously L2 functions
and it holds for φ ∈ D0 (H) also
Hφ ∈ H .
We recall that once given the definition domain of H, the domain of its adjoint is completely defined and in our case it is easy to see that
D0 (H + ) ⊃ D0 (H) = C0∞ .
33
34
CHAPTER 5. PROPERTIES OF THE HAMILTONIAN OPERATOR
We need now to extend the domain D0 (H) to a domain D(H) such that
D(H + ) = D(H)
and such that on D(H) we have H = H + . Only in this case we can speak of H as a selfadjoint operator. If such a process can be carried out in more than one way we shall have
more self-adjoint extensions and to each of them there corresponds a different physical
problem.
To start we prove the following result: If a self-adjoint extension is possible, D(H) (which
coincides with D(H + )) cannot contain functions which diverge at a point (e.g. the origin)
like r −1 or faster than r −1 .
In fact let ψ ∈ D(H) and φ ∈ D0 (H) = C0∞ ⊂ D(H). H = H + implies
(ψ, Hφ) = (Hψ, φ)
i.e.
0=
Z
∗
2
2
∗
3
(ψ (x)∇ φ(x) − ∇ ψ (x)φ(x))d x ≡
Z
∇(ψ ∗ (x)∇φ(x) − ∇ψ ∗ (x)φ(x))d3 x .
Computing the last term using Gauss theorem and the fact that φ has compact support
Z
n · (ψ ∗ (x)∇φ(x) − ∇ψ ∗ (x)φ(x))dΣr
0 = lim
r→0
Σr
But if ψ ≈ r −α the first term is O(r −α+2) while the second is O(r −α−1+2)φ(0). Choosing
φ(0) 6= 0 we see that for α ≥ 1 the result in non zero or even divergent.
Such a result is important to exclude e.g. in the treatment of the hydrogen atom square
integrable solutions ψ which behave at the origin like 1/r. Notice that this is due to the
fact that we chose as initial domain D0 (H) = C0∞ . Had we chosen for D0 (H) the functions
C0∞ whose support excludes the origin we would have had the possibility of more than
one self-adjoint extensions. Even such self-adjoint extensions, which do not give rise to
Bohr spectrum have a physical interpretation.
5.2
Some properties of the spectrum
Let us use the trial functions
ψ(r) =
1 r0 r
f( )
r r0
3/2
r0
with f (x) ∈ C0∞ with a support which exclude the origin and
Z ∞
4π
f 2 (x)dx = 1 .
0
5.3. LOWER BOUNDEDNESS OF THE SPECTRUM
We have
Tψ = −
~2 1 ∂ 2
(rψ)
2m r ∂r 2
and
(ψ, T ψ) = cT
while for r0 sufficiently small
35
1
,
r02
cT > 0
1
.
r0s
< 0 the spectrum of the Hamiltonian cannot be
(ψ, V ψ) = cV
which proves that for s > 2 and cV
bounded from below.
If at infinity V ≈ −c
1
rs
with s < 2 and c > 0, H has an infinite number of bound states.
In fact starting from an f (x) with support (1, 2) we can construct an infinite sequence of
states ψ1 , ψ2 , ψ3 . . . whose wave functions have disjoint supports and on which the mean
value of H is negative.
Moreover we have (ψn , ψm ) = δmn and (ψn , Hψm ) = 0 for m 6= n. Let
ψn = φ(n) + ζ (n)
where φ(n) are combinations of φn and ζ (n) combinations of ζE with
Hφn = En φn ,
En ≤ 0
HζE = EζE , E > 0 .
P +1
(n)
If the φn are finite in number N, then N
= 0 and
n=1 αn φ
ψ=
N
+1
X
n=1
αn ψn =
Z
β(E)ζE dE 6= 0
because the ψn are all orthogonal.
The last structure gives us (ψ, Hψ) ≥ 0 while the first gives us (ψ, Hψ) < 0, which is
contradictory.
5.3
Lower boundedness of the spectrum
Let us come now to the lower boundedness of the spectrum in the case in which |V (r)| <
const r −α with α < 3/2 for r → 0 i.e. the problem of stability.
∂
xn
We consider the three operators Kn =
+ β 2 and let be φ ∈ C0∞ . We have
∂xn
r
Z
X
β2
β
0≤
(Kn φ, Kn φ) = φ∗ (−∇2 − 2 + 2 )φ d3 x
r
r
n
36
CHAPTER 5. PROPERTIES OF THE HAMILTONIAN OPERATOR
and for β = 1/2
0≤
Z
φ∗ (−∇2 −
1
)φ d3 x .
2
4r
It follows that
~2
1
~2 1
~2 2
2
φ) ≥
∇ + V (r) φ) = (φ,
(−∇ − 2 )φ) + (φ, V (r) +
(φ, −
2m
2m
4r
2m 4r 2
~2 1
min V (r) +
r
2m 4r 2
which is a finite energy.
One can show that the closure of H defined on D0 (H) = C0∞ is the unique self-adjoint
extension of H and, as under closure the lower bound is left unchanged, we have that the
operator
~2 2
−
∇ + V (r)
2m
with V (∞) = 0 and |V (r)| < const r −3/2+ε for r → 0, is lower bounded.
If H is lower bounded, as it is the rule in quantum mechanics, one can replace the
spectral analysis of the unbounded self-adjoint operator H with the simpler analysis of
the bounded self-adjoint positive operator (H + cI)−1 with H + cI > c1 I > 0. In fact we
have
(H + cI)
−1
=
Z
1/c1
λ dEλ
0
from which
H=
Z
∞
c1
140523
µ dE
1
µ+c
.
Chapter 6
Description of composite systems
We recall that if A and B are compatible observables we can write for any |ψi ∈ H
X
|ψi =
cm
ab |a, b, mi
abm
where |a, b, mi are eigenstates of A to the value a and of B to the value b, and m takes
into account the possible residual degeneracy.
X
m
cm
ab |a, b, mi
is an eigenstate of A to the value ai and eigenstate of B to the value bj . Thus we can
rewrite
|ψi =
X
ab
cab |a, b, (ψ)i
where |a, b, (ψ)i are eigenstates of A and B relative to different pairs of values a, b.
Repeating the argument of Chapter 2 on the positivity of probabilities, performed for an
observable we have that
(ab)
ProbAB (ψ) = |cab |2 = |ha, b, (ψ)|ψi|2 .
(6.1)
We come now to composite systems. As it happens in classical mechanics, frequently one
is faced with a system which is the composition of two subsystem.
Let system S be described by the Hilbert space HS and system M by the Hilbert space
HM .
We want to justify the following principle: The Hilbert space which describes the composite system S + M is given by the tensor product space HS ⊗ HM .
Let us consider first the case in which the two system are prepared independently and let
A be an observable of S and B an observable of M. We have when system S is described
37
38
CHAPTER 6. DESCRIPTION OF COMPOSITE SYSTEMS
by the vector |S1 i and system M by the vector |M1 i
(ab)
(a)
(b)
ProbAB (S1 M1 ) = ProbA (S1 )ProbB (M1 ) = |ha, (S1 )|S1 i|2 |hb, (M1 )|M1 i|2
(6.2)
due to the independence of the two systems.
The tensor product of the two Hilbert spaces is defined considering the pairs of vectors,
one of HS and the other of HM and their linear combinations. One demands that the
scalar product of two vectors be bilinear hermitean in the component vectors. The scalar
product of two pairs of vectors is defined as the product of the two scalar products in HS
and HM
(hS1 |hM1 |)(|S2 i|M2 i) = hS1 |S2 ihM1 |M2 i
which is extended by linearity to all vectors. The product of the null vector of HS and
of any vector of HM (and viceversa) is defined as the null vector in the tensor product
space.
One verifies immediately that such a space is a pre-Hilbert space. Its completition is the
Hilbert space HS ⊗ HM tensor product of HS and HM .
One defines the action of the operators on HS , on the tensor product as follows
A(|S1 i|M1 i) = (A|S1 i)|M1 i
and the same for the operators B which act on HM . Then we have
[A, B] = 0
and thus two observables A and B relative to distinct subsystems are always compatible.
A|a, mi|b, ni = a|a, mi|b, ni
B|a, mi|b, ni = b|a, mi|b, ni .
As |a, mi is a complete system in HS and |b, ni a complete system in HM we have that
|a, mi|b, ni
is a complete system in HS ⊗ HM . We can write the result (6.2)
(ab)
(a)
(b)
ProbAB (S1 M1 ) = ProbA (S1 )ProbB (M1 ) = |ha, (S1)|hb, (M1 )|S1 iM1 i|2
which is consistent with the result (6.1).
Wave functions.
39
If qk is a complete set of compatible observables in HS and Qk a complete set of compatible
observables in HM we have that |qi|Qi is a complete set vectors in the tensor product
space and
hq|hQ|S1i|M1 i = hq|S1 ihQ|M1 i
i.e. the wave function of |S1 i|M1 i is the product of the wave functions.
This happens only for factorized states. The wave function of a generic state will be a
generic function of q and Q i.e. Ψ(q, Q).
140501
40
CHAPTER 6. DESCRIPTION OF COMPOSITE SYSTEMS
Chapter 7
The density operator
7.1
Introduction
Often we are in a situation in which the preparing apparatus does not produce a well
defined quantum state but a statistical distribution of states.
There exists an analogue situation in classical physics when we describe the state of the
system by a statistical ensemble, i.e. by means of a density function ρ(q, p, t) in phase
space and the mean values of the dynamical variables are given by
Z
Z
F̄ = F (q, q)ρ(q, p, t)dq dp/ ρ(q, p, t) dq dp .
Sometime F may depend explicitly on time and in general one takes
R
ρdq dp = 1.
We recall that from the conservation of the number of points in phase space (determinism
in both time directions) it follows that
Z
Z
d
ρ(q, p, t)dq dp = − n · ρ vdΣ
dt V
Σ
with v = (q̇1 q̇2 . . . q̇n ṗ1 ṗ3 . . . ṗn ) and thus
Z
Z
∂
ρ(q, p, t)dq dp = −
∇ · (ρv) dq dp
V ∂t
V
i.e.
∂
ρ(q, p, t) = −∇ · (ρv) = −ρ∇ · v − v · ∇ρ.
∂t
Using Hamilton equations it follows ∇ · v = 0 and at last we have
∂
ρ(q, p, t) = −v · ∇ρ = −[ρ, H]P P .
∂t
41
42
CHAPTER 7. THE DENSITY OPERATOR
In the case of quantum mechanics we are in the situation in which we have more than one
P
state ψj which occur with probability ρj con j ρj = 1. The mean value of the measures
of the observable F on the ensemble of states is
X
F̄ =
ρj (ψj , F ψj ) .
j
This can be written in more concise way as
F̄ = Tr (F W )
with W the density operator (sometime called also density matrix)
W =
X
j
ρj ψj ◦ ψj
,
X
ρj = 1,
ρj > 0.
(7.1)
j
W is a bounded hermitean positive operator. The trace on an Hilbert space is defined by
Tr (A) =
X
(ζk , Aζk )
k
with ζk a complete orthonormal set in Hilbert space. For operators of the type W B
with B bounded one can show that the trace Tr(W B) is independent of the choice of the
orthonormal set in Hilbert space. This because W = V V with Tr V V = 1 is HilbertSchmidt and thus also W B is Hilbert-Schmidt.
We have
Tr (F W ) =
XX
k
(ζk , F ψj )ρj (ψj , ζk ) =
j
XX
j
X
ρj (ψj , ζk )(ζk , F ψj ) =
k
ρj (ψj , F ψj ) .
j
Notice that
Tr (W ) =
X
ρj = 1 .
j
7.2
Time evolution of the density operator
The individual states evolve as
ψk (t) = e−iHt/~ ψ(0)
from which
W (t) = U(t)W (0)U + (t) .
7.2. TIME EVOLUTION OF THE DENSITY OPERATOR
43
Taking the derivative w.r.t. time we have in the Schrödinger picture
1
∂
W (t) = − [W (t), H(t)] .
∂t
i~
We have the following correspondence between classical and quantum mechanics.
Z
dq dp ↔ Tr
ρ(q, p, t) ↔ W (t)
1
[ , ]P P ↔ [ , ] .
i~
Pure states are represented by
W =ψ◦ψ .
All other density operators are obtained by convex combination of pure states. Pure
states are extremal in the ensemble of the convex set of density operators. If the state is
pure we have
W2 = W .
This is necessary but also sufficient for W to represent a pure state. In fact as TrW = 1
we cannot have W = 0; thus there exists a φ such that
W φ 6= 0 .
From W 2 = 1 we have
W W φ = W φ ≡ φ1 6= 0
and we choose (φ1 , φ1 ) = 1. Let us construct an orthonormal base in Hilbert space
φ1 , φ2 , . . . . Being W + = W we have
1 = Tr (W ) = (φ1 , W φ1)+(φ2 , W φ2 )+(φ3 , W φ3)+· · · = 1+(W φ2, W φ2)+(W φ3 , W φ3 )+. . .
and thus it follows that W φk = 0 for k ≥ 2. It follows that W − φ1 ◦ φ1 = 0 as it is seen
by applying such an operator to the base φk .
By performing a sufficient number of measures is is possible to distinguish a statistical
ensemble from pure states. Let us consider in fact the observables Z = ζ ◦ ζ. We have
Z̄ = Tr (ZW ) = (ζ, W ζ).
with (φj , φj ) = 1 and (ξj , ξk ) = 0 for j 6= k. Let us pose (ξj , ξj ) = ρj . We have
X
X
1 = (ψ, ψ) =
(ξj , ξj ) =
ρj .
j
j
44
CHAPTER 7. THE DENSITY OPERATOR
If we measure an observable F relative to the subsystem S, as F do not operate on HM
we have
F̄ = (ψ, F ψ) =
X
j
where W =
P
j
(ξj , ξj )(φj , F φj ) =
X
ρj (φj , F φj ) = Tr
j
HS (W F )
ρj φj ◦φj is a density operator which acts on HS and the trace is performed
on the Hilbert space HS .
The Hamiltonian of the system S + M is given by
H = HS + HM + HI
where HS is the Hamiltonian of S and as such acts on HS , HM is the Hamiltonian of M
and as such acts on HM while HI is the interaction Hamiltonian which acts on H.
If the initial state ψ(0) is such that for t > 0 the interaction is negligible (in a decay
process e.g.)
ψ(t) = e−i(HS +HM +HI )t/~ ψ(0) = e−i(HS +HM )t/~ ψ(0)
we have for t > 0
F̄ (t) = (ψ(t), F ψ(t)) = (e−i(HS +HM )t/~ ψ(0), F e−i(HS +HM )t/~ ψ(0)) =
(e−iHS t/~ e−iHM t/~ ψ(0), F e−iHS t/~ e−iHM t/~ ψ(0)) =
=
X
(e−iHM t/~ ξj (0), e−iHM t/~ ξj (0))(e−iHS t/~ φj (0), F e−iHS t/~ φj (0))
j
=
X
(ξj (0), ξj (0))(e−iHS t/~ φj (0), F e−iHS t/~ φj (0)) =
j
X
j
ρj (φj (t), F φj (t)) = Tr HS (F W (t))
where W (T ) is the density operator evolved according the Hamiltonian HS .
140502
Chapter 8
Measure in quantum mechanics
We recall the results obtained in Chapter 2: Given a state ψ if we perform the measure
of the observable F on ψ the probability to obtain the value fi (of the spectrum) is given
by |ci |2 being ci the coefficient which appears in the expansion of ψ in eigenstates F
P
P (n) (n)
i.e. ψ = i ci φi . Using the expansion in standard eigenstates ψ = i ci φi such a
P (n)
probability is given by n |ci |2 .
We justified such a formula generalizing Born hypothesis. A fundamental problem which
arises is: What happens after the measure, i.e. which is the vector that describes the
system after the measure if we have obtained as the result of the measure the value fi .
A first classification of the measuring processes is the distinction between repeatable and
non repeatable measures. We shall say that the measure of a certain quantity repeatable
if repeating the measure of the same quantity on the system immediately after the first
measure we obtain with certainty the result previously obtained.
Example of a repeatable measure: The Stern-Gerlach experiment.
Example of a non repeatable measure: The measure of the energy of a particle through
ionization.
Whether an apparatus effects a repeatable measure can be experimentally ascertained.
If the value for which we select the system is a non degenerate value of the spectrum and
the measure is repeatable we conclude that the selected system after the measure is in
the eigenstate of F relative to the measured value fi of the spectrum.
Thus in the case of repeatable measure in which we select the system if we measure a
value fi of the spectrum which is non degenerate, the measure operation coincides with
that of the preparation of the state.
This can be expressed by saying that in such a measuring process the evolution ψ → φi
has taken place; if we insist that such evolution is linear (we do not want to give up
linearity in quantum mechanics) we have that such a transition is given by ψ → φi = Pi ψ
45
46
CHAPTER 8. MEASURE IN QUANTUM MECHANICS
being Pi the projector φi ◦ φi . Notice that such a linear transformation does not conserve
the norm and as such it is not unitary. We recall also from Chap.4 that it is not possible
to perform a further transformation which brings back the norm to 1 without loosing
linearity. Notice also that the probability to obtain fi is given by (ψ, Pi ψ).
More subtle is the problem when the measured value fi is degenerate. In fact in this case
it is not enough to say that the measure is repeatable to know the state of the system
after the measure.
We shall introduce the notion of strongly repeatable measure (or better of an apparatus which performs strongly repeatable measurements) characterized by the following
property: If the initial state is an eigenstate of F , then the measure leaves such state
unchanged. Also this is a property experimentally ascertainable.
If this happens we shall say that our experimental apparatus performs a strongly repeatable measurement.
Given an arbitrary initial state we denote by Ai the linear transformation which corresponds to the selection of the final states for which the result of the measure of F gave
fi (by the way: these will be eigenstates of F to the value fi ). Example: The Stern and
Gerlach experiment in which one keeps open just one slit.
On the eigenstates of F to the value fi , Ai acts as follows (strongly measurable measurement)
Ai φ = α(φ)φ .
If we do not want to give up linearity of the transformation we have α(φ) = α (independent
of φ) and thus Ai /α is the identity operator on this subspace. On the eigenvectors of F
to a different value fj of the spectrum we have
Ai φj = 0
otherwise a state which gives with certainty the value fj would pass the selection test Ai
which is contradictory with the probability formula for the measure of fi .
We conclude that Ai apart for a factor is the projection operator Pi on the eigensubspace
of F to the eigenvalue fi . Such a projection property of the strongly repeatable measures
is called for obvious reasons, the property of least disturbance.
Notice again that the probability to find the value fi is given by Prob(fi ) = (ψ, Pi ψ).
If our system is described by a density operator it is easy to find how this changes due
to a measuring process. If we select the systems for which the measuring apparatus has
measured the value fi , as any state ψ is changed into Pi ψ/||Pi ψ|| (strongly repeatable
measure) and this happens with probability (Pi ψ, Pi ψ) we have that the density operator
47
preparing
apparatus
observable F
fi
Ψ
f1
Ai Ψ
fn
Figure 8.1: Generalized Stern-Gerlach apparatus: the measuring apparatus passes only
those systems for which the value fi of the spectrum of F has been measured.
after the measure is given Pi W Pi . Notice that such a density operator is no longer
normalized i.e. in general we have Tr Pi W Pi < 1. This is due to the fact that a certain
number of systems are lost. We can normalize by dividing by its trace i.e. the new
normalized density operator is given by
W′ =
Pi W Pi
.
Tr (W Pi )
The probability to find fi in the measure of F on W is given by
Tr (W Pi ).
P
j
ρj (ψj , Pi ψj ) i.e. by
We want now, given two observables F and G, compute the composite probability to find
in two successive measurements first fi and then gj . Such a probability is given by
Prob(fi , gj ) = tr(W Pi )tr(W ′ Qj ) = tr(Pi W Pi Qj ).
We could proceed with many measures, but we shall limit ourselves to two measures.
Instead the probability to find on W first gj and thereafter fi is given by
Prob(gj , fi ) = tr(W Qj )tr(W ′ Pi ) = tr(Qj W Qj Pi ).
In general these two probabilities are different, contrary to what we know to happen
experimentally in the classical case.
It is easy to see that if Pi commutes with Qj then the two composite probabilities are
equal: In fact in this case Pi Qj Pi = Pi2 Qj = Pi Qj = Qj Pi Qj .
Let us suppose now to find experimentally that the two above mentioned composite
probabilities are equal, whatever the state W and for all indices i and j. Given the
arbitrariness of the state W , (think e.g. to the pure states) we have that Pi Qj Pi = Qj Pi Qj
for all choices of indices i and j. It is easy then to see that this implies [Pi , Qj ] = 0, for
48
CHAPTER 8. MEASURE IN QUANTUM MECHANICS
all pairs i, j. In fact
Pi = Pi2 =
X
k
from which we derive
Pi Qk Pi =
X
Qk Pi Qk
k
Pi Qj = Qj Pi Qj = Qj Pi
which is what we wanted to prove. If we now consider the spectral representations of the
operators
F =
X
fi Pi
G=
i
X
gj Qj
j
we have that [F, G] = 0. Viceversa if F and G commute we have that there exist a
complete set φijn of eigenstates common to F and G i.e.
F φijn = fi φijn
Gφijn = gj φijn
from which
Pi =
X
k,n
and
φikn ◦ φikn
Pi Qj =
X
n
Qj =
X
h,m
φhjm ◦ φhjm
φijn ◦ φijn = Qj Pi .
In conclusion: The temporal commutativity of the measure of F and G at the statistical level implies the mathematical commutativity of the operators which represent the
observables F and G and viceversa.
References
[1] K. Gottfried : Quantum Mechanics, W. A. Benjamin, 1966 New York.
[2] E.B. Davies: Quantum mechanics of open systems, Academic Press, 1976 London.
[3] J.S. Bell : Speakable and unspeakable in quantum mechanics, Cambridge University
Press, Cambridge 1993.
[4] J.A. Wheeler and W.H.Zurek: Quantum theory of measurement, Princeton University
Press, 1983 Princeton.
140502
Chapter 9
Symmetry transformations
9.1
Space translations
As an introduction we want to find the generator of translations. Let us consider a system
described by the vector ψ and let
ψε = (1 − iεt̂k )ψ
be the state vector which describes the same system translated by ε in the positive direction along the k axis. We must have
(ψε , q̂m ψε ) = (ψ, q̂m ψ) + εδkm
and thus
[q̂m , t̂k ] = iδmk .
(9.1)
Similarly
(ψε , pm ψε ) = (ψ, pm ψ)
that is
[p̂m , t̂k ] = 0.
(9.2)
The first tells us that t̂k = p̂k /~+fk (q̂), where fk (q) are three functions of the coordinates,
while the second tells us that fk are constants. Thus we have proved that
t̂k = p̂k /~ + ck
where the ck if we want (1 − iεt̂k ) to conserve the norm of vectors, must be real numbers.
The finite translation is given by
e−ip̂k b/~ e−ick b .
49
50
CHAPTER 9. SYMMETRY TRANSFORMATIONS
Being the group of translations abelian translations are easily composed.
One could even ask whether we cannot define the momentum as the generator of translations and similarly angular momentum as the generator of rotations. The advantage to
free angular momentum from the expression of the orbital angular momentum, is to allow
more general form of angular momenta i.e. the spin.
To carry on such a program is necessary to develop some general considerations on symmetry transformations.
9.2
Symmetry transformations
Let us consider for concreteness the orthogonal transformation q′ = γ(q) with γ a proper
orthogonal matrix.
We shall assume the active viewpoint: Rotating the system (with respect to the “fixed
stars” or better with respect to the cosmic microwave radiation) the physical points move
from the positions of coordinates q to those of coordinates q′ = γ(q).
For microscopic systems this is obtained by rotating by γ the preparing apparatus.
In presence of invariance under rotations (i.e. in absence of external fields) the invariance
of the transition probability tells us
|(φ, ψ)|2
|(T φ, T ψ)|2
=
(φ, φ)(ψ, ψ)
(T φ, T φ)(T ψ, T ψ)
(9.3)
Notice that (9.3) is an experimentally verifiable relation.
A set of transformation T which admit inverse and for which (9.3) holds, form a group
and such a group is called a symmetry group.
The transformation T can always be written such that it conserves the norm as up to
now T is a general, not necessarily linear, transformation; i.e. we shall set
(T ψ, T ψ) = (ψ, ψ).
(9.4)
Clearly a change of type
T → T′
defined by T ′ ψ = exp(iα(ψ))T ψ
(9.5)
leaves invariant the relations (9.3) and (9.4) and thus we can exploit such arbitrariness to
reduce the operator T to a more conventional form. Wigner [1] [2] shows that it is always
possible to choose the phases in (9.5) in such a way that the operator T becomes either
linear or anti-linear (not both cases can be realized starting from a given transformation
9.2. SYMMETRY TRANSFORMATIONS
51
T ). In the linear case (9.4) tells us that L (the name we give to this linear operator) is
isometric i.e. L+ L = 1 and as T has inverse we have also LL+ = 1 i.e. L is unitary.
Let us examine now the anti-linear case. The definition of anti-linear operator is
A(αψ + βφ) = α∗ Aψ + β ∗ Aφ.
(9.6)
If (Aψ, Aψ) = (ψ, ψ) from the definition of anti-linear operator we have
(A(αψ + βφ), A(αψ + βφ)) = |α|2(ψ, ψ) + β ∗ α(φ, ψ) + α∗ β(ψ, φ) + |β|2 (φ, φ) =
|α|2(Aψ, Aψ) + β ∗ α(Aψ, Aφ) + α∗ β(Aφ, Aφ) + |β|2(Aφ, Aφ)
(9.7)
from which given the arbitrariness of α and β we have (Aψ, Aφ) = (φ, ψ) which defines
and anti-isometric operator. The complex number (ψ, Aφ)∗ = (Aφ, ψ) depends linearly
from φ and thus can be written due to Riesz theorem as (ζ, φ) i.e.
(ψ, Aφ) = (φ, ζ)
with ζ depending anti-linearly from ψ and thus we can introduce the anti-linear operator
A+ which we shall call the adjoint of A defined by
A+ ψ = ζ .
The invariance of the norm of ψ tells us that such operator A is anti-isometric i.e. A+ A = 1
while again the invertibility hypothesis tell us that also that AA+ = 1 and thus A is called
anti-unitary. Such arguments hold for all symmetry operations. We prove now [1] that
for all symmetry operations which projectively commute with the time evolution and
transform energy eigenstates in energy eigenstates with the same value of the energy the
only possibility, consistent with the superposition principle is the unitary case. Let us
consider in fact the superposition of two eigenstates of the energy ψ1 and ψ2 relative
to different values of the energy E1 and E2 and suppose by absurd that the symmetry
transformation is anti-unitary. The state α1 ψ1 + α2 ψ2 at time 0 evolves at time t into the
state
α1 exp(−iE1 t/~)ψ1 + α2 exp(−iE2 /~)ψ2 .
(9.8)
Instead the transformed state at t = 0 i.e. α1∗ Aψ1 + α2∗ Aψ2 evolves at time t into the state
α1∗ exp(−iE1 /~)Aψ1 + α2∗ exp(−iE2 /~)Aψ2 .
(9.9)
If we now transform according the operator A the state (9.8) we must find the state (9.9)
and thus the vector
α1∗ exp(iE1 /~)Aψ1 + α2∗ exp(iE2 /~)Aψ2
52
CHAPTER 9. SYMMETRY TRANSFORMATIONS
may differ at most by a phase factor from
α1∗ exp(−iE1 /~)Aψ1 + α2∗ exp(−iE2 /~)Aψ2 .
Bur being the two states Aψ1 and Aψ2 orthogonal and E1 6= E2 this cannot be valid for
all t. Thus with the above accepted hypothesis a anti-unitary transformation gives rise
to a contradiction. In particular the two statements [A, H] = 0 and exp(−iHt/~)A =
A exp(−iHt/~) with A anti-unitary are in contradiction. On the contrary the following
two relations are consistent [A, H] = 0 and exp(iHt/~)A = A exp(−iHt/~), with A
anti-linear, actually from the first we obtain the second (e.g. by expanding in series the
exponential and recalling that A anti-commutes with i). If there exists an A anti-linear
with the property of commuting with H then we say that A is an operation of inversion
of time and that the system described by H is invariant under time reversal.
Let us consider a group of symmetry transformations represented by unitary operators
and let by γ1 and γ2 represented by the unitary operators U(γ1 ) and U(γ2 ). To the
product transformation γ2 γ1 there corresponds the unitary operator U(γ2 γ1 ). But the
sequence of the two transformations γ1 , γ2 is physically equivalent to the transformation
γ2 γ1 and thus we must have
U(γ2 γ1 )ψ = α(γ2 , γ1, ψ)U(γ2 )U(γ1 )ψ
where α is a phase factor which in principle can depend both on γ1 , γ2 and on the state
vector ψ.
A simple reasoning shows that α cannot depend on the vector ψ. In fact let us consider
two unitary operators U and V such that for any vector ψ holds Uψ = αV ψ with α in
general dependent on ψ. Defined K = V + U and given two linearly independent vectors
ψ1 and ψ2 we have Kψ1 = α1 ψ1 and Kψ2 = α2 ψ2 and also
K(a1 ψ1 + a2 ψ2 ) = a1 Kψ1 + a2 Kψ2 = a1 α1 ψ1 + a2 α2 ψ2 =
= α3 (a1 ψ1 + a2 ψ2 ) = a1 α3 ψ1 + a2 α3 ψ2
and being ψ1 and ψ2 independent, we must have α3 = α1 and α3 = α2 , i.e. α1 = α2 =
const. Thus
U(γ2 γ1 ) = α(γ2, γ1 )U(γ2 )U(γ1 ).
(9.10)
A correspondence γ → U(γ) which satisfies (9.10) is called a representation up to a phase,
or a projective representation, of the symmetry group.
The following theorem by Bargmann holds [3]: Given a projective representation of a
compact group i.e. which satisfies (9.10), continuous in a neighborhood of the identity it
9.3. ROTATIONS
53
is possible to perform a phase transformation on the U i.e. U → ω(U)U with ω phase
factor, such that in a neighborhood of the identity it remains continuous and in such
neighborhood becomes a true representation i.e. (9.10) holds with α ≡ 1. The same
cannot be say for the representation in the large.
In fact we can assure that (9.10) holds with α ≡ 1 for any pair of transformations only if
the group is simply connected. SO(3) is not simply connected and in general we cannot
redefine the phases in such a way as to have a continuous representation with α ≡ 1 in
(9.10) in the large.
9.3
Rotations
Let us specify our considerations for the group of rotations i.e. SO(3). The main feature
which distinguishes this group from the group of translation is its non commutativity.
Given a real antisymmetric transformation in three dimensions α = −αT , let us consider
eα . We have (eα )T = e−α = (eα )−1 , i.e. eα is an orthogonal transformation and eφα is
a one parameter group of orthogonal transformations connected with the identity. We
recall that in three dimensions all proper rotations i.e. the elements of SO(3) are rotations
around an axis.
Given two infinitesimal rotations characterized by the antisymmetric transformation α
and β we want to compute the “commutator” of the transformations generated by α and
β i.e. exp(−εβ) exp(−εα) exp(εβ) exp(εα) keeping up to the second order terms . Using
twice the Campbell-Baker-Hausdorff formula exp(A) exp(B) = exp(A+B+[A, B]/2)+O3 )
we obtain
e−εβ e−εα eεβ eεα = e−εβ−εα+ε
2 [β,α]/2+O(ε3 )
eεβ+εα+ε
2 [β,α]/2+O(ε3 )
= eε
2 [β,α]+O(ε3 )
(9.11)
(notice that also [β, α] is an antisymmetric transformation). According to Wigner’s theorem, given a rotation, it is always possible to represent it in Hilbert space by means of
a unitary transformation and according to Bargmann theorem the U(γ) can be chosen as
to form a true representation in a neighborhood of the identity i.e. such that
U(γ2 )U(γ1 ) = U(γ2 γ1 ).
(9.12)
According to Stone theorem the one parameter group of unitary operators U(eφα ) can be
written in the form
U(eφα ) = e−iφr(α)/~
(9.13)
54
CHAPTER 9. SYMMETRY TRANSFORMATIONS
with r(α) self-adjoint operator and the same for β. Using again Campbell-Baker-Hausdorff
formula we obtain
U(e−εβ )U(e−εα )U(eεβ )U(eεα ) = eε
2 [−ir(β)/~,−ir(α)/~]+O(ε3 )
which according to (9.11) and to (9.12) must be equal to
e−iε
2 r[β,α]/~+O(ε3 )
i.e.
[r(α), r(β)] = i~r([α, β]).
(9.14)
Let us specify now the transformations α and β ... as infinitesimal rotations around the
coordinate axes.
α1 = A1 ,
α2 = A2 ,
α3 = A3
with A1 , A2 , A3 given by

0 −1 0



0 0 A3 = 1 0 0 .
0 0 0
−1 0 0



0
0 0 0



A1 = 0 0 −1 A2 =  0
0 1
0
0 1
E.g. α3 gives rise to the infinitesimal transformation



q ′ = q1 − ǫ3 q2

 1
q2′ = q2 + ǫ3 q1



q ′ = q
3
3


.
The three matrices A satisfy the commutation relations
Ai Aj − Aj Ai = ǫijk Ak e.g. A1 A2 − A2 A1 = A3 .
(9.15)
Setting now
r(Aj ) = Mj
and using (9.14) and (9.15) we obtain
[Mj , Mk ] = i~ǫjkl Ml
(9.16)
which are identical to the commutation rules of the orbital angular momentum which can
be derived by the transcription of the classical angular momentum of a particle (coordinate
representation).
9.4. THE EUCLIDEAN GROUP
55
However presently we reached (9.16) without the need to specify the nature of the vector
we are transforming.
To start, let us consider the case in which the state is represented by the wave function
ψ(q). The simplest and most natural way to transform the wave function under rotations
is obtained imposing the invariance in value of the wave function i.e.
ψ ′ (q′ ) = ψ ′ (γ1 (q)) = ψ(q) i.e. ψ ′ (q) = ψ(γ1−1 (q)) .
(9.17)
Under the sequence of two transformations γ1 and then γ2 we have
ψ ′′ (q) = ψ ′ (γ2−1 (q)) = ψ(γ1−1 γ2−1 (q)) = ψ((γ2 γ1 )−1 (q)).
It is immediate that transformation (9.17) is unitary insofar
Z
Z
∗ −1
−1
ψ (γ (q))φ(γ (q)) dq = ψ ∗ (q)φ(q) dq
due to the fact that the Jacobian of an orthogonal transformation equals 1. The we
have ψ ′ (q) = ψ(γ1−1 (q)) ≡ U(γ1 )ψ(q) and U(γ2 γ1 ) = U(γ2 )U(γ1 ) without any additional
phase. Thus the transformation in value of the wave functions realizes completely the
program to obtain for SO(3) a true representation of the whole group SO(3). Let us see
the explicit form of the generator of the rotations around the axis 3.
(1 − iεr(A3 )/~)ψ(q) = (1 − iεM3 /~)ψ(q) =
= ψ(q − εA3 (q)) = ψ(q1 + εq2 , q2 − εq1 , q3 ) = ψ(q) + ε(q2
∂
∂
− q1
)ψ(q)
∂q1
∂q2
∂
∂
i
+ i~q2
)
= ψ(q) − ε(−i~q1
~
∂q2
∂q1
i.e.
M3 = q1 p2 − q2 p1
which is exactly the expression of the orbital angular momentum.
9.4
The euclidean group
We want now to give the complete treatment of the euclidean group [4].
The translations
qk′ = qk + ck
combined with the rotations
qk′ = γkl ql
56
CHAPTER 9. SYMMETRY TRANSFORMATIONS
form obviously a group which is called the euclidean group.
If we denote by Tk the generators of translations and Jk those of the rotations (Jk ≡ Ak )
we obtain easily the following commutation rules
[Jk , Jl ] = εklm Jm
[Jk , Tl ] = εklm Tm
[Tk , Tl ] = 0 .
The last one tells us simply that the translations commute among themselves.
In quantum mechanics the symmetry group “euclidean group” can be projectively represented by unitary transformations (Wigner theorem) whose generators are self-adjoint
operators (Stone theorem). The group however is not compact and thus we cannot exploit
Bargmann theorem to reach a true representation. If we denote by Jˆk and T̂k the operators which generate the rotations and the translations on the vectors in Hilbert space,
due to the fact that in quantum mechanics what are essential are the rays and not the
vector we can only write
[Jˆk , Jˆl ] = i~(εklm Jm + akl )
[Jˆk , T̂l ] = i~(εklmTm + bkl )
[T̂k , T̂l ] = i~ckl
where due to the unitarity of the transformations akl , bkl , ckl have to be real.
Notice that being the commutators antisymmetric we have that akl = −aml , ckl = −clk
while the same cannot be said for the bkl . Given the antisymmetry we can write
akl = εklm am ; ;
ckl = εklmcm .
Jacobi identity
[Jˆk , [P̂l , P̂m ]] + cyclic = 0
furnishes
(εmkr εlrs + εklr εmrs )cs = 0
i.e. cs = 0. Decomposing bkl in antisymmetric and symmetric part
bkl = εklmbm + b(kl)
Jacobi identity
[Jˆk , [Jˆl , P̂m ]] + cyclic = 0
9.4. THE EUCLIDEAN GROUP
57
furnishes
εlmr b(kr) + εmkr b(lr) − εklr b(kr) = 0 .
(9.18)
Anti-symmetrizing (9.18) in lm we obtain
εlmr b(kr) = 0;
i.e.
b(lr) = 0 .
Thus we reached
[Jˆk , Jˆl ] = i~εklm(Jˆm + am )
[Jˆk , T̂l ] = i~εklm(T̂m + bm )
[T̂k , T̂l ] = 0 .
Defining J˜k = Jˆk + ak , T̃k = T̂k + bk we have
[J˜k , J˜l ] = i~εklmJ˜m
(9.19)
[J˜k , T̃l ] = i~εklmT̃m
(9.20)
[T̃k , T̃l ] = 0 .
(9.21)
The (9.19,9.20,9.21) are called a canonical representation of rotations and translations.
Eq.(9.19) gives rise to the quantization of angular momentum as already discussed (and
experimentally observed) while the (9.21) combined with
[q̂k , q̂l ] = 0;
[q̂k , T̂l ] = iδkl
can be used as the general definition of momentum: p̂k = ~T̂k .
Summing up: We define momentum and angular momentum of a quantum system the
canonical representation of the generators of the euclidean group.
References
[1] E.P. Wigner: Group theory, Academic Press (1959) pag. 233
[2] S. Weinberg: The quantum theory of fields I, Cambridge University Press, pag. 91
[3] V. Bargmann, Ann.of Math. 59 1 (1952))
[4] G. Sartori: Lezioni di meccanica quantistica, Edizioni Libreria Cortina, Padova.
140504
58
CHAPTER 9. SYMMETRY TRANSFORMATIONS
Chapter 10
The spin
10.1
Realization of half-integer angular momenta
We look for an Hilbert space H on which three hermitean operators si , i = 1, 2, 3 are
defined which satisfy the angular momentum commutation relations i.e.
sj sk − sk sj = i
and such that also
P
j
X
ǫjkl sl
l
s2j = 1/2(1/2 + 1). It is useful to set s = 12 σ from which
σj σk − σk σj = 2 i
X
ǫjkl σl .
(10.1)
l
Let χ+ and and χ− be the eigenvectors of σ3
σ3 χ− = −χ− .
σ3 χ+ = χ+ ;
Clearly we have (χ− , χ+ ) = 0 and (σ1 + iσ2 )χ+ ≡ σ+ χ+ = 0 and we shall define the phase
of χ− by means of
(σ1 − iσ2 )χ+ ≡ σ− χ+ = 2χ− .
From this it follows
(χ+ , σ3 χ+ ) = 1;
(χ− , σ3 χ+ ) = 0;
(χ+ , σ3 χ− ) = 0
(χ− , σ3 χ− ) = −1
while the matrix elements of σ+ and σ− are all zero except
(χ− , σ− χ+ ) = 2,
(χ+ , σ+ χ− ) = 2 .
59
60
CHAPTER 10. THE SPIN
Thus we have the following two-dimensional representation
!
!
!
0 1
0 −i
1 0
σ1 =
σ2 =
σ3 =
1 0
i 0
0 1
called Pauli matrices. Notice that σj2 = 1 which tells us simply that every σk has eigenvalues ±1.
From the commutation relations of the σj and from σj2 = 1 it follows also that σj obey a
Clifford algebra
{σj , σk } = 2 δjk
where {, } denotes the anticommutator.
This is found e.g. by summing to the (10.1) multiplied on the right by σk , the same
multiplied on the left by σk (k 6= j).
According to eq.(9.13) the infinitesimal rotation around an axis n of an angle ǫ is given
by
′
a
a
a
=
(1
−
i
σ
·
n
ǫ/2)
=
(1
−
i
M
·
n
ǫ/~)
b
b
b′
and the finite rotation by
′
a
a
a
σ · n sin(φ/2))
σ · n φ/2)
= (cos(φ/2) − iσ
= exp(−iσ
′
b
b
b
′
a
a
2
σ · n) = 1. We see that for a rotation of 2π around any axis we have
;
=−
as (σ
′
b
b
this does not contrast with the physical interpretation of the state vector.
The matrices 2 × 2
σ · n sin(φ/2)
U = cos(φ/2) − iσ
(10.2)
are all the elements of the group SU(2) i.e. the group of the unimodular (determinant 1)
unitary transformations in two dimensions.
In fact introducing the identity matrix in two dimensions σ0 we have that writing the
most general two-dimensional matrix in the form aσ0 + b · σ its determinant is given by
a2 − b2 . The inverse of the above written unimodular matrix is given by aσ0 − b · σ as is
immediately verified using Clifford algebra.
If we impose now unitarity i.e. that the inverse equals the adjoint we have that a = a∗
and b = −b∗ . Thus the most general SU(2) matrix is written in the form
aσ0 + ib · σ
with real a and b and a2 + b2 = 1 i.e. in one to one correspondence with the point of the
surface of a four-dimensional sphere of radius 1 which is a simply connected set.
10.1. REALIZATION OF HALF-INTEGER ANGULAR MOMENTA
61
In the parametrization (10.2) the angle can be chosen 0 ≤ φ < 2π.
In order for the correspondence we found between the elements of SO(3) and those of
SU(2) to be a projective representation of the group SO(3) it is necessary to show that
given two elements of SO(3), γ1 and γ2 , the related elements of SU(2), U(γ1 ) and U(γ2 )
must be such that
U(γ2 γ1 ) = α(γ2 , γ1)U(γ2 )U(γ1 )
(10.3)
with α(γ2, γ1 ) a phase factor. Given an element of SO(3), i.e. the rotation around an
axis n of an angle φ we consider the element of SU(2) related according to formula (10.2)
which we already saw to be determined up to a sign. Let us show that
σ U(γ) = γ(σ
σ) .
U + (γ)σ
(10.4)
If we perform a rotation of an angle φ around the z axis we have
(cos(φ/2) + iσz sin(φ/2))σ1 (cos(φ/2) − iσz sin(φ/2)) = cos(φ)σ1 − sin(φ)σ2
(cos(φ/2) + iσz sin(φ/2))σ2 (cos(φ/2) − iσz sin(φ/2)) = sin(φ)σ1 + cos(φ)σ2
(cos(φ/2) + iσz sin(φ/2))σ3 (cos(φ/2) − iσz sin(φ/2)) = σ3
i.e. we found that
σ (cos(φ/2) − iσz sin(φ/2)) = γ(σ
σ)
(cos(φ/2) + iσz sin(φ/2))σ
It is not difficult to prove that (10.4) holds for the rotation of an angle φ around whatever
axis n.
In fact under the rotation around n of an angle φ we have
q → q′ = γ(q) =
= n(q · n) + cos(φ)(q − n(q · n)) + sin(φ)n ∧ q .
Using the relation σ · n σk σ · n = −σk + 2 nk σ · n which is easily derived from Clifford
algebra we find
σ(cos(φ/2) − i n · σ sin(φ/2)) = γ(σ
σ) .
(cos(φ/2) + i n · σ sin(φ/2))σ
Thus given two elements of SO(3), γ1 and γ2 and considered their product γ2 γ1 we have
σU(γ2 )U(γ1 ) = U + (γ1 )γ2(σ
σ )U(γ1 ) =
U + (γ1 )U + (γ2 )σ
σ ) = U + (γ2 γ1 )σ
σ U(γ2 γ1 ) .
= γ2 γ1 (σ
62
CHAPTER 10. THE SPIN
Thus we have that the unitary operator V = U(γ2 γ1 )U + (γ1 )U + (γ2 ) is such that
V +σ V = σ
(10.5)
σV = V σ .
(10.6)
or
Being V an element of SU(2) we easily verify that it implies V = 1 or V = −1. We can
then say that (10.2) gives a projective representation of SO(3) i.e. (10.3) holds with the
phase factor α which can take only the values ±1. This tells us also that if we have a
sequence of transformations of SO(3) whose product equals the identity, under the related
SU(2) transformations the spinor is transformed either in itself or just changes sign.
Viceversa given an element U of SU(2) we can write
+
U σj U =
3
X
Γjl σl
(10.7)
l=1
as the trace of the first member is zero. Being the first member an hermitean operator we
have that the elements Γjl are real. Taking the product of two such relations and taking
the trace we have
δjm =
3
X
Γjl Γml
l=1
from which we conclude that the Γjl are elements of the real orthogonal group.
Moreover if we take the trace of U + σ1 UU + σ2 UU + σ3 U we find
X
2i = 2i
ǫjkl Γ1j Γ2k Γ3l
jkl
i.e. det Γ = 1. U and −U through (10.7) generate the same Γ.
Viceversa if U and V generate the same Γ due to the reasoning of eq.(10.5,10.6)) we have
U = ±V . Thus to each element of SU(2) there corresponds an element of SO(3) while
to each element of SO(3) there correspond two elements SU(2) given by ±U.
SU(2) is a simply connected group which is the universal covering group of SO(3). This
is seen by the fact that it is topologically equivalent to the surface of a four dimensional
sphere which is also topologically equivalent to two tridimensional balls of radius 1 having
in common the external surface.
Instead SO(3) is topologically equivalent to a ball of radius π with the diametrically
opposite points of the surface of such a ball, identified.
This because each element of SO(3) is the rotation around an axis and the rotation
around an axis by π is the same transformation which we obtain rotating by π around
the opposite direction. SO(3) is not simply connected.
Chapter 11
Identical particles
11.1
Introduction
There are no observable differences when we exchange two identical particles. In quantum
mechanics however some phenomena arise which have no counterpart in classical physics.
Is it possible two distinguish two states which differ by the exchange of two particles?
Classically yes because the two particles can be distinguished by the different initial
conditions. This presupposes the possibility to follow the particles along their trajectories.
This we know is not possible in quantum mechanics and this gives rise to the operative
impossibility to distinguish two states in which e.g. two electrons have been exchanged.
In a surprising (and also satisfactory) manner such the indistinguishable nature is more
than operative but results is some new axioms in the theory which restrict the possible
states when we are in presence of identical particles.
Let us consider n identical particles: we can describe the first particle with some dynamical
variables ξ1 (e.g q1 , p1 , s1 ) the second with ξ2 etc... . As the refer to different systems
we shall have that the ξ1 commutes with the ξ2 etc... . The fact that the particles are
identical is translated into the fact that the Hamiltonian will be a symmetric function
of the variables ξ1 , ξ2 etc.. i.e. invariant under the exchange of any pair of particle and
this must happen whatever physical perturbation the system is subject to (this is true
(1)
(1)
obviously also in classical physics). If we denote with ψa , ψb .... a base of vectors which
(2)
(2)
describe the first particle we shall have analogous vectors ψa , ψb ... which describe the
second particle etc.... The Hilbert space Hn which describes the system of n particles is in
natural manner given by the tensor product of the Hilbert spaces of the single particles;
(1)
(2)
(3)
a base of such tensor product is given by ψa ψb ψc .... and such a basis is called (Dirac)
a symmetric base. If we exchange two particles e.g. 1 con 2 the previously described
(2)
(1)
(3)
state changes into ψa ψb ψc ..... The means values of physical observables (which are
63
64
CHAPTER 11. IDENTICAL PARTICLES
symmetric under exchange of the dynamical variables ξ1 ,..... ξn ) do not change. Clearly
factorized vectors of the above type do not exhaust the Hilbert space, in the sense that
such a space will be formed by arbitrary superposition of such factorized vectors. The
exchange of two particles generates a linear transformation in Hilbert space and it is
immediate to realize that such operator which represents the exchange does not depend
on the particular symmetric base chosen in Hn .
Thus given any vector ψ ∈ Hn we can consider the action of whatever permutation on
it P ψ and we shall have a linear representation of the permutation group. Among the
vectors of Hn there are some which under the action of P changes only by a phase, i.e.
ψ and P ψ represent the same state whatever P ∈ Sn being Sn the group of permutation
of n objects. In this case given such a ψ, the one dimensional space of vectors {P ψ} is
the base of a one dimensional representation of Sn . It is a simple exercise to prove that
there are only two possibilities: a) P ψ = ψ for all P ∈ Sn ; b) P ψ = δP ψ being δ = ±1
according to the fact that P be an even or odd permutation and we have δP Q = δP δQ .
In the first case the vector ψ will be called symmetric and in the second will be called
antisymmetric. The important fact is that such symmetry or antisymmetry property are
conserved in time due to the symmetry with respect to permutations of the evolution
operator, i.e. of the Hamiltonian H.
In fact as P H = H P whatever P , we have P Hψ = H P ψ = ±H ψ and thus if P ψ = ±ψ
ǫ
ǫ
we have P (ψ + Hψ) = ±(ψ + H ψ). Thus a state which is initially symmetric, stays
ih
ih
symmetric, while one which is antisymmetric stays antisymmetric.
Thus there is the consistent possibility that for certain types of particles only symmetric
states are realized in nature, and for others only antisymmetric states are realized. This
by the way would realize a very elegant formal peculiarity: indistinguishable states are
described by the same ray in Hilbert space; in fact the exchange operator P change at
most the sign of the vector.
This is what happens in nature. One finds in fact that all experimental data agree with
the following axiom (of statistics): Particle with half-integer spin, called fermions, are
described by antisymmetric vectors and those of integers spin called bosons, by vectors
symmetric under the exchange of particles.
Let us see now how one can build in the Hilbert space Hn symmetric or antisymmetric
(1)
(2)
(n)
states. If we starts from the vectors ψa , ψb ,......ψg , with a, b, c . . . not necessarily
different among themselves, a symmetric vector is easily constructed with the operation
X
(2)
ψS =
P ψa(1) ψb ....ψg(n)
P
For a such a state we can only say that there are na particles in the state a, nb particles
11.1. INTRODUCTION
65
in the state b etc.. without being able to specify which particle occupy such states. An
antisymmetric vector is easily built in analogous manner
ψA =
X
(2)
δP P ψa(1) ψb ....ψg(n)
which is equivalent to the determinant
ψ (1) ψ (2)
a
a
(1)
(2)
ψb
ψb
... ...
(1)
ψg ψg(2)
(n)
. . . ψa (n) . . . ψb .
. . . . . . (n) . . . ψg An immediate result is that if two indices e.g. a and b are equal, due to the antisymmetry
we have ψA = 0. This can be also seen from the fact that we must have at the same time
P12 ψA = ψA
and
P12 ψA = −ψA
Thus there cannot be two identical fermions in the same state. The states described above
are not the only symmetric or antisymmetric states in Hn .
Linear combinations of symmetric or antisymmetric states of the type above described
cannot be written in the above form even if they maintain the property of symmetry or
antisymmetry. The physical Hilbert space is given by HnS or by HnA whether the particles
are boson or fermions.
This axiom has very important consequences on the statistics of particle systems, i.e. on
the counting of states and thus on the computations of the mean values which occur in
statistical thermodynamics.
In fact according to the particles being bosons or fermions, the traces which give rise the
statistical mean values of observables, are to be taken on the symmetric subspace of Hn
i.e. HS , or on the antisymmetric subspace HA .
hF i = TrHS (F e−βH )/TrHS (e−βH );
hF i = TrHA (F e−βH )/TrHA (e−βH ) .
Let us notice that in completely general way, i.e. independently of the principle of spin
and statistics, a permutation P ∈ Sn induces a unitary transformation in the Hilbert
space Hn ; this is seen from how P operates on the vectors of a symmetric basis for which
(1)
(2)
(n)
(2)
(1)
(2)
(n)
(2)
(P ψa′ ψb′ ....ψg′ , P ψa(1) ψb ....ψg(n) ) = (ψa′ ψb′ ....ψg′ , ψa(1) ψb ....ψg(n) ) .
66
CHAPTER 11. IDENTICAL PARTICLES
It is known that the interaction of the spin of a particle with the other degrees of freedom
and with the spin of other particles is a relativistic phenomenon and that in light atoms
1 2
such interaction is estimated of the order ( 137
) . Thus it is a very good procedure to
neglect in zero order approximation, the spin in the Hamiltonian and to introduce the
interactions due to the spin later as a perturbation.
Given now a Hamiltonian which depends only on the coordinates and conjugate momenta,
we want to examine the symmetry properties under permutations of the energy eigensolutions of the Schrödinger equation, independently of the principle of spin and statistics.
The results we will obtain will be combined later with the spin degrees of freedom is such
a way as to obtain symmetric or antisymmetric state vectors.
11.2
Two particle system
Let us consider for the case of two particles a solution of the Schrödinger equation
Hψ(q1 , q2 ) = Eψ(q1 , q2 ) .
The identity of the two particle implies that the Hamiltonian commutes with the elements
of the permutation group which in this simple case are two i.e. the identity and P12 . Such
a symmetry allows us to derive from ψ(q1 , q2 ) an other solution. In fact
P12 Hψ(q1 , q2 ) = HP12 ψ(q1 , q2 ) = EP12 ψ(q1 , q2 ) .
2
Let us notice that such a vector P12 ψ(q1 , q2 ) cannot be the null vector as P12
= 1. If
the new vector P12 ψ(q1 , q2 ) is not proportional to ψ(q1 , q2 ) we have at our disposal a two
dimensional subspace of solutions to the energy E. It is very useful to see such a two
dimensional space as the direct sum of two one dimensional spaces invariant under the
group P . In fact after forming the two vectors
ψ(q1 , q2 ) + P12 ψ(q1 , q2 )
ψ(q1 , q2 ) − P12 ψ(q1 , q2 )
we have that
P12 (ψ(q1 , q2 ) ± P12 ψ(q1 , q2 )) = ±(ψ(q1 , q2 ) ± P12 ψ(q1 , q2 ))
and thus the two one dimensional spaces generated by ψ(q1 , q2 )±P12 ψ(q1 , q2 ) are invariant.
Obviously it may happen that the starting ψ(q1 , q2 ) be already symmetric or antisymmetric with respect to the exchange in which case one of the two vectors ψ(q1 , q2 )±P12 ψ(q1 , q2 )
11.3. THREE PARTICLE SYSTEM
67
vanishes and we have only one one-dimensional space. In the general case of n identical
particles given a solution of the Schrödinger equations
Hψ(q1 , q2 , . . . qn ) = Eψ(q1 , q2 , . . . qn )
we can construct a linear space of solutions by applying to this solution the permutation
group
P ψ(q1 , q2 , . . . qn ) .
In this way we obtain a space which is at most n! dimensional.
Again it is important to decompose such a linear space into subspaces invariant under the
permutation group.
11.3
Three particle system
Let us consider the simple but not trivial case of three particles. An arbitrary wave
function ψ(q1 , q2 , q3 ) can be rewritten as the sum of the following four wave functions
ψ(q1 , q2 , q3 ) =
1 2
3
ψ(q1 , q2 , q3 )
ψ(q1 , q2 , q3 )
+
+
1 3
2
ψ(q1 , q2 , q3 )
+
ψ(q1 , q2 , q3 ) =
1
[(123) + (132) + (213) + (231) + (312) + (321)]+
6
1
+ [(123) + (213) − (321) − (231)]+
3
1
+ [(123) + (321) − (213) − (312)]+
3
The symbol
1
[(123) − (132) − (213) + (231) + (312) − (321)] .
6
is the symmetrizer, the symbol
is the antisymmetriser while the
symbol 1 2 means, first symmetrize with respect to 1 and 2 and then antisymmetrize
3
68
CHAPTER 11. IDENTICAL PARTICLES
the found result with respect to 1 and 3. In analogous manner the symbol 1 3 means,
2
first symmetrize with respect to 1 and 3 and then antisymmetrize the found result with
respect to 1 and 2.
If we apply an element of the permutation group to
and thus {P }
ψ we find the same vector
ψ is a one dimensional space. Similarly {P }
ψ gives rise to a
one dimensional representation
Q
X
P
δP P ψ =
X
δP QP ψ = δP
P
X
δP Q QP ψ = δQ
P
X
δP P ψ.
P
I.e. if we apply an element of the permutation group we find the same vector multiplied
by 1 or −1 for even or odd permutation.
Let us examine now the subspace {P }α with α = (123) + (213) − (321) − (231).
We have
P12 α = P12 [(123) + (213) − (321) − (231)] =
= (213) + (123) − (312) − (132) ≡ β
P12 β = P12 [(213) + (123) − (312) − (132)] =
= (123) + (213) − (321) − (231) = α
2
as P12
= 1.
P13 α = P13 [(123) + (213) − (321) − (231)] =
= (321) + (231) − (123) − (213)] = −α
P13 β = P13 [(213) + (123) − (312) − (132)] =
= (231) + (321) − (132) − (312)] = −α + β .
P12 and P13 generate the whole permutation group of three elements: in fact
P12 (123) = (213); P13 (213) = (231); P12 (231) = (132);
P13 (132) = (312); P12 (312) = (321) .
In the base v1 = α, v2 = β we have the representation
X
P vi =
vj P (j, i)
j
11.3. THREE PARTICLE SYSTEM
with
P12 =
69
!
0 1
P13 =
1 0
−1 −1
0
1
!
.
From these by multiplication we can compute the representatives of all permutations.
In similar way we can compute the representation generated by 1 3
2
ψ ≡ γ.
P12 γ = −γ, P12 ǫ = −γ + ǫ, P13 γ = ǫ, P13 ǫ = γ
and we find the representation
′
P12
=
−1 −1
0
1
!
′
P13
=
0 1
1 0
!
.
The two representations are equivalent by a similitude transformation. In fact given
!
1
0
S=
−1 1
we have P S = SP ′ which is enough to verify for P12 and P13 . This is a general fact:
representations generated by Young tables having the same Young diagram are equivalent.
It is simple exercise to show that the space spanned by α is β either two-dimensional or
it is the null space, i.e. if we set β = cα we have α = 0 an that these two representations
(i.e. α, β and γ, ǫ), are irreducible, i.e. if the subspace α + cβ were invariant we have
α = β = 0. This is a general result: spaces generated applying the permutation group
to a vector symmetrized by means of a Young symmetrizer are irreducible. The two
subspaces (α, β) and (γ, ǫ) are not always independent, in the sense that one can be the
null subspace, or it may happen that the two subspaces coincide.
One verifies however that vectors belonging to spaces generated by different Young diagrams (i.e. symmetric, antisymmetric and mixed) are orthogonal. This is also a general
result.
Note: The fact that wave functions belonging to irreducible inequivalent representations,
i.e. to different Young diagrams are orthogonal is a general consequence of the orthogonality of inequivalent representations
P
(P ψi , P φj ) X X X ∗
P (k, i)P ′ (h, j)(ψk , φh )
(ψi , φj ) = P
=
n!
P
k
h
70
CHAPTER 11. IDENTICAL PARTICLES
where P (k, i) is the representation to which ψi belong and P ′ (k, i) is the representation
to which φj belong. But the theorem on the orthogonality of the matrix elements of
irreducible inequivalent representations tells us
X
P ∗ (k, i)P ′ (h, j) = 0
P
We saw that given a generic wave function, by applying to it the permutation group we
obtain an n! dimensional space; in our example a 6 dimensional space. Notice that we have
three inequivalent representations: two one dimensional (inequivalent) and two equivalent
two dimensional representations. Even this is a general result: In the decomposition of the
generic vector in irreducible representations each representation appears as many times
as the dimension of the representation itself and thus (Burnside theorem) the sum of the
squares of the dimension of inequivalent representations equals the dimension of the group
i.e. n! in the case of Sn .
In the case in which the wave function is not generic but e.g. a stationary solution of
Schrödinger equation, is not granted that the space generated by applying the permutation
group be n! dimensional, in our case 6 dimensional.
We want now to examine which are the possible dimensions of such a space for a generic
Hamiltonian invariant under the permutation group. I.e. let us suppose that initially the
space of an energy level be effectively 6 times degenerate.
We ask in how many levels and with which degeneracy such a level splits under the
action of an arbitrary perturbation whose only requirements will be to be hermitean and
invariant under permutations.
If a perturbation existed non invariant under permutations we would have discovered a
method to distinguish the particles which would be no longer identical.
We have the following simple result: If {P } is a group of unitary transformations in H
and V is a subspace invariant under such group of transformations, also its orthogonal
complement V⊥ is invariant under such a group of transformations. In fact if w ∈ V⊥ for
any v ∈ V and any P , we have (v, P w) = (P + v, w) = (P −1 v, w) = 0 because P −1v ∈ V
and thus P w ∈ V⊥ .
In this way we can decompose H successively in orthogonal invariant subspaces to reach
at the end a decomposition in invariant orthogonal irreducible subspaces.
Moreover if V is invariant the projector on V , PV commutes with all the P . In fact
PV P x = PV P (PV x + (1 − PV )x) = PV P PV x
11.3. THREE PARTICLE SYSTEM
71
while
P PV x = P PV (PV x + (1 − PV )x) = P PV x = PV P PV x .
The simplest Hamiltonian which breaks the degeneration is given by a proper combinations of projectors on the invariant orthogonal subspaces. We saw how the symmetric
vector generates an invariant subspace which is orthogonal to the one generated by the
antisymmetric vector which at the same time are orthogonal to the four dimensional space
generated by the two mixed Young tables.
Let us consider within such four dimensional space the vector aα + a′ α′ .
Applying to it all elements of the permutation group we generate the two dimensional
space (aα + a′ α′ , aβ + a′ β ′ ).
Let us consider now the orthogonal complement, within this four dimensional space, of
such two dimensional space.
In full generality, if we have a subspace V invariant under the permutation group the
projector on such subspace commutes with all permutations.
In fact chosen an orthogonal basis v1 in such subspaces such projector can be written as
P
P = i vi · vi and we have
P PP −1 = P PP + =
X
i
P vi · P vi =
X
i
vi · vi
because the unitary transformation P transforms an orthonormal system of V in an
orthonormal system of V .
In this way the four projectors on the described invariant orthogonal subspaces constitute
four hermitean operators, which commute with all permutations and thus an acceptable
perturbation Hamiltonian is
c1 P1 + c2 P2 + c3 P3 + c4 P4
with real arbitrary ci . Such a perturbation is apt to break the degenerate level into four
levels.
Thus a generic perturbation Hamiltonian, invariant under the permutation group breaks
the original 6 times degenerate level in four distinct levels, of which two non degenerate,
i.e. the symmetric and the antisymmetric ones, and two doubly degenerate.
It is a consequence of Schur lemma that such residual degeneracy cannot be further
removed. In fact according to perturbation theory we have to compute the eigenvalues of
(vi , HI vj ). But we must have P HI = HI P and we have
(vi , HI P vj ) =
X
k
(vi , HI vk )P (k, j)
72
CHAPTER 11. IDENTICAL PARTICLES
which taking into account the unitarity of P gives
X
X
(vi , P HI vj ) = (P −1 vi , HI vj ) = (P + vi , HI vj ) = (
P ∗ (i, l)vl , HI vj ) =
P (i, l)(vl , HI vj )
l
l
and thus for Schur lemma we have (vi , HI vj ) = cδij and it is not possible to remove the
degeneracy.
Notice that the above analysis is completely independent of the axiom of spin and statistics.
It is now important to study the structure of the state vector in presence of such axiom.
I.e. we must find which vector in the space of three identical particles are totally symmetric
or totally antisymmetric.
If the particles are of spin zero and as such bosons of the studied wave functions the only
acceptable one is
ψ.
We consider now the case of three identical particles of spin spin 1/2 (fermions).
A spinor of total spin 3/2 is symmetric under the exchange of particles and thus the only
way to form a totally antisymmetric vector is to multiply it by the unique antisymmetric
function we have at our disposal i.e.
ψ.
Suppose now to have a combination of vectors α and β with spinors such that the vector
is totally antisymmetric.
Without loosing in generality we can consider states with well defined Sz . In fact we
can project our state on the eigensubspaces of Sz using the projectors on such subspaces
e.g. the projector −(Sz − 3/2)(Sz + 1/2)(Sz + 3/2)/2, which being symmetric in the
spin operators does not alter the symmetry nature of the vector. We know already that
Sz = ±3/2 cannot be used as completely symmetric.
Thus we use Sz = ±1/2, referring for concreteness to the case Sz = 1/2. The most general
vector with Sz = 1/2 is
ψ = [a| + +−i + b| + −+i + c| − ++i]α + [d| + +−i + e| + −+i + f | − ++i]β.
Being P12 odd, we must have
P12 ψ = [a| + +−i + b| − ++i + c| + −+i]β + [d| + +−i + e| − ++i + f | + −+i]α = −ψ
and thus comparing the two terms −a = d, −c = e, −b = f, −d = a, −e = c, −f = b.
It follows
ψ = [a| + +−i + b| + −+i + c| − ++i]α − [a| + +−i + c| + −+i + b| − ++i]β.
11.3. THREE PARTICLE SYSTEM
73
Moreover
P13 ψ = −[a|−++i+b|+−+i+c|++−i]α−[a|−++i+c|+−+i+b|++−i](−α+β) = −ψ
from which −a = −c + b, − c = −a + a = 0, a = −b, which gives us at last
ψ = [| + +−i − | + −+i]α − [| + +−i − | − ++i]β.
As from P12 and P13 we can compose all permutations the total antisymmetry of ψ is
proven.
Notice that
S+ (| − ++i − | + −+i) = 0
as well
S+ (| + +−i − | − ++i) = 0
i.e. we have the well known result that both spin vectors are eigenstates of total spin
S = 1/2.
Thus we found that for two or three electrons (particles of spin 1/2) to each value of the
total spin there corresponds a well defined irreducible representation of the permutation
group.
This is also a general result for n particles of spin 1/2 which by the spin-statistics theorem
obey Fermi statistics.
140424
74
CHAPTER 11. IDENTICAL PARTICLES
Chapter 12
The formal theory of scattering
12.1
Introduction
The elementary treatment of potential scattering is performed in the time independent
method: one looks for a stationary solution of the Schrödinger equation which obeys to
proper boundary conditions. The computation of the current at large distances allows to
go over from the so computed wave function to the cross section. There exists a variety
of more complex scattering problems for which such a formulation is not apt; the time
dependent method in which one explicitly considers the evolution of the state appears
more useful, both in deriving general properties of the scattering states and of transitions
amplitudes and to perform approximate computation, like perturbative expansions. In
such time dependent methods one tries to reproduce in a mathematical language a situation more or less similar the the experimental setting, which is characterized by the
fact the before and after the scattering process, the intervening particles (subsystems)
are free. This can be obtained in two ways 1) Construct wave packets (obviously of not
perfectly defined energy) and then take the limit in which the energy becomes perfectly
defined. 2) Use plane wave (states of perfectly defined energy) and switch off artificially
the interaction before and after a certain time interval and then have such time interval
tend to infinity as to have a process of well defined energy. Clearly the second method is
more artificial than the one dealing with wave packets, but formally simpler. After the
switching off of the interaction the particle is again free and the momentum is a constant
of motion; one analyzes such state in momentum eigenstates so obtaining the transition
amplitude.
In the present treatment we use the first method, following the line of the classical paper
by Lippmann and Schwinger [1]; the treatment by means of wave packets has been given
by Gell-Mann and Goldberger [2] and in mathematical rigorous way by Jauch [3].
75
76
12.2
CHAPTER 12. THE FORMAL THEORY OF SCATTERING
The Lippmann-Schwinger equation
Let us consider the evolution in time of a system compose of two interacting parts. Let
H be the hamiltonian of the system H = H0 + V where H0 is the free hamiltonian and
V the interaction.
To remove the time dependence associated to H0 , (as we want to describe the effect of V )
let us perform on the generic state vector |ψ s (t)i, which satisfies Schrödinger’s equation
(t in the following stays for t/~)
i
∂ s
|ψ (t)i = (H0 + V )|ψ s (t)i
∂t
the transformation
|ψ(t)i = eiHo t |ψ s (t)i
thus obtaining
i
∂
|ψ(t)i = H1 (t)|ψ(t)i,
∂t
H1 (t) = eiHo t V e−iHo t .
The evolution operator defined by
|ψ(t)i = U(t, t0 )|ψ(t0 )i
with the property U(t, t) = 1 satisfies
i
∂
U(t, t0 ) = H1 (t)U(t, t0 ) .
∂t
(12.1)
Such a differential equation can be rewritten in the form of an integral equation
Z t2
U(t2 , t1 ) = 1 − i
H(t′ )U(t′ , t1 )dt′
t1
which includes the condition U(t1 , t1 ) = 1. The main properties of U(t2 , t1 ) have been
discussed in Chapter 5.
Eq.(12.1) can be rewritten as
U(t, −∞) = 1 − i
Z
t
−∞
H1 (t′ )U(t′ , ∞)dt′
(12.2)
provided the limit for t1 → −∞ exists. We are interested in the transition between
eigenstates of the free hamiltonian |φa i, |φbi with
H0 |φa i = Ea |φa i,
H0 |φbi = Eb |φb i
a 6= b .
(12.3)
12.2. THE LIPPMANN-SCHWINGER EQUATION
77
We shall introduce following the general ideas discussed at the beginning of this section,
an adiabatic switch-off factor, V → e−ε|t| and thus we have
Z +∞
Tba = hφb | − i
eiH0 t e−ε|t| V e−iH0 t U(t, −∞)dt|φa i .
−∞
A slow switching on and off of the interacrion is necessary because only with an adiabatic
process the energy of the final state will equal that of the initial state, something which
is not true for a sudden switch-on. We shall show this in the following. We have
Tba = −ihφb |V |Ψ+
a (Eb )i
with
|Ψ+
a (Eb )i
=
Z
∞
−∞
(12.4)
e−ε|t| eiEb t e−iH0 t U(t, −∞)|φa idt .
We want to obtain an operator equation for such a state which will be at the center of
our interest. From eq.(12.2) we obtain
Z +∞
Z
+
−ε|t| iEb t −iH0 t
′
|Ψa (Eb )i =
e
e e
|φa idt − i
−∞
= 2πδε (Eb − Ea )|φa i
Z ∞
Z
iEb τ −iH0 τ −ετ
− i
dτ e
e
e V
0
+∞
−∞
+∞
−∞
e−ε|t| eiEb t |Ψ+
a (Eb )i
(12.5)
′
′
′
′
eiEb t e−iHo t [eετ e−ε|τ +t | e−ε|t | ]U(t′ , −∞)|φaidt′
obtained by setting τ = t − t′ and
δε (x) =
ε
1
.
2
π x + ε2
′
The switching-off factor in the square bracket differs from e−ε|t | for the fact that it stays
constant equal to 1 in the interval −τ < t′ < 0. It is an assumption of the treatment that
for ε → 0 the two (adiabatic) switching off factors are equivalent. We have then
|Ψ+
a (Eb )i = 2πδε (Eb − Ea )|φa i +
1
V |Ψ+
a (Eb )i .
Eb − H0 + iε
To understand the nature of this equation let us take the scalar product with a complete
set of vectors |ni.
hn|Ψ+
a (Eb )i
1
= 2πδε (Eb − Ea )hn|φa i +
Eb − En + iε
Z
hn|V |mihm|Ψ+
a (Eb )idm .
This is an equation of type
1
F (x) = f (Ea , Eb )G(x) +
Eb − E(x) + iε
Z
V (x, x′ )F (x′ )dx′
78
CHAPTER 12. THE FORMAL THEORY OF SCATTERING
where f (Ea , Eb ) is a multiplicative factor inessential to the solution. Therefore it is
meaningful to divide by δε (Eb − Ea ), i.e. consider a new vector |ψa+ i given by
+
|Ψ+
a (Eb )i = 2πδε (Eb − Ea )|ψa i
(12.6)
to obtain
|ψa+ i = |φa i +
1
V |ψa+ i .
Ea − H0 + iε
(12.7)
This is the Lippmann-Schwinger equation. The δε (Eb − Ea ) in eq.(12.6) clearly indicates
energy conservation in the limit ε → 0. The obtained |ψa+ i is a solution of Schrödinger
equation at the energy Ea . In fact
(Ea − H0 )|ψa+ i = (Ea − H0 )|φa i + V |ψa+ i
i.e.
(Ea − H)|ψa+ i = 0 .
Thus starting from a time dependent equation we obtain in the limit ε → 0 a time
independent equation which defines a particular solution of the Schrödinger equation.
|ψa+ i satisfies also an other equation which will become useful for several formal develop-
ments. Multiplying |ψa+ i by V we obtain
V |ψa+ i = V |φa i + V
(1 − V
1
V |ψa+ i
Ea + iε − H0
1
)V |ψa+ i = V |φa i
Ea − H0
or
(Ea − H + iε)
1
V |ψa+ i = V |φa i
Ea − H0 + iε
i.e.
|ψa+ i = |φa i +
1
V |φa i .
Ea − H + iε
We prove now the following orthogonality relation for the |ψa+ i
1
1
)(|φa i +
V |ψa+ i
Eb − H − iε
Ea − H0 + iε
1
1
|ψ + i + hφb |
|ψ + i = hφb |φa i .
= hφb|φa i + hφb |
Ea − H0 + iε a
Eb − H + iε a
hψb+ |ψa+ i = (hφb | + hφb|V
12.3. THE ADIABATIC THEOREM
12.3
79
The adiabatic theorem
We prove now the “adiabatic theorem”: The solution of Schrödinger equation, |ψ + i,
characterized by
|ψ + i = |φi +
1
V |ψ + i,
E + iε − H0
(H − E)|ψ + i = 0
when the coordinate representation make sense, satisfy the usual boundary conditions
which we impose in the elementary treatment of non relativistic scattering, i.e. at space
infinity only outgoing waves are present. To this end we calculate
Z
1
hr|k′ihk′ |V |ψ + i ′
+
hr|
V |ψ i =
dk
k′2
E + iε − H0
E + iε − 2m
Z ∞
Z +1
k ′2 dk ′
1
′
eik rζ f (ζ, k ′)dζ
=
′2
3/2
k
(2π)
E + iε − 2m −1
0
with
hr|ki =
k2
k2
=
=E
2m
2m
1
eik·r ,
3/2
(2π)
and
′
f (ζ, k ) =
Z
ζ=const
(12.8)
hk′ |V |ψ + idϕ
having assumed the direction of r as polar axis.
We have for eq.(12.8)
Z ∞ ik′ r
Z 1 ik′ rζ
′
1
e−ik r
k ′2 dk ′
e
∂f (ζ, k ′ )
e
′
′
f
(1,
k
)
−
f
(−1,
k
)
−
dζ
′
(2π)3/2 0
ik ′ r
ik ′ r
∂ζ
E + iε −
−1 ik r
If
∂f (ζ,k ′ )
∂ζ
k ′2
2m
.
is L1 (ζ), as r tends to infinity for the Riemann-Lebesgue theorem the term
R1
−1
goes to zero more rapidly than 1/r. Thus it is of no interest in the discussion of the
asymptotic wave. We are left with
Z ∞ ik′ r
′
e f (1, k ′) − e−ik r f (−1, k ′ ) ′ ′
1
k dk =
k ′2
(2π)3/2 ir 0
E + iε − 2m
Z ∞ ik′ r ′ ′
Z ∞ ik′ r
e k dk
1
e (f (1, k ′ ) − f (1, k))k ′dk ′
f
(1,
k)
+
k ′2
k ′2
(2π)3/2 ir
E + iε − 2m
E + iε − 2m
0
0
Z ∞ −ik′ r ′ ′
Z ∞ −ik′ r
e
k dk
e
(f (−1, k ′ ) − f (−1, k))k ′ dk ′
− f (−1, k)
−
. (12.9)
k ′2
k ′2
E + iε − 2m
E + iε − 2m
0
0
Again if
(f (±1, k ′ ) − f (±1, k))k ′
k ′2
E + iε − 2m
80
CHAPTER 12. THE FORMAL THEORY OF SCATTERING
are L1 , by Riemann-Lebesgue theorem the two integral go to zero for r going to infinity
and in such limit we have that the above expression goes over to
Z ∞ −χr
Z ∞ −χr
e χdχ
e χdχ
1
ikr
f (1, k) − me 2πi −
+ f (−1, k)
χ2
χ2
(2π)3/2 ir
E + 2m
E + 2m
0
0
where we applied the Jordan lemma [6] to the contours depicted in figure 12.1.
+i oo
+i oo
+
+
0
0
−ioo
0
0
−i oo
Figure 12.1: Contours in the k complex plane
By Riemann-Lebesgue theorem also the terms
Z ∞ −χr
e χdχ
0
E+
χ2
2m
vanish for large r and we are left with
hr|
1
eikr 2πim
(2π)2 m eikr
√
V |ψ + i ∼ −
f
(1,
k)
=
−
hφk |V |ψ + i
3/2
E + iε − H0
r (2π) i
2π r
where
hr|φk i = p
1
(2π)3
eik·r ,
|k| =
r√
2mE .
r
12.4. “IN” AND “OUT” STATES
81
Recalling the elementary treatment of scattering we have for the cross section
σ(θ, φ) = (2π)4 m2 hφk |V |ψ + i|2
with
hr|φk i = p
for the initial state.
12.4
(12.10)
1
eikz
(2π)3
“In” and “out” states
We proceed in the interpretation of |ψa+ i establishing the fundamental relation
|ψa+ i = U(0, −∞)|φa i .
We obtain this result by noticing that
Z +∞
+
|Ψa (Eb )i =
eiEb t e−ε|t| e−iHo t U(t, −∞)dt = 2πδε (Eb − Ea )|ψa+ i
−∞
is the Fourier transform of A = e−ε|t| e−iH0 t U(t, −∞)|φa i at the frequency Eb which gives
a 2πδε (Eb − Ea ) iff
e−iH0 t U(t, −∞)|φa i = e−iEa t |ψa+ i .
For t = 0 we have
U(0, −∞)|φa i = |ψa+ i .
It means that if we start from a plane wave |φa i at t = −∞ and we switch on the
interaction up to time 0 we obtain the scattering wave. We can also perform the symmetric
treatment from +∞ to −∞ defining
Z ∞
−
|Ψa (Ea )i =
eiEb t e−ε|t| e−iH0 t U(t, +∞)|φa idt .
−∞
We shall call ψ + “in state” and ψ − “out state”; ψ − is obtained replacing −∞ with +∞
in the definition of the ψ + . The out states satisfy an equation which is similar to the
Lippmann-Schwinger equation (12.7) replacing iε with −iε as can be verified by repeating
the procedure.
Summing up we have
1
V |ψa± i
Ea ± iε − H0
1
V |φa i .
|ψa± i = |φa i +
Ea ± iε − H
|ψa± i = |φa i +
82
CHAPTER 12. THE FORMAL THEORY OF SCATTERING
The orthogonality relation (18) can be extended to the out states
hψb± |ψa± i = hφa′ |φa i .
We compute now hφb |U(+∞, −∞)|φai
1
hφb |U(+∞, −∞)|φa i =
= hφb| + hφb |V
|ψa+ i
Eb − iε − H
1
1
V |ψa+ i + hφb |V
|ψ + i
= hφb |φa i + hφb |
Ea + iε − H0
Eb − iε − H a
= hφb |φa i − 2πiδ(Eb − Ea )hφb|V |ψa+ i
(12.11)
hψb− |ψa+ i
insofar
|ψb− i = U(0, +∞)|φbi,
|ψa+ i = U(0, −∞)|φa i .
The S-matrix will be defined by
hφb |S|φai = hφb |U(+∞, −∞)|φai = hψb− |ψa+ i .
12.5
The Moller wave function operators and the S
matrix
Let us introduce two operators, the “Moller wave function operators”
Ω(+) = U(0, −∞),
Ω(−) = U(0, +∞)
|ψa+ i = Ω(+) |φa i
(in potential scattering ψ + is represented by a plane wave plus an outgoing spherical
wave)
|ψa+ i = Ω(+) |φa i
(in potential scattering ψ − is represented by a plane wave plus an incoming spherical
wave)
Properties of the operators Ω(±) .
The operator Ω(+) is isometric i.e.
+
Ω(+) Ω(+) = I
In fact
+
hφb |Ω(+) Ω(+) |φa i = hψb+ |ψa+ i = hφb|φa i
(12.12)
12.6. THE UNITARITY OF THE S MATRIX AND THE OPTICAL THEOREM
83
from which eq.(12.12) follows, being |φa i a complete set.
+
Ω(+) Ω(+) = 1 − Σ|ψB ihψb | ≡ 1 − B
(12.13)
being |ψB i the bound states. In fact
+
Ω(+) Σa |φa ihφa |Ω(+) = Σa |ψa+ ihψa+ | = I − Σ|ψB ihψB |
Similarly
+
Ω(−) Ω(−) = I
+
Ω(−) Ω(−) = I − B
In the proof we used the fact that the set of (outgoing) incoming solutions plus the bound
stated form a complete set. We examine now the unitarity of the S-matrix.
+
+
+
+
+
+
+
+
S + S = Ω(+) Ω(−) Ω(−) Ω(+) = Ω(+) (I − B)Ω(+) = I − Ω(+) BΩ(+) = I
SS + = Ω(−) Ω(+) Ω(+) Ω(−) = Ω(−) (I − B)Ω(−) = I − Ω(−) BΩ(−) = I
+
The fact that Ω(+) BΩ(+) equals zero is proved by applying such an operator to the
generic plane wave |φa i. In fact
Ω+ |φa i = |ψa+ i
and
hψB |ψa+ i = 0
being |ψB i and |ψa+ i eigenstates of H to different values of the energy.
12.6
The unitarity of the S matrix and the optical
theorem
We can express the unitarity of the S-matrix in terms of the matrix T defines by the
relation
S = I − iT
We have
S + S = I + i(T + − T ) + T + T = I
from which
i(T − T + ) = T + T .
We define T by
hφb |T |φa i = 2πδ(Eb − Ea )hφb|T |φa i .
84
CHAPTER 12. THE FORMAL THEORY OF SCATTERING
From eq.(12.11) we have
hφb |T |φa i = hφb |V |ψa+ i .
Simplifying the notation we obtain
+
hb|T |ai − hb|T |ai = −i
Z
hb|T + |nihn|T |aidn
i.e.
+
2πiδ(Eb − Ea )(hb|T |ai − hb|T |ai) = (2π)
with
i.e.
∗
i[hb|T |ai − (ha|T |bi) ] = 2π
2
Z
ha|T + |nihn|T |biδ(Ea − Eb )δ(Ea − En )dn
Z
|nihn|dn = I
Z
(hn|T |bi)∗ hn|T |aiδ(Ea − En ) dn .
(12.14)
For |ai = |bi we obtain the optical theorem
Z
−2 Imha|T |ai = 2π |hn|T |ai|2δ(En − Ea )dn .
If T is symmetric under the exchange of |ai with |bi the first term in eq.(12.14) will be
equal to −2 Imhb|T |ai. This occurs in a very important case i.e. when the theory is
invariant under time reversal and parity and when we adopt for the vectors |ai and |bi
special phase conventions. We want to see how this happens in the simple case of non
relativistic potential scattering. To say that the theory is invariant under time reversal
means that (in absence of spin)
i
∂ψ
= Hψ
∂t
implies
∂ψ ∗
= Hψ ∗
∂t
where we have H = H0 + V . Clearly a local and thus real potential gives rise to an
−i
hamiltonian invariant under time reversal. A non local potential V (x, x′ ) can give a
counterexample. In order for V to be hermitean we need V (x, x′ ) = V ∗ (x′ , x). To be the
hamiltonian invariant under time reversal we need
∗ Z
Z
′
′
′
V (x, x )ψ(x )dx
= V (x, x′ )ψ ∗ (x′ )dx′
i.e. V must be real and thus symmetric. Let us examine the consequences of such a
potential on the related T -matrix
hφb |S|φai = hφb |φa i − 2πiδ(Eb − Ea )hφb |T |φa i
12.6. THE UNITARITY OF THE S MATRIX AND THE OPTICAL THEOREM
hφb |T |φai = hφb |V |ψa+ i = hφb|V |φa i + hφb|V
which we can write as
85
1
T |φai
Ea + iε − H0
1
T
E + iε − H0
1
T =V +V
V
E + iε − H
1
(E + iε − H)
T =V .
E + iε − H0
Let us consider now the symmetry properties of the matrix elements of V
Z
Z
′
−ikb ·x
′ ika ·x′
′
hφb|V |φa i = e
V (x, x )e
dx dx = eika ·x V (x, x′ )e−ikb ·x dx dx′
T =V +V
i.e.
hk|V |k′ i = h−k′ |V | − ki
in the representation in which
hx|ki =
eik·x
.
(2π)3/2
For the kinetic energy we have
hk|p2 |k′ i = k2 δ(k − k′ ) = h−k′ |p2 | − ki
and as a consequence we have
′
′
hk|T |k i = hk|V |k i +
Z Z
hk|V |hidhhh|
1
|h′ idh′ hh′ |V |k′ i
E + iε − H
1
|h′ idh′ hh′ |V | − ki
h−k |T | − ki = hk|V |k i +
h−k|V |hidhhh|
E + iε − H
Z Z
1
| − h′ idh′ h−h′ |V |k′ i = hk|T |k′i .
= hk|V |k′ i +
hk|V | − hidhh−h|
E + iε − H
′
′
Z Z
The interpretation of this equation is clear.
Figure 12.2: Time reversed process
86
CHAPTER 12. THE FORMAL THEORY OF SCATTERING
The events depiceted in Figure 12.2 have the same amplitude. Moreover if the theory is
invariant under parity
hk|V |k′ i = h−k|V | − k′ i = hk′ |V |ki
(12.15)
and it follows
hk|T |k′ i = hk′ |T |ki .
Such a symmetry however depend in essential way on the used representation. If e.g. we
change over to
|ki → eiα(k |ki
we have
′
hk′ |V |ki → ei(α(k )−α(k) hk′ |V |ki
′
= ei(α(k)−α(k )
6= hk|V |k′ i .
12.7
Transition probabilities per unit time
The expression
w = |hφb |U(t, −∞)|φa i|2
(12.16)
gives us the probability to find at time t the particle with momentum pb . Being the
process stationary
∂w
∂t
gives us the transition probability per unit time to the state with
momentum pb .
∂U(t, −∞)
∂w
= hφb |
φa ihφb |U(0, −∞)|φa i∗ + c.c. =
∂t
∂t
−ihφb |V |ψa+ i
Using Lippmann-Schwinger equation
hφb |V |ψa+ i∗
+ c.c.
Ea − Eb + iε
∂w
= 2πδ(Eb − Ea )|hφb|V |ψa+ i|2 = 2πδ(Eb − Ea )|hφb|T |φa i|2 .
∂t
(12.17)
From this one goes over to the cross section with standard methods. The differntial cross
section is defined as the transition probability from an initial state φa , to all final states
whose the final k/k lies in a given solid angle ∆Ω and dividing by the initial flux. Thus
hx|φa i = p
1
(2π)3
eka ·x
with flux
ka
(2π)3 m
(12.18)
12.8. EXAMPLE ON PARTIAL WAVES
87
and for |φb i we can use the δ(k − k′ ) normalization (or any other normalization)
hx|φbi = p
1
(2π)3
eka ·x .
(12.19)
Integrating (12.17) in dk with k/k lying in ∆Ω we have we have
dσ(θ, φ)
∆Ω = (2π)4 m2 |hφb |V |ψa+ i|2 ∆Ω = (2π)4 m2 hφb |T |φa i|2∆Ω
dΩ
in agreement with eq.(12.10).
12.8
Example on partial waves
We mentioned at the beginning how there exist a relation between the S matrix and
the asymptotic behavior of the wave function in coordinate space, or the singularities of
the wave function in momentum space. Using the Lippmann-Schwinger equation we can
analyze this problem more completely. The scattering wave is
|ψa+ i = |φa i +
1
V |ψa+ i
Ea + iε − H0
hφb |V |ψa i = hφb |T |φa i .
Analyzing |ψa i in eigenstates of H0 we have
hφb |ψa+ i = hφb |φa i +
1
hφb |V |ψa+ i
Ea + iε − Eb
1
hφb |T |φa i .
Ea + iε − Eb
It is not necessary for |φa i in the Lippmann-Schwinger equation to be a plane wave; it
= hφb |φa i +
can also be a spherical wave of given angular momentum, provided it is an eigenstate of
H0 . Let us apply as an instructive example the developed formalism to the s-wave. For
the free wave we have
1
hr|φa i =
π
r
m sin kr
2k r
hφb|φa i = δ(Eb − Ea ) .
The Lippmann-Schwinger equation is the same
|ψa+ i = |φa i +
1
V |ψa+ i
Ea + iε − H0
and recalling eq.(12.11)
hφb |(S − 1)|φa i = −2πiδ(Ea − Eb )hφb |V |ψa+ i = −2πiδ(Ea − Eb )hφb |T |φai .
(12.20)
88
CHAPTER 12. THE FORMAL THEORY OF SCATTERING
As given l, the energy is the only variable we have from the unitarity of the S matrix
hφb|S|φa i = δ(Eb − Ea )eiα
with a real α. From the Schrödinger equation we know that (for l = 0)
hr|ψa+ i ∼
sin kr
eikr
sin(kr + δ)
+A
= const
r
r
r
with
A=
e2iδ − 1
= sin δ eiδ .
2i
Using eq.(12.20) we have
Z
2m
sin k ′ r eikr 2
√
hφ |ψ i − hφ |φi = −
A
r dr
r
r
π kk ′
r
r
k′
e2iδ − 1
1
1
A k′
=
−
.
=−
′
π k E − E + iε
2πi
k E − E ′ + iε
T is defines only on shell and thus we have
′
+
′
hφb |(S − 1)|φa i = −2πiδ(Eb − Ea )hφa |T |φai
hφa |T |φa i = −
e2iδ − 1
2πi
i.e. α = 2δ.
References
[1] B.A. Lippmann and J. Schwinger, Variational principles for scattering processes I
Phys.Rev. 79 3 (1950)
[2] M. Gell-Mann and M.L. Goldberger, The formal theory of scattering Phys.Rev. 91 2
(1953)
[3] J.M. Jauch, Theory of the scattering operator Helv.Phys.Acta 31 127 (1958)
[4] E.T. Whittaker and G.N. Watson, A course of modern analysis Chapt.VI
170228
Chapter 13
Scattering from a central potential
13.1
Introduction
In this chapter we shall examine in details some properties of the function S0 (k) i.e. of the
S-matrix in the partial wave l = 0, in the model of non relativistic potential scattering.
The reduced radial wave function obeys Schrödinger equation
−ϕ′′ + U(r)ϕ = k 2 ϕ,
with U =
2mV
.
~2
We shall confine ourselves to potentials which at the origin do not diverge more rapidly
than
1
,
r 2−ε
ε > 0. In this case from the indicial equation one deduces that one of the
possible solution vanishes for r → 0 and has derivative ϕ′ (0) 6= 0 and finite. The condition
ϕ(0) = 0 chooses among the the solutions the physically acceptable one (see Chapter V).
We shall suppose furthermore that our potential vanishes sufficiently fast (faster than 1r )
for r → ∞. The solution will have the asymptotic behavior
ϕ ∼ e−ikr − S(k)eikr
where the coefficient S(k) is the element of the S matrix insofar
e−ikr − S(k)eikr = −2ieiδ sin(kr + δ),
S(k) = e2iδ .
S is completely determined by the condition ϕ(0) = 0. Let us suppose that the solutions
of the Schrödinger equation are meaningful also for complex k. This will be rigorously
shown in the following.
Then the function S will be defined also for non physical values of k. For this S(k) some
general properties hold. For example for k → k ∗ the solution will be ϕ∗ , being U real.
∗
ϕ∗ (k, r) ∼ (eikr − S(k)eikr )∗ = eik r − S(k)∗ e−ik
89
∗r
90
CHAPTER 13. SCATTERING FROM A CENTRAL POTENTIAL
from which it follows
S ∗ (k) =
For k → −k we have
1
.
S(k ∗ )
ϕ(−k, r) = eikr − S(−k)e−ikr
from which it follows
1
S(k)
(13.2)
1
= S(−k ∗ ) .
∗
S(k )
(13.3)
S(−k) =
Thus
S ∗ (k) =
(13.1)
Notice that while (13.2) is general (13.3) is related to the fact that being U local, it is
real. Such a property is related to the invariance of the theory under time reversal (see
Chap. 12). Thus we have the following scheme
S(k)*
S(k)
−1
S(k)
( S(k)
−1 *
)
Figure 13.1: Inversion and conjugation
For pure imaginary values of k, Schrödinger equation can describe bound states. In fact
let k = iχ; to have a bound state of energy −χ2 it is necessary that the wave function ϕ,
13.1. INTRODUCTION
91
regular at the origin behaves at infinity like e−χr . I.e.
eχr − S(iχ)e−χr = ce−χr ,
c = const
which means that S(k) must have a pole for k = iχ and thus a zero for k = −iχ
as S −1 (k) = S(−k). For a pole for negative χ we obtain a solution which explodes
at infinity. These solutions take the name of anti-bound states or virtual states; their
occurrence gives rise to physically observables effects even though they do not correspond
to physical states.
To eventual poles in the complex plane there corresponds solution with a pure (complex)
exponential behavior at infinity. Such poles can exist only in the lower half plane. In fact
let ϕ be a solution corresponding to a pole of S in k = R + iI, with I > 0. We have
−ϕ′′ + Uϕ = k 2 ϕ
−ϕ∗ ′′ + Uϕ∗ = k 2 ϕ∗
from which
∂ ′ ∗
[ϕ ϕ − ϕ∗ ′ ϕ] = [k 2 − k ∗ 2 ]ϕ∗ ϕ.
(13.4)
∂r
Being I > 0 ϕ∗ ϕ goes to zero exponentially at infinity, |ϕ| ∼ e−rImk and ϕ(0) = 0.
−
Integrating (13.4) we have
2
∗2
(k − k )
Z
0
∞
ϕ∗ ϕdr = [ϕ∗ ϕ′ − ϕ∗ ′ ϕ]∞
0 = 0
so that k 2 −k ∗ 2 = 0, i.e. poles in the upper half plane can be located only on the imaginary
axis. The behavior of S for real k, k → ∞ can be investigated from
Z ∞
k sin δ0 = −
U(r)ϕϕ0 dr
0
which is easily obtained from the Schrödinger equation. In the Born approximation it
gives
k sin δ0 = −
and thus for k → ∞
Z
1
δ0 (k) = −
2k
∞
0
Z
U(r)ϕ20 dr
∞
U(r)dr.
0
For potentials with finite integral we shall have δ0 (∞) = 0 and S(∞) = 1. Similar
considerations can be carried out also for certain potentials whose integral in r does not
converge. Summing up in a figure the distributions of poles (×) and of zeros (◦) is of the
type
92
CHAPTER 13. SCATTERING FROM A CENTRAL POTENTIAL
Figure 13.2: Singularities of the S matrix
13.2
Jost functions
We can deal systematically the problems of the proceeding paragraph introducing some
particular solutions of the Schrödinger equation
−f ′′ (k, r) + U(r)f (k, r) = k 2 f (k, r)
with asymptotic behavior f (k, r) ∼ e−ikr for r → ∞ and as such do not necessarily satisfy
the boundary condition f (k, 0) = 0. Such solutions are called the Jost functions [1]. In
addition to f (k, r) also f (−k, r) is a solution which is independent of f (k, r); its wronskian
with f (k, r) is 2ik. The regular solution ϕ(k, r) with ϕ(k, 0) = 0 and ϕ′ (k, 0) = 1 is not
linearly independent of f (k, r) and f (−k, r) i.e.
ϕ(k, r) = Cf (k, r) + Df (−k, r) .
To compute C and D we consider the wronskian computed at r = 0
1
In this paragraph we follow the line of the paper by Regge [2]
(13.5)
1
13.2. JOST FUNCTIONS
93
w(ϕ, f ) = ϕ′ (k, 0)f (k, 0) − ϕ(k, 0)f ′ (k, 0) = f (k, 0)
and that derived from (13.5)
w(ϕ, f ) = C w(f (k, r), f (k, r)) + D w(f (−k, r), f (k, r)) = 2ikD
and thus
D=
f (k, 0)
,
2ik
Thus
ϕ(k, r) =
and
C=−
f (−k, 0)
.
2ik
1
[f (k, 0)f (−k, r) − f (−k, 0)f (k, r)]
2ik
and being
ϕ(k, r) ≈
1
[f (k, 0)eikr − f (−k, 0)e−ikr ]
2ik
we shall have
S(k) =
f (k, 0)
.
f (−k, 0)
From the definition of f (k, r) and from the uniqueness of the solutions it follows that
f (−k ∗ , r) = f ∗ (k, r) and thus f (−k ∗ , 0) = f ∗ (k, 0) and from these the symmetry properties (13.2, 13.3) of S(k). We study now the mathematical properties of f (k, r). It is
useful to transform the differential equation into an integral equation, which includes the
conditions on the asymptotic behavior. The following Volterra equation is particularly
apt
−ikr
f (k, r) = e
1
+
k
A possible derivation is the following
Z
∞
r
U(x) sin k(x − r)f (k, x)dx .
(13.6)
f ′′ + k 2 f = Uf
d2
sin k(x − r) + k 2 sin k(x − r) = 0
dr 2
U(x)f (k, x) sin k(x − r) = f ′′ (k, x) sin k(x − r) − f (k, x)
=
d2
sin k(x − r)
dx2
d ′
d
[f (k, x) sin k(x − r) − f (k, x) sin k(x − r)]
dx
dx
(13.7)
and integrating we have eq.(13.6).
Rewriting now (13.6) with f (k, r) = e−ikr g(k, r) we obtain
1
g(k, r) = 1 +
k
Z
r
∞
U(x)
1 − e−2ik(x−r)
g(k, x)dx .
2i
(13.8)
94
CHAPTER 13. SCATTERING FROM A CENTRAL POTENTIAL
One can solve such an equation by iteration
∞
X
g(k, r) =
gn (k, r)
0
g0 (k, r) = 1
Z ∞
1
1 − e−2ik(x−r)
g1 (k, r) =
U(x)
dx
k r
2i
Z
1 − e−2ik(x−r)
1 ∞
U(x)
gn (k, x)dx .
gn+1 (k, r) =
k r
2i
Let us analyze now the conditions for the convergence of the above series. We take k
complex and we shall distinguish two cases 1) Imk = b ≤ 0; 2) Imk = b > 0.
1) Imk = b ≤ 0 with k 6= 0.
We have
|
from which
1 − e−2ik(x−r)
1
e2b(x−r)
1
|≤|
|+|
|≤
2ik
2ik
2ik
|k|
1
gn+1 (k, r)| ≤
|k|
and thus
Z
r
∞
|U(x)||gn (k, x)|dx
Z ∞
1
1
|g1(k, r)| ≤
|U(x)|dx =
M(r)
|k| r
|k|
Z ∞
Z ∞
M(r)2
1
dM(x)
1
M(x)dx =
|U(x)|M(r)dx = − 2
|g2 (k, r)| ≤ 2
|k| r
|k| r
dx
2|k|2
and by induction
M n (r)
|gn (k, r)| ≤
n!|k|n
|g(k, r) − 1| ≤ e
M (r)
|k|
−1
Thus we have:
a) If b ≤ 0, k 6= 0 and
uniformly.
R∞
r
|U(x)|dr < ∞ the series is a solution of (13.8) as it converges
b) Each term of the series is analytic in k provided its integral converges.
c) If b 0, k 6= 0 the series converges on a closed contour. Then by Weierstrass theorem
the series is analytic in k.
d) For k → ∞ on the real axis and in the lower half-plane we have g(r, k) → 1
e) If M(0) < ∞ all the statements holds also for g(k, 0) = f (k, 0).
A sufficient condition for having M(0) < ∞ is that the potential at infinity be bounded
by r −1−ε and at the origin by r −1+ε .
13.2. JOST FUNCTIONS
95
A different way to proceed, which includes k = 0 is to use the inequality
Z
1 − e−2ik(x−r) x−r −2iηk =
e
dη ≤ (x − r) < x
2ik
0
(13.9)
The changes in the proof are
a) k is absorbed in the majoring factor
b) U(x) is multiplied by x. Thus we have
N (r)
|g − 1| ≤ e
−1
with N(r) =
Z
∞
x|U(x)|dx
r
1) Imk = b > 0 with k 6= 0
|
with
1 − e−2ik(x−r)
e2b(x−r)
|≤
because e2b(x−r) ≥ 1
2ik
|k|
Z ∞
e2b(x−r)
e−2br
g1 (k, r) ≤
|U(x)|
dx ≤
P (r)
|k|
|k|
r
P (r) =
|g2 (k, r)| ≤ |
Z
∞
2b(x−r)
U(x)
e
r
e−2br
|
|k|2
Z
|k|
∞
Z
∞
r
|U(x)|e2bx dx
e−2br
P (x)dx| ≤
|
|k|
|k|2
−2bx
e
U(x)P (x)dx| = P (r)
r
|gn (k, r)| ≤ P (r)
But we also have
Z
∞
r
U(x)P (x)dx| ≤
e−2br
|M(r)
|k|2
e−2br M n−1 (r)
|
|k|n (n − 1)!
|g(k, r) − 1| ≤ P (r)
1 − e−2ik(x−r)
|
|=|
2ik
Z
e−2br M|k|(r)
e
.
|k|
x−r
e−2ikη dη| ≤ (x − r)e2b(x−r) ≤ xe2b(x−r) .
0
In this way we reach the result
|gn (k, r)| ≤ Q(r)
with
Q(r) =
and
Z
∞
r
2bx
x|U(x)|e
dx,
N n−1 (r) −2br
e
(n − 1)!
N(r) =
Z
r
|g(k, r) − 1| ≤ Q(r)e−2br eN (r) .
∞
x|U(x)|dx
(13.10)
96
CHAPTER 13. SCATTERING FROM A CENTRAL POTENTIAL
Then the Jost function f (k, r) is an analytic function of k for fixed r for Im k < µ/2 being
µ/2 the upper limit of the values of b for which
Z
∞
x|U(x)|e2bx dx < ∞ .
r
For r 6= 0 or in the cases in which M(0) < ∞ we have that for |k| → ∞ the function
f (k, r) goes to 1 in the region Im k < µ/2 − η.
We have shown that the S(k) = f (k, 0)/f (−k, 0) is meromorphic in the strip |Im k| < µ/2
called the Bargmann strip [3]. The poles of S(k) in the interior of this strip do not arise
from singularities of f (k, 0) but from zeros of f (−k, 0). To these, for k = iχ, χ > 0 there
correspond bound states because f (−k) = 0 and from (8) we have
ψ(k, r) =
f (k, 0)eikr
f (k, 0)e−χr
f (k, 0)f (−k, r)
∼
=−
.
2ik
2ik
2χ
The Bargmann strip extends to the whole k-plane for the so called finite-range potentials; by these we understand those potentials which decreases at infinity faster than any
exponential, e.g. a gaussian potential; in these cases b is arbitrary.
The analyticity region of S(k) can be notably extended if the potential in question of
yukawian i.e. the superposition discrete or continuous of Yukawa potentials
U(r) =
Z
∞
m
e−µr
σ(µ)dµ .
r
Several condition can be imposed on σ(µ) e.g
Z
∞
m
|σ(µ)dµ < ∞ .
The domain of analyticity of f (k, 0) is extended to the whole complex plane except the
cut (i m/2, +i∞).
The structure of the S(k) matrix for yukawian potentials thus becomes The cut is not
necessarily of continuous structure. There exist potentials like e.g. the Hultèn potentials
U(r) = λ
e−µr
1 − eµr
(13.11)
for which such a cut reduces to an infinite number of poles.
One notices as the distance of the cut from the origin im/2 is directly related the the
“range” 1/m of the potential in question.
13.3. PHYSICAL INTERPRETATION OF THE SINGULARITIES
97
Figure 13.3: Singularity structure of the S matrix for Yukwian potentials
13.3
Physical interpretation of the singularities
Let us examine which is the the physical meaning of the position and of the residues of
the poles of the S matrix. We already saw how the position is related in simple way to the
binding energy of the state in question. As far as the residue is concerned it is useful to
exploit the relation S(k) = 1/S(−k) and examine the zeros of S in the lower half plane.
To compute the residue at the pole it is necessary to obtain the expression of ∂S/∂k.
With standard techniques, from the Schrödinger equation we have
Z a
∂ϕ(k, a) ∂ϕ(k, a)
∂ 2 ϕ(k, a)
− ϕ(k, a)
= 2k
ϕ2 (k, r)dr
∂a
∂k
∂k∂a
0
Recalling that ϕ(k, r) ∼ e−χr − S(−iχ)eχr we have
∂S(−iχ)
+ e−2χa − e2χa S 2 (−iχ)] ∼ −2iχ
i[4ξaS(−iχ) + 2χ
∂χ
Z
a
ϕ2 (k, r)dr,
0
Taking the limit for a → ∞ and recalling that S(−iχ0 ) = 0 we have
Z ∞
∂S
=i
ϕ2B (r)dr
∂k
0
(k = −iχ)
98
CHAPTER 13. SCATTERING FROM A CENTRAL POTENTIAL
being ϕB the real wave function of the bound state with asymptotic behavior ϕB (r) ∼
e−χ0 r . In the neighborhood of the pole then we have
Z ∞
1
−iα
−i
S(k) =
ϕ2B (r)dr ≡
≈
.
S(−k)
k − iχ0 0
k − iχ0
The fact that the residue has to be pure imaginary is seen also from S(k) = S ∗ (−k ∗ ). In
the variable k 2 we have
S(k 2 ) ∼
2αχ0
−i2iαχ0
= 2
.
(k − iχ0 )(k + iχ0 )
k + χ2
We can now relate the residue α with the asymptotic normalization function: If
Z
ϕ2 (r)dr = 1
and βϕB (r) = ϕ(r) ∼ βe−χ0 r
Z
ϕ2B (r)dr =
1
1
= 2
α
β
−iβ 2
.
k − iχ0
If the pole is near the origin with respect to the other poles we can use the approximation
S(k) =
f (−k) ≈ ic(k − iχ0 )
where c is real for unitarity. It follows that
S(k) ≈ −
k + iχ0
k − iχ0
from which β 2 ≈ 2χ0 and thus for χ0 → 0 we have β 2 → 0. It is possible to relate these
results with the cross section at low energies.
e2iδ =
χ0 − ik
χ0 + ik
from which
tan δ = −
k
.
χ0
In our case δ ∼ −k/χ0 and recalling δ0 = ak with a the scattering length we have
σs (0) = 4πa2 =
4π
.
χ20
This is the case for deuteron in the p − n scattering. In the case of an antibound state we
have the situation
S(k) ≈ −
k − iχ0
k + iχ0
13.4. LEVINSON THEOREM
99
and thus δ ≈ k/χ0 and
4π
.
χ20
One sees that there is no difference in the cross section but the sign of the scattering
σs (0) = 4πa2 =
length is opposite. Let us consider now a pole in the lower half plane of position kR − iΓ
with Γ << kR . To this pole there corresponds a resonance. The Jost function in the
proximity of such a point will have the form f (k) ≈ c(k − (kR + iΓ)) from which
S=
and we derive
f (k)
f (k)
c k − kR − iΓ
k − kR − iΓ 2iδ0
= ∗ ∗ ≈ ∗
≈
e
f (−k)
f (k )
c k − kR + iΓ
k − kR + iΓ
Γ
.
k − kR
Thus we see that we are in presence of a resonance and that passing through the point kR
δ(k) = δ0 − arctan
the phase-shift increases by π. It is worth noticing that being Γ > 0, in a neighborhood
of kR the phase shift is always increasing. To such a phase-shift there corresponds the
well known Breit-Wigner cross section which for δ0 = 0 is
So − 1 2 4π
Γ2
=
.
σ = 4π 2ik k 2 (k − kR )2 + Γ2
It is interesting to examine the structure of the scattering wave function when the energy
has a value near the real part of a resonance pole
ϕ(k, r) =
Due to the wronskian property,
1
[f (k)f (−kr) − f (−k)f (kr)] .
2ik
∂ϕ(k,r)
|r=0
∂r
= 1 while for large r, ϕ(k, r) ∼
For k = kR +iΓ, f (k) = 0 while we always have
∂ϕ(k,r)
|r=0
∂r
|f (k)|
k
sin(kr + δ).
= 1. In proximity of a resonance
with small Γ we are in presence of a wave function which is large near the origin with
an asymptotic normalization factor very small. The intuitive picture is that of a particle
which interact for a long time with the scattering center.
13.4
Levinson theorem
The following remarkable theorem holds: Given a potential U(r) the difference between
the phases of scattering at zero energy and at infinite energy equals nπ where n is the
number of bound states. Such a theorem hold for all partial waves; here we prove it for
the s-wave.
We know that S(0) = 1 because δ ≈ ak (unless we have a zero energy bound state in
which case we have S(0) = −1 which we exclude) and also that S(∞) = 1. Let us set by
convention δ(∞) = 0 and define then δ by continuity.
100
CHAPTER 13. SCATTERING FROM A CENTRAL POTENTIAL
We know that f (k) for k → ∞ and Im k ≤ 0 tends to 1.
Let us consider now the function log f (k); we have that
Z 0+ ′
f (k)
log f (0+) − log f (0−) =
dk
0− f (k)
where the integral is performed along the contour represented in figure. *** If we nor-
−0 +0
Figure 13.4: Integration contour
malize the logarithm so that log f (∞) = 1 we have that
log f (0+) − log f (0−) = 2iδ(0) .
The integrand is analytic except for the poles at the values of k corresponding to a zero of
f (k); these poles have residue 1 being simple poles [6]. As the zeros of f (k) in the lower
half plane represent the bound states of the system we have
2iδ(0) = 2πinB .
13.5
Construction of special potentials
It is instructive to examine potentials which give rise to particularly simple S matrices.
Let us try to construct and S matrix of type
S=−
k + iχ
k − iχ
(χ real )
13.5. CONSTRUCTION OF SPECIAL POTENTIALS
101
with the condition S(0) = 1. This is obtained by imposing that the wave function be free
except for the linear condition at the origin
ϕ′ (0)
=c.
ϕ(0)
(13.12)
The from
ϕ(r) = e−ikr − S(k)eikr
it follows
Thus
ϕ′ (0)
e−ikr + S(k)eikr
= −ik −ikr
|r=0 = c
ϕ(0)
e
− S(k)eikr
c + ik
c − ik
which as we saw before corresponds for c < 0 to a bound state and for c > 0 to an
S(k) =
antibound state.
A boundary condition of this type can be approximated by an attractive potential well of
short range and large depth. One has simply to see whether, as the radius tends to zero,
is it possible to change the depth in such a way as to obtain in the limit eq.(13.12).
Let us note that if the depth of the well is very large the wave number inside the well
does not depend on the particle energy. We have
ϕ′ (a)
k cos ka
=
.
ϕ(a)
sin ka
If we send a to zero keeping between a and the the relation
a=
we have
π
c
− 2
2k k
(13.13)
cos( π2 − kc )
ϕ′ (a)
= lim k
lim
=c.
a→0 sin( π − c )
a→0 ϕ(a)
2
k
A potential which matches the boundary condition (13.12) inside the well, is given for
small a, by
~2 k 2
θ(a − r)
2m
where k is a solution of eq.(13.13) with k > 0.
V (r) = −
The behavior of the wave function in relation to the potential is and the phase shifts are.
The S-matrix so constructed in the limit case of a potential problem.
We want now to examine the problem of the construction of potentials which give rise to
simple S matrices in a systematic way [Bargmann].
102
CHAPTER 13. SCATTERING FROM A CENTRAL POTENTIAL
Let us try to obtain Jost functions such that g(k, r) = f (k, r)eikr is a simple algebraic
function.
Setting ϕ = e−ikr χ(k, r) from Schrödinger equation −ϕ′′ + Uϕ = k 2 ϕ, similarly to what
done for Jost functions, we obtain the equation for χ(k, r)
χ′′ − 2ikχ′ = Uχ .
Let χ be a solution of such equation with the property that there exist the limit, for
r → ∞, of χ(k, r) and that such limit exists an be finite. We have
f (k, r) = e−ikr
and thus
f (k, 0) ≡ f (k) =
The S-matrix is then given by
S(k) =
χ(k, r)
χ(k, ∞)
χ(k, 0)
.
χ(k, ∞)
f (k)
χ(k, 0) χ(−k, ∞)
=
f (−k)
χ(k, ∞) χ(−k, 0)
We start from the assumption that ξ(k, r) be a polynomial ink
0) Zero degree polynomial
χ(k, r) = a(r)
This gives rise to a Jost function which does not depend of k. From
a′′ − 2ika′ = Ua
it follows that, given the independence of k, a′ = 0, a′′ = 0. Then Ua = 0. If U 6= 0 we
cannot construct the Jost function because we have a = 0. If U = 0 we have χ(k, r) =
const and f (k, r) = e−ikr which is the Jost function of the free particle: S(k) = 1.
1) First degree polynomial
χ(k, r) = b(r)k + a(r) .
From
χ′′ − 2ikχ′ = Uχ
it follows
kb′′ + a′′ − 2ik(b′ k + a′ ) = U(kb + a) .
In order this to be true as k varies, it is necessary that b′ = 0 i.e. b = const. We can then
write
χ(k, r) = 2k + ia(r)
13.5. CONSTRUCTION OF SPECIAL POTENTIALS
103
ia′′ + ka′ = U(2k + ia)
and we have
U = a′ ,
a′′ = a′ a .
(13.14)
The correspond Jost functions
f (k, r) = e−ikr
2k + ia(r)
2k + ia(∞)
f ∗ (k, r) = f (−k ∗ , r)
imposes
a∗ (r) = a(r),
a∗ (∞) = a(∞)
i.e. a(r) as to be real. We must then look for a real solution of eq.(13.14). The first
integration is immediate
1
a′ − a2 = 2c .
2
To obtain a linear equation it is convenient to pose
Z
1 r
w = exp(−
a(r ′ )dr ′),
with
2 0
and we have
2w ′
,
w
In terms of w our equation becomes
a=−
a′ = −2
w>0
w ′′ w − w ′2
.
w2
w ′ + cw = 0 .
In order for w to be real also c has to be real. We recall furthermore that we are interested
in solution with w(r) > 0 for r > 0.
i) c = 0
w ′′ = 0,
from which
W = α + r,
1
α + r = exp(−
2
Z
α>0
r
a(r ′ )dr ′)
0
2
α+r
2
U(r) = a′ (r) =
.
(α + r)2
a(r) = −
The Jost function is given by
f (k) =
k − i/α
k
104
CHAPTER 13. SCATTERING FROM A CENTRAL POTENTIAL
from which
k − i/α
.
k + i/α
S(k) =
Such a S matrix has a single pole for k = −i/α, which being α > 0 is an antibound state.
The fact that f (k) has a pole for k = is not surprising as we have shown that necessary
and sufficient condition for k = 0 to be an analyticity point of the Jost function f (k) is
Z
N(0) =
∞
0
x|U(x)|dx < ∞
condition which is clearly violated by the potential in question.
ii) c > 0
We obtain an oscillating solution and thus w cannot be positive definite.
iii) c < 0
Let us write
c = −λ2 /4
and we have
w ′′ − λ2
w
= 0,
4
λr
w = e 2 + βe
−λr
2
.
β has to be ≥ −1 otherwise there a point in which w is negative.
a=−
βe−λr − 1
2w ′
= λ −λr
w
βe
+1
from which
f (k) =
k + i λ(β−1)
β+1
k − i λ2
≡
k + i ν2
k − i λ2
where we set
ν=λ
β−1
.
β+1
Notice that ν has the sign of β − 1 and ν < λ for β ≥ 1.
U(r) = a′ = −2λ2 β
e−λr
(1 + βe−λr )2
(Eckart potential)
U is finite at the origin and at infinity behaves as e−λr .
(13.15)
13.6.
DISCUSSION OF THE SINGULARITIES OF THE S-MATRIX
13.6
105
Discussion of the singularities of the S-matrix
The S-matrix in the case of Eckart potential is given by
S(k) =
k + iν/2 k + iλ/2
f (k)
=
.
f (−k)
k − iν/2 k − iλ/2
We notice that the Jost function (26) has a pole for k = iλ/2. Thus we expect that the
Bargmann strip be of half-width λ/2 which is consistent with the asymptotic behavior
e−λr of the potential. Thus this pole of the S-matrix at k = iλ/2 is a typical interaction
pole which lies at the boundary of the Bargmann strip. The Jost function ha a zero at
the point
ν
λβ −1
= −i
2
2β+1
to which there corresponds a pole of S at the point
k = −i
k = iχ = i
λβ −1
.
2β+1
For β > 1 we have χ > 0 and thus we have a bound state. If on the contrary −1 < β < 1,
χ is negative and we have an antibound state. The residues at the two poles k = iλ/2
and k = iν/2 are respectively
iλ
and
λ+ν
= iλβ
λ−ν
β−1
λ+ν
= −iλ β
λ−ν
β+1
The phase shift is immediately obtained from
−iν
k cot δ = Re
2ik
1 1
νλ
2
= + rk 2 = −
+
k2 .
S(k) − 1
a 2
2(ν + λ) ν + λ
Thus the phase-shift due to the Eckart potential obeys the effective range formula. We
can summarize the situation in the following table where the third anf fourth column give
the sign of −i times the residues of the respective poles
Thus for −1 < β < 0 we have a repulsive potential which admits and antibound state.
For 0 < β < 1 we have a weakly attractive potential with an antibound state within the
Bargmann strip. For β > 1 we have strong attraction with the consequent formation of
the bound state. The scattering length a (δ ≈ ak) is always negative except in the case
0 < β < 1. Levinson theorem is always satisfied as one sees from (13.15).
Of the four possibilities a > 0, a < 0, r > 0, r < 0 in the effective range formula all can
be realized by the potential in question, except the a > 0, r < 0. To this there correspond
a resonant phase-shift with δ(∞) − δ(0) = π which would violate Levinson theorem.
106
CHAPTER 13. SCATTERING FROM A CENTRAL POTENTIAL
β
ν
1<β
0<β<1
−1 < β < 0
iλ/2
iν/2
0<ν<λ
+
–
–
+
bound
−λ < ν < 0
+
+
+
+
antibound
–
–
–
–
antibound
ν < −λ
a
r
state
Table 13.1: ǫc is the rigorous lower limit on the convergence radius. The physical region
for the coupling is 0 < ǫ ≤ 1/4. B is the parameter appearing in eq.(??)
References
[1] R. Jost, Über die falschen Nullstellen der Eigenverte der S-matrix Helvetica Physica
Acta 20, 256 (1947)
[2] T. Regge, Mathematical Theory of potential scattering Theoretical Physics IAEA 1963
[3] V. Bargmann, On the connection between phase shifts and scattering potentials Rev.
Mod. Phys. 21 488 (1949)
[4] Witthaker and G.N. Watson, A Course of Modern Analysis Chapt. X
[5] E.C. Titchmarsh, Theory of functions Chapt. V
[6] R.G. Newton, Analytic properties of radial wave functions J. Math. Phys. 1, 319
(1960)
Chapter 14
Functional formulation of quantum
mechanics
14.1
Introduction
Quantum mechanics in its final form was discovered in two different mathematical schemes
which were soon proved to be equivalent. We can call it the operator formulation in so
far as the dynamical variables are represented by linear operators which act on the linear
space of states.
The quantization rule is provided by the correspondence principle i.e. replacing Poisson brackets of classical mechanics with commutators divided by i~. We saw that this
statement has to be taken with great care.
The Hamiltonian formulation of classical mechanics is not explicitly relativistically invariant; in fact the H (energy) is not a relativistic invariant but the zero component of a
four-vector.
On the contrary the classical lagrangian approach is explicitly covariant insofar through
the lagrangian on succeeds to define an action which is is an invariant:
Z qt
L(q, q̇)dt
S=
q0 t0
has the same value in all reference frames. The vanishing of the variation of S, δS = 0,
provides the equations of motion which being S an invariant are themselves covariant.
E.g. for a charged particle in an external electromagnetic field we have
Z qt
Z qt
ds e dxµ
ds e dxµ
(−mc
)dt =
+ Aµ
)dλ
(−mc + Aµ
S=
dt c
dt
dλ c
dλ
q0 t0
q0 t0
√
with ds = c2 − v 2 dt and Aµ dxµ = Al dxl − V cdt; Aµ = (A, −V ).
107
108 CHAPTER 14. FUNCTIONAL FORMULATION OF QUANTUM MECHANICS
The search for a lagrangian formulation of quantum mechanics (and thus the possibility
to give a relativistically invariant formulation of the same) was the main motivation of
Feynman [1]. It was inspired by some remarks by Dirac [2].
The main characteristics are
1. It is more intuitive (as far as quantum mechanics may be intuitive).
2. Requires mathematics of higher level (concept of integration on a space of functions)
3. Few problems can be solved directly in this approach.
4. It has a great heuristic and formal power:
4.1. Possibility to give a relativistically invariant formulation (lagrangian, not hamiltonian)
4.2 A strong guide on how to organize the perturbative series.
4.3 Strong analogies with statistical mechanics.
4.4 As a method to develop new approximation schemes.
I won’t follow the line historically developed by Feynman, because it would take too much
time; instead I will follow the inverse path i.e. how from the usual operator hamiltonian
formulation one can reach Feynman’s one.
I will refer to the one dimensional case (it is not difficult to go to higher dimensions) and
the object we take into consideration is the propagator which we already studied
i
G(q, t; q0 , t0 ) = hq|e− ~ H(t−t0 ) |q0 i
using Dirac’s notations, where H is supposed time independent.
It is known that the Green function gives us the probability amplitude for a particle which
at time t0 is in q0 to be found at time t in q. The probability is given by
|G(q, t; q0 , t0 )|2 .
Such a Green function allows more generally to compute transition amplitudes between
states
− ~i H(t−t0 )
hφ|e
|ψi =
=
Z
Z Z
i
hφ|qidqhq|e− ~ H(t−t0 ) |q0 idq0 hq0 |ψi =
φ∗ (q)dqG(q, t; q0 , t0 )dq0 ψ(q0 )
ant thus solves the dynamical problem of quantum mechanics.
Notice moreover that inserting a complete set of eigenstates of the energy
Z
i
G(q, t; q0 , t0 ) = dEhq|EihE|q0ie− ~ E(t−t0 )
14.1. INTRODUCTION
109
and from Fourier analysis of G(q, t; q0 , t0 ) one succeeds in recovering the energy levels and
the relative wave functions.
Notice that the same informations can be obtained from
Z
τ
− τ~ H
|q0 i = dEe− ~ E hq|EihE|q0i
G̃(q, τ ; q0 , 0) = hq|e
and thus in term of an anti-Laplace (Mellin) transform.
Given a time t1 with t > t1 > t0 we can also rewrite
Z
− ~i H(t−t1 ) − ~i H(t1 −t0 )
G(q, t; q0 , t0 ) = hq|e
e
|q0 i = G(q, t; q1 , t1 )dq1 G(q1 , t1 ; q0 , t0 ) . (14.1)
Such an expression is reminiscent of the formula for probabilistic processes i.e.
Z
P (q, t; q0 , t0 ) = P (q, t; q1 , t1 )dq1 P (q1 , t1 ; q0 , t0 ).
(14.2)
On the other hand in quantum mechanics
P (q, t; q0, t0 ) = |G(q, t; q0 , t0 )|2
and thus if eq.(14.1) is true the (14.2) cannot be true. Actually (14.2) is true if we
replace in the l.h.s. P (q, t; q0, t0 ) with Pq̂t1 (q, t; q0 , t0 ), in different words if at instant t1 we
perform a measure of the coordinate, o better if an instrument is present apt to measure
the coordinate.
We can now insert many times (e.g. equally spaced) between t0 and t and we have
Z
Z
G(q, t; q0 , t0 ) = . . . G(q, t; qn−1 , tn−1 )dqn−1 G(qn−1 , tn−1 ; qn−2 , tn−2 )dqn−2 . . .
. . . G(q2 , t2 ; q1 , t1 )dq1 G(q1 , t1 ; q0 , t0 )
with tk = t0 + k∆ and ∆ = (t − t0 )/n ≡
T
.
n
What did we gain with this decomposition?
If the Hamiltonian is of form H = V + K with V = V (q̂) and K = p̂2 /2m we have
i
G(qk , tk ; qk−1 , tk−1) = hqk |e− ~ (V +K)∆ |qk−1 i =
i
i
= hqk |e− ~ V ∆ e− ~ K∆ |qk−1 i + o(∆)
where qualitatively o(∆)/∆ → 0 for ∆ → 0.
Actually there exists a theorem (see appendix) known as Trotter formula which says that
i
i
T
i
T
e− ~ HT = s − lim (e− ~ V n e− ~ K n )n
n→∞
110 CHAPTER 14. FUNCTIONAL FORMULATION OF QUANTUM MECHANICS
where s − lim stay for strong limit. These results can be qualitatively understood from
the Baker-Campbell-Haussdorf formula
1
eA eB = eA+B+ 2 [A,B]+O3 .
Being the error due to the separation of the two exponential of second order, in the limit
in which n goes to infinity such error induced in the product goes to zero.
E.g. for bounded operators
keA+B − eA k ≤ ekAk+kBk − ekAk .
Moreover for the accumulated error in the multiplication of n terms we have |(1+ nε )n −1| ≤
e|ε| − 1 which goes to zero for ε → 0. The main difficulty in proving Trotter formula is in
dealing with unbounded operators whose appearance is the rule in the Hamiltonians of
quantum mechanics. We have
G(q, t; q0 , t0 ) =
= lim
n→∞
Z
...
Z
i
i
i
i
hq|e− ~ V ∆ e− ~ K∆ |qn−1 idqn−1 hqn−1 |e− ~ V ∆ e− ~ K∆ |qn−2 idqn−2 . . .
i
i
. . . dq1 hq1 |e− ~ V ∆ e− ~ K∆ |q0 i.
Let us consider the generic term
− ~i V (q̂)∆ − ~i
′′
hq |e
=
from which
Z
e
i
e− ~ V (q
′′ )∆
p̂2
∆
2m
′
|q i =
Z
i
e− ~ V (q
′′ )∆
i p̂2
hq ′′ |e− ~ 2m ∆ |p′ idp′ hp′ |q ′i =
′2
i p
e− ~ 2m ∆ hq ′′ |p′ idp′hp′ |q ′ i = (
im(q ′′ −q ′ )2
i
1
m
′′
) 2 e− ~ V (q ) e 2~∆
2πi~∆
G(q, t; q0 , t0 ) =
= lim (
n→∞
n
m
)2
2πi~∆
Z
i
e− ~ V (q)∆ e
im(q−qn−1 )2
2~∆
i
i
dqn−1 e− ~ V (qn−1 )∆ e
dq1 e− ~ V (q1 )∆ e
im(q1 −q0 )2
2~∆
im(qn−1 −qn−2 )2
2~∆
dqn−2 . . .
.
Given now the partition of the time interval t − t0 , with the times tn = t, tn−1 , . . . , t1 , t0 ,
taken arbitrary values of qn−1 , . . . q1 let us consider the motion given by the piecewise
linear curve drawn in figure. We want to compute the classical action relative to such
motion. We have
Z
qt
q0 t0
L(q, q̇)dt =
X
k
" Z
−
#
qk − qk−1
m qk − qk−1 2
V (qk−1 +
(t − tk−1 ))dt + (
)∆ ≈
∆
2
∆
tk−1
tk
14.1. INTRODUCTION
111
≈
n X
k=1
m (qk − qk−1 )2
−V (qk )∆ +
2
∆
were we performed an error of order of ∆2 replacing
Z tk
V (q(t))dt with V (qk )∆.
tk−1
Such error is irrelevant in the limit of n → ∞. The limit for n → ∞ which we know
to exist and we know rigorously equal to the Green function, is the definition of the
functional integral on the paths which connect the event (q0 , t0 ) to the event (q, t) and is
written
G(q, t; q0 , t0 ) =
Z
i
D[q(t)]e ~ S[q(t)] .
Notice that L is not an operator but the classical Lagrangian of the system. Thus we do
not have operators but functional integrals (∞−dimensional integrals).
All paths are possible and all equally probable, i.e. all contribute with the same weight
which is simply a phase. The path which are near the classical path, being on the classical
path the action stationary give the most important contribution while those far away from
the classical path tend to give contribution which cancel by interference.
In completely analogous way one can compute
− τ~ H
G̃(q, τ ; q0 , 0) = hq|e
m n
|q0 i = lim (
)2
n→∞ 2π~∆
Z
e−
V (q)
∆
~
e−
m(q−qn−1 )2
2~∆
dqn−1 . . .
(14.3)
We see that
−
e
V (qk )
∆
~
−
e
m(qk−qk−1 )2
2~∆
≈e
1
~
Rτ
k
τk−1
[− m2 (q̇(τ ))2 −V (q(τ ))]dτ
=e
1
~
Rτ
k
τk−1
LE (q,q̇)dτ
where the “euclidean Lagrangian” LE (q, q̇) is defined by
LE (q,
∂q
∂q
) = L(q,
)
∂τ
∂(−iτ )
i.e. it formally obtained by replacing t with −iτ (Wick rotation). The euclidean functional
integral turns out to be a powerful calculational tool of the partition function of statistical
mechanics of the system described by that Hamiltonian H as
Z
Z
Z
R
1 q,~β
−βH
Z(β) = Tr (e
) = dq G̃(q, ~β; q, 0) = dq D[q(τ )]e ~ q,0 LE (q,q̇)dτ .
There exists also an interpretation of eq.(14.3) at the level of classical statistical mechanics.
Eq.(14.3) before taking the limit for n → ∞ can be seen as the partition function of a one
dimensional chain of classical degrees of freedom (like a spin chain) subject to the external
potential V (q) and first neighbors interaction with interaction energy c(qk − qk−1 )2 being
c a positive constant.
112 CHAPTER 14. FUNCTIONAL FORMULATION OF QUANTUM MECHANICS
14.2
Trotter formula
We want to prove here without any pretense of rigor (for a rigorous proof see the Appendix
to this Chapter) Trotter formula, i.e.
i
i
i
e− ~ (K+V )∆ ψ(q) = e− ~ V ∆ e− ~ K∆ ψ(q) + O(∆2 ) .
We shall use the following results
Z
Z
We have
e
iαq 2
2
dq =
r
2πi
α
iαq 2
2
q 2n+1 dq = 0
r
Z
iαq 2
i 2πi
2
e 2 q dq =
α
α
r
Z
iαq 2
3 2πi
e 2 q 4 dq = − 2
α
α
r
Z
iαq 2
1
1
.
e 2 q 2n dq = c(n) n
α
α
e
i
i
i
e− ~ V ∆ e− ~ K∆ ψ(q) = (1 − V (q)∆ + O(∆2 ))×
~
Z
′ )2
im(q−q
(q ′ − q)2 ′′
m 1/2
′
′
′
2~∆
e
)
ψ (q) + . . . =
dq ψ(q) + (q − q)ψ (q) +
×(
2πi~∆
2
i
i~∆ ′′
2
2
(1 − V (q)∆ + O(∆ )) ψ(q) +
ψ (q) + O(∆ ) =
~
2m
i
i
~2 ′′
i
ψ (q) + O(∆2 )) = e− ~ H∆ ψ(q) + O(∆2 ).
ψ(q) − V (q)ψ(q)∆ − (−
~
~ 2m
In this way we have reproduced the Schrödinger evolution equation.
We add now a few notes
1. As mentioned the same informations on the bound states and on the wave functions of
τ
the bound states can be obtained by studying the “euclidean” propagator e− ~ H , i.e. by
computing the inverse Laplace transform (Mellin).
From the computational viewpoint the euclidean propagator is much simpler especially
R
at the numerical level. This because the exp( 1~ LE (q, q̇)dτ ) has now the nature of a
probability and thus one can apply to it all the results of statistical mechanics. Formally
the transition from L to LE is obtained setting x4 = ix0 = it.
2. Configurations which are not minimal but are stationary, can give particularly relevant
contributions to the functional integral and thus can be taken as starting point of new
methods of approximation.
14.3. APPENDIX: PROOF OF TROTTER FORMULA
14.3
113
Appendix: Proof of Trotter formula
Let A and B two self-adjoint operators on an separable Hilbert space H and let A + B
p̂2
2m
be defined on D(A) ∩ D(B) also self-adjoint (e.g. A =
p̂2
2m
or A =
origin).
2
2
∇
= − 2m
e B = V (q) bounded;
∇
= − 2m
and B = V (q) bounded at infinity with a Coulomb singularity at the
Theorem [6]:
t
t
eit(A+B) = s − lim (ei n A ei n B )n
n→∞
If A and B are lower bounded it holds also
t
t
e−t(A+B) = s − lim (e− n A e− n B )n
n→∞
where s − lim stays for strong limit.
Proof: Let us define St = eit(A+B) , Vt = eitA , Wt = eitB , Ut = Vt Wt . Also let us define the
time evolved of ψ by ψt = St ψ.
We must prove that
lim ||(S nt − U nt )ψ|| = 0
n→∞
n
(14.4)
n
but as all intervening operators are of norm 1, it is sufficient to prove eq.(14.4) for all φ
of a dense set and we shall choose φ ∈ D(A) ∩ D(B).
We have
||(S nt − U nt )ψ|| = ||
n
n
e.g.
n−1
X
(n−j−1)
Ut
j=0
(S t − U t )S jt ψ||
n
n
n
(14.5)
n
SSSS − UUUU = (S − U)SSS + U(S − U)SS + UU(S − U)S + UUU(S − U) .
From Stone theorem we have for a vector φ ∈ D(A) ∩ D(B)
Ss − 1
φ = i(A + B)φ
s→0
s
lim
Us − 1
Ws − 1
Vs − 1 φ = lim iVs Bφ + Vs (
− iB)φ +
φ = i(A + B)φ .
s→0
s→0
s
s
s
Thus on a fixed vector φ ∈ D(A) ∩ D(B) we have with ∆n ≡ n(S nt − U nt )
lim
lim ∆n φ = 0.
n→∞
We can rewrite eq.(14.5) as
n
||(St − U t )φ|| = ||
n
n−1
X
j=0
(n−j−1)
Ut
n
(S nt − U nt )φsj || ≤
(14.6)
114 CHAPTER 14. FUNCTIONAL FORMULATION OF QUANTUM MECHANICS
≤ n sup ||(S nt − U nt )φsj || = sup ||∆n φsj || ≤ sup ||∆n φs ||
j
j
(14.7)
0≤s≤t
where sj = j nt .
If the vector φs were fixed, i.e. independent of s, the theorem would be already proven.
The problem is that φs varies and an excessive non uniformity of the limit eq.(14.7) could
jeopardize the result i.e. it would be not assured that for n → ∞ the last member of
eq.(14.7) tends to zero.
One notices however that
1. D ≡ D(A) ∩ D(B), with the metric
|||φ||| = ||φ|| + ||(A + B)φ||
which is stricter than the one of the Hilbert space, has the structure of a Banach space.
Note that D with the Hilbert metric || || is not a Banach space as it is not complete (in
general). Instead D with the metric ||| ||| is complete, being A + B a closed operator as
a result of being A + B self-adjoint.
2. Each ∆n , considered as a linear operator from the Banach space D to H is bounded
and we have for every φ ∈ D, fixed
sup ||∆n φ|| < ∞
n
and thus for the principle of uniform boundedness (Banach-Steinhaus theorem) we have
||∆n φ|| < C|||φ|||
which implies that the limit (14.6) is uniform over compact sets of D.
Given φ ∈ D, for s varying in [0, t] φs varies continuously in D with the metric |||
because
(A + B)(ei(A+B)s − 1)φ = (ei(A+B)s − 1)(A + B)φ → 0
for
|||
s→0
(again a consequence of Stone theorem) and thus φs describes a compact in D and as a
result the last member of eq.(14.7) for n → ∞ goes to zero..
References
[1] R.P. Feynman Space-Time Approach to Non-Relativistic Quantum mechanics, Rev.
Mod. Phys.20 1948 367
[2] P.A.M. Dirac The principles of quantum mechanics, Oxford University Press, par.32
[3] W. Pauli Lectures on theoretical physics, Boringhieri
14.3. APPENDIX: PROOF OF TROTTER FORMULA
115
[4] T.D.Lee Particle physics and introduction to field theory Harwood Academic Publishers, New York
[5] S. Weinberg The quantum theory of fields Vol.I Cambridge University Press
[6] B. Simon Functional integration and quantum physics Academic Press, New York, 1979
140506
116 CHAPTER 14. FUNCTIONAL FORMULATION OF QUANTUM MECHANICS
Chapter 15
Bell inequalities
15.1
Introduction
Let us consider a pair on non identical particles of spin 1/2 which are produced in the
singlet state and thus with wave function
1
|si = √ (|+i1|−i2 − |−i2 |+i1 )ψ(q1 , q2 )
2
where
ψ(q1 , q2 , t) = ψ1 (q1 , t)ψ2 (q2 , t)
represents two wave packets which move apart in opposite directions.
(1)
After the wave packets have separated to large distance we perform the measure of sz .
We know that the possible results are +1/2 and −1/2 which in the following we shall
denote as + and −. The probability to obtain such results are hs|P1+ |si and hs|P1− |si
where P1+ = |+i1h+|1 and P1− = |−i1 h−|1 .
According to the theory of measure if we obtain as a result + the state of the system after
the measure is represented by the vector P1+ |si =
√1 |+i1 |−i2 ψ(q1 , q2 , t)
2
which in order to
perform further calculations is to be normalized to 1.
(2)
If we perform now a measure of sz we obtain with certainty, i.e. with probability 1 the
(1)
value −. As when we performed the measure of sz
on particle 1 such a particle i.e.
counter (1), was very far from particle 2, it is reasonable to maintain that such measuring
process does not influence the system “particle 2”.
We can thus assert according to EPR (Einstein, Podolsky, Rosen) that there exists an
(2)
element of physical reality which corresponds to the physical quantity sz . In different
words as during the process of measure of 1 we have not disturbed particle 2, one can
(2)
maintain that particle 2 had the value − for sz even before the measure of the spin of
particle 1.
117
118
CHAPTER 15. BELL INEQUALITIES
In fact the criterion of EPR is the following: If, without in any way disturbing a system,
we can predict with certainty (i.e. with probability equal to unity) the value of a physical quantity , then there is an element of physical reality corresponding to this physical
quantity. According to EPR this is a sufficient criterion of reality.
Form such a criterion we conclude that the description of the state by means of the vector
(2)
|si is not complete because the value − for sz
of particle 2, even if it has a physical
reality, cannot be fore-casted from the structure of the state |si.
It appears then that the state vector gives a incomplete description of the system, a
description of statistical character as it happens e.g. in classical statistical mechanics
where the density function ρ(q, p) in phase space gives an incomplete description of a
classical physical system. EPR in fact adopt the following criterion of completeness of
a physical theory: every element of the physical reality must have a counterpart in the
physical theory.
Such considerations may put in doubts the completeness of quantum mechanics as theory;
they do not put in doubts the correctness of the predictions, in general of statistical
character, of quantum mechanics.
However J.S. Bell in 1964 has shown that the predictions of quantum mechanics violate
in quantitative manner, certain relations imposed by the idea of reality given by EPR
(which is a refinement of the usual idea of reality as something which exist independently
of the observer) coupled with the requirement of locality.
Thus while EPR leaves open the possibility of a completion (actually they consider it as
possible and welcome) of quantum mechanics in the sense of a realistic local theory, Bell’s
analysis denies such a possibility.
Experiments confirm the predictions of quantum mechanics.
15.2
Clauser-Holt-Horne-Shimony inequality
The analysis according to the most general local realistic scheme of the above described
experiment is the following: The preparing apparatus produces a pair of systems with
characteristics described by a parameter λ which will change for every emission with a
probability distribution f (λ).
It will be useful to consider the following product of difference of probabilities
(P1 (yes, a, λ) − P1 (no, a, λ)) × (P2 (yes, b, λ) − P2 (no, b, λ))
where P1 (yes, a, λ) is the probability to find the answer “yes” when the system 1 emitted by
the producing apparatus enters the counter 1 set on the characteristic a (e.g. orientation)
15.2. CLAUSER-HOLT-HORNE-SHIMONY INEQUALITY
119
and so on.
It is useful to write
A1 (a, λ) = P1 (yes, a, λ) − P1 (no, a, λ)
A2 (b, λ) = P2 (yes, b, λ) − P2 (no, b, λ)
and as probabilities are positive numbers between 0 and 1 we have
|Ai (b, λ)| ≤ 1 .
Consider now the mean value of the product of the above quantities i.e.
Z
E(a, b) = dλf (λ)A1 (a, λ)A2 (b, λ) =
=
Z
dλf (λ)(P1(yes, a, λ) − P1 (no, a, λ))(P2 (yes, b, λ) − P2 (no, b, λ)) =
= P (yes, a, yes, b) + P (no, a, no, b) − P (yes, a, no, b) − P (no, a, yes, b) .
We have
′
E(a, b) + E(a, b ) =
and thus
Z
′
dλf (λ)A1(a, λ)(A2 (b, λ) + A2 (b′ , λ))
|E(a, b) + E(a, b )| ≤
Z
dλf (λ)|A2(b, λ) + A2 (b′ , λ)|
and changing the setting of the counters and taking the difference instead of the sum
Z
′
′ ′
|E(a , b) − E(a , b )| ≤ dλf (λ)|A2(b, λ) − A2 (b′ , λ)|
and adding
because
Z
|E(a, b) + E(a, b′ )| + |E(a′ , b) − E(a′ , b′ )| ≤
dλf (λ)|A2(b, λ) + A2 (b′ , λ)| + |A2 (b, λ) − A2 (b′ , λ)| ≤ 2
|A2 (b, λ) + A2 (b′ , λ)| + |A2 (b, λ) − A2 (b′ , λ)| ≤ 2.
This holds for any pair of produced systems 1 and 2 and for whatever type of counters 1
and 2. Notice that such inequality holds even if the fundamental theory were of statistical
nature but always local. In fact the result of the measure of system 1 by counter 1 will
still be described by a probability function P1 (x, a, λ). Thus system 1 carries with himself
through the value of λ, the imprinting of how even stochastically, it has been produced
but the result of the measure by the counter 1 will not necessarily be well determined.
120
15.3
CHAPTER 15. BELL INEQUALITIES
Violation of Bell (CHHS) inequality in quantum
mechanics
Let us consider the state
1
√ (|+i1 |−i2 − |−i1 |+i2).
2
Let us specify the experiment as follows: The produced system is the one described at the
beginning of section (15.1), the two counters are Stern-Gerlach polarimeters. x is the axis
along which the two counters and the preparation device are located. The parameters a
and b of the counters 1 and 2 are the rotation angles of the two polarimeters along the
x axis, with respect to a reference direction z. “Yes” means that + has been measured,
“no” means that − has been measured.
As |si is invariant under rotations as far as the spin state vector is concerned, P (· a; · b)
can depend only on the difference of the two angles b and a. We have
P (yes a; yes b) = |hs|[cos(
b−a
b−a
1
b−a
)−iσx(2) sin(
)]|+i1|+i2 |2 = sin2 (
) = P (no a; no b)
2
2
2
2
and similarly we find
P (yes a; no b) =
1
b−a
b−a
1
cos2 (
) = P (no a; yes b) = cos2 (
).
2
2
2
2
Let us come now to the E.
E(a, b) = 2 sin2 (
b−a
)−1 .
2
We shall choose a = 0, a′ = π/2, b = π/4, b′ = −π/4 i.e. the individual counter can
assume two positions one orthogonal to the other; the position of the counter 2 are rotated
of π/4 with respect to those of counter 1.
We see that the combination
√
|E(a, b) + E(a, b′ )| + |E(a′ , b) − E(a′ , b′ )| = 2(1 + sin2 (3π/8) − 3 sin2 (π/8)) = 2 2 > 2 (!)
violates the CHHS inequality. This shows that quantum mechanics predicts correlations
which are higher that those we can expect from any realistic local theory. If one relaxes
the principle of locality it is easy to show with examples that the CHHS is no longer valid
and actually one can supply examples in which the violation of such inequalities is even
stronger.
The results of quantum mechanics, confirmed by experiments, teach us that, as maintained
by Bohr, the two particles 1 and 2 even when the wave packets ψ1 (q1 , t) and ψ2 (q2 , t) are
15.3. VIOLATION OF BELL (CHHS) INEQUALITY IN QUANTUM MECHANICS121
far apart, cannot be considered as two independent systems but have to be considered as
a unique system; this is true until we perform an operation of measure.
These experiments can be performed even “outside the light-cone” i.e. in such a way to
exclude, according the principle of relativity, a transfer of information from counter 1 and
counter 2 which would invalidate the proof of CHHS inequality.
Bell has shown, in the quantum field theory framework, that such correlations (higher
that those which we can admit in any local realistic theory) do not allow to transmit information with a speed higher that that of light (impossibility to construct a superluminal
telegraph). At the basis of such impossibility lies the statistical nature of the relationship
between the state vector and the results of an experiment.
Even more stringent inequalities, which are violated by the predictions of quantum mechanics, can be deduced by studying triple correlations [4].
References
[1]A. Einstein, B. Podolsky, N. Rosen Can Quantum-Mechanical Description of Physical
Reality Be Considered Complete? Phys. Rev. 47 (1935) 777.
[2] J.S. Bell Speakable and Unspeakable in quantum mechanics Cambridge University
Press. (1987).
[3] B. d’Espagnat The quantum theory and reality, Scientific American p.128 November
1979.
[4] N.D. Mermin Quantum misteries revisited, Am.J.Phys. 58 731 (1990).
140507