Download Chapter 1. Fundamental Theory

Chapter 1. Fundamental Theory
1.1 A Very Brief Introduction
Let’s begin with classical physics:
1. A set of dynamical quantities with well-defined (definite) values at each instant in space.
2. Mathematically, they satisfy a system of deferential equations.
3. Universe = matter + radiation.
4. Matter consists of particles satisfying Newton’s laws.
5. Radiation is described by the dynamics of field (wave), e.g., electromagnetic wave
satisfying Maxwell’s equations.
6. A particle is discrete, whose dynamical variables are position and momentum.
7. A field is continuous, described by infinite number of dynamic variables.
8. Classical physics agrees with our normal experience and intuition.
Although classical physics was extremely successful in 19th century, new phenomena
discovered in later 19th and early 20th centuries cannot be explained within the existing
theoretical framework. For example,
1. Black Body Radiation
2. Photoelectric Effect
Both phenomena require field quantization: E = ω with the reduced Planck’s constant:
 = 1.054 ×10−34 J ⋅ s . Thus, the electromagnetic wave has particle properties. On the other
hand, matter particles also have wave properties, the so-called De Broige wave.
1
Field quantization → duality → uncertainty principle → quantum mechanics
In order to describe the wave-particle duality, wave function ψ(r,t) is the basic quantity in
quantum mechanics. Whether ψ(r,t) is a pure math tool or it has a one-to-one correspondence to
physical reality is still in debate. The physical meaning of the single-particle wave function:
2
P(r,t) = ψ(r,t) = ψ*(r,t)ψ(r,t) ,
(1.1)
i.e., probability density P(r,t) is the square of the modulus of wave function, which is
normalized as
1 = ∫ P(r,t)d r = ∫ ψ*(r,t)ψ(r,t)d r .
(1.2)
Classically: ψ(r,t) = δ(r − vt) , i.e., the exact location and velocity at any given time t is known.
Quantum Mechanically: Δpx ⋅ Δx ≥  (Heisenberg’s Uncertainty Principle).
Connection: (1) when  → 0 , quantum mechanics (QM) reduces to classical mechanics.
(2) Correspondence principle: QM must approach classical theory asymptotically at the limit of
large quantum numbers.
One can easily extend one-particle wave function ψ(r,t) to many-particle wave function
Ψ(r1, r2,..., rN ;t) :
2
P(r1, r2,..., rN ;t) = Ψ(r1, r2,..., rN ;t) ,
(1.3)
1 = ∫ P(r1, r2,..., rN ;t)d r1dr2 ...drN .
(1.4)
Once wave function is known, everything about this system is known. Determination of
Ψ(r1, r2,..., rN ,t) by solving the many-particle Schrödinger Equation:
i
∂Ψ(r1, r2,..., rN ;t)
∂t
= H Ψ(r1, r2,..., rN ;t) ,
(1.5)
where H is the Hamiltonian Operator
⎡  2∇ 2
⎤
N
⎢
⎥
H = ∑ ⎢−
+U(rj ,s j )⎥ + ∑V(ri − rj ) ,
2m j
j =1 ⎢
⎥⎦ i>j
⎣
N
2
(1.6)
with the single-particle potential U(rj ,s j ) depending on position and spin (s), while V is the
many-particle potential: V = 0 → single-particle system; V ≠ 0 → many-particle system.
This class is essentially dedicated to solving the Schrödinger Equation.
(1) Single-particle quantum system (QM I, II)
A. Free particles (U = 0 ):
Planewave; Momentum; Angular momentum, orbital and spin.
B. Exactly solvable potentials:
Square potentials; Simple harmonic oscillator; H atom.
C. Approximate methods:
WKB approximation; Variational method
Perturbation theory: Time-independent and time-dependent
D. Applications:
Interaction between charged particles and electromagnetic radiation
Quantum Scattering; Addition of angular momentum
(2) Many-particle quantum system (QM III)
A. Basic theory and practical methods:
Second quantization
Mean-field methods (reduces to single-particle):
Hartree-Fock approximation, Density functional theory, etc
Many-particle perturbation theory: Green’s function and the Feynman diagram
B. Application: Condensed Matter Physics
Electronic structures, phonons, electron-phonon interaction, optical properties
Superconductivity (BCS theory), superfluid, Bose-Einstein condensation (BEC)
(3) Relativistic quantum mechanics (QFT)
A. Dirac equation, Quantum Electrodynamics (QED), Quantum Chromodynamics (QCD)
B. Application: particle physics, etc.
3
1.2 The Schrödinger Equation
Consider a free particle with mass m and momentum p, its wave function can be written as
ψ(r,t) = exp[i(k ⋅ r − ωt + ϕ)]
(1.7)
where the wave vector k = p /  and angular frequency ω = E /  .
The phase velocity
ω E
= ,
k
p
(1.8)
dω dE
=
.
dk
dp
(1.9)
p
p
, vg = .
2m
m
(1.10)
E
c 2p
, vg =
.
p
E
(1.11)
vph =
while the group velocity
vg =
Non-relativistically, E =
p2
, thus
2m
vph =
Relativistically, E = p2c 2 + m 2c 4 , then
vph =
We find that (1) relativistically vph ⋅ vg = c 2 ; (2) in both cases vg corresponds to the classical
velocity.
Eq. (1.7) indicates that
−i∇ψ(r,t) = pψ(r,t) ,
(1.12)
∂
ψ(r,t) = E ψ(r,t) .
∂t
(1.13)
i
In the non-relativistic case, E =
p2
, so
2m
E ψ(r,t) =
i
p2
1
ψ(r,t) =
(−i∇)2 ψ(r,t)
2m
2m
∂
2 2
ψ(r,t) = −
∇ ψ(r,t) .
∂t
2m
4
(1.14)
(1.15)
Then more generally, for a particle with mass m subject to potential V(r,t) , E =
p2
+V(r,t) ,
2m
Eqs. (1.14) and (1.15) suggest that
i
⎡ 2 2
⎤
∂
ψ(r,t) = ⎢⎢−
∇ +V(r,t)⎥⎥ ψ(r,t) ,
∂t
⎢⎣ 2m
⎥⎦
(1.16)
which is the Schrödinger Equation.
On the other hand, in the relativistic case, E = p2c 2 + m 2c 4 , or E 2 = p2c 2 + m 2c 4 , one can
derive the Klein-Gordon Equation,
− 2
∂2
ψ(r,t) = ⎡⎢− 2c 2∇2 + m 2c 4 ⎤⎥ ψ(r,t) .
2
⎣
⎦
∂t
(1.17)
But this is not a good relativistic QM equation because of second-order derivatives in both time
and space. A successful relativistic treatment is the Dirac Equation,
i

∂
ψ(r,t) = ⎡⎢cα ⋅(−i∇) + βmc 2 ⎤⎥ ψ(r,t) ,
⎣
⎦
∂t
(1.18)
which is of first order in both time and space.
Let’s get back to Schrödinger Equation (1.16), which is time-dependent. For timeindependent potential V(r) , it is straightforward to derive the time-independent Schrödinger
Equation,
⎡ 2 2
⎤
Eψ(r) = ⎢⎢−
∇ +V(r)⎥⎥ ψ(r) ≡ H ψ(r) .
⎢⎣ 2m
⎥⎦
Here the Hamiltonian operator
H=
with momentum operator
p̂2
+V(r) ,
2m
p̂ = −i∇ .
(1.19)
(1.20)
(1.21)
A solution to Eq. (1.16) or Eq. (1.19), ψ(r,t) or ψ(r) , respectively, corresponds to a
quantum state denoted by ψ(t) or ψ using Dirac’s notation; explicitly, a wave function
ψ(r) is a representation of a quantum state ψ in real space: ψ(r) = r ψ .
5
1.3 Quantum States
The first postulate of quantum mechanics is that a physical state can be represented by a vector
in an infinite-dimensional complex vector space, known as Hilbert space.
In Dirac’s notations, a physical state is described by a ket vector α in a ket space. Let (a,
b, c, ….) be complex numbers (scalars, or the field) and ( α , β , γ , …) be kets. Then
(1) a α is another ket; however, α and a α correspond to the same quantum state if a ≠ 0 .
When a = 0 , a α becomes a null ket.
(2) Linear combination of kets forms another ket, for example,
ξ = a α +b β .
(1.22)
The inner (scalar) product is defined between a ket and a bra vector, written as α , which
is a dual vector to ket α . The dual correspondence is expressed as
α
DC
↔ α
and
DC
α a* ↔ a α .
(1.23)
The inner product is denoted as α β , a complex number, having the property
*
αβ = β α ,
(1.24)
so that α α is real and positive (except for the null state) by definition. Then the norm of a ket
α (or bra α ) is defined as
α α ; we can normalize a ket by
α
 =
α
αα
,
(1.25)
α
 = 1 . Two kets α and β are orthogonal if α β = 0 .
with α
An orthonormal basis consists of a complete set of kets,
{ n } , n = 1, 2,... , satisfying
n m = δnm , that span the Hilbert space in a sense that an arbitrary ket can be express as a
linear combination of basis kets:
6
ψ = ∑ cn n ,
(1.26)
n=1
where cn = n ψ . If ψ is normalized, then
ψ ψ = 1 ⇒ ∑ |cn |2 = 1 .
(1.27)
ψ =∑ n ψ n =∑ n n ψ .
(1.28)
n=1
Eq. (1.26) also indicates that
n=1
n=1
1.4 Quantum Operator
Eq. (1.28) can be re-written as
(
ψ =∑ n n
n=1
) ψ = ∑P
n=1
n
ψ ,
(1.29)
where Pn ≡ n n , the projection operator:
Pn ψ = n n ψ = cn n ,
(1.30)
1 = ∑ Pn = ∑ n n .
(1.31)
n=1
n=1
A quantum operator A converts a ket α to another ket β :
A α = const ⋅ β .
Linear operator:
(
) = aA α
+bA β
(1.33)
(
)=a A α
+b*A β
(1.34)
A a α +b β
Anti-linear operator:
(1.32)
*
A a α +b β
Hermitian adjoint:
DC
αA ↔ Aα
†
(1.35)
A is a Hermitian operator if A = A† .
Two operators A and B are equal if A α = B α for any ket α .
Commutative relations:
A+B = B +A
AB ≠ BA
7
(1.36)
(1.37)
1.5 Matrix Representation
Let’s only consider linear operator A. For a basis set { n } , with n = 1, 2,... , using Eq. (1.31), we
obtain A in terms of its matrix elements Amn = m A n :
A = 1 ⋅ A ⋅1 = ∑ m m A n n = ∑ Amn m n .
m,n
(1.38)
m,n
(
Amn = m A n = m A n
( (
= n A† m
))
*
) = ( m A) n
= n A† m
*
(1.39)
†
= (Anm
)*
Let’s emphasize that
*
† *
m A n = n A† m ; i.e., Amn = (Anm
)
(1.40)
For Hermitian operator
*
*
m A n = n A m ; i.e., Amn = Anm
(1.41)
Thus, explicitly we can write down the matrix representation for A:
⎛ A A  ⎞⎟
⎜⎜ 11
⎟⎟
12
⎜⎜
⎟
A  ⎜⎜ A21 A22  ⎟⎟⎟ .
⎟
⎜⎜
  ⎟⎟⎠
⎝ 
(1.42)
The basis kets n (bras n ) in matrix representation:
⎛
⎜⎜
⎜
n  ⎜⎜⎜
⎜⎜
⎜⎜
⎝
0

1

⎞⎟
⎟⎟
⎟⎟
,
⎟⎟
⎟⎟ n-th element
⎟⎟
⎠
n  (0, ,
)
1,
n-th element
(1.43)
For an arbitrary ket state, α = ∑ cn n , its matrix representation:
n=1
⎛
⎜⎜
⎜⎜
α  ⎜⎜⎜
⎜⎜
⎜⎜
⎜⎝
⎛
⎞
c1 ⎞⎟⎟ ⎜⎜ 1 α ⎟⎟⎟
⎟⎟ ⎜⎜
⎟⎟
⎟⎟
 ⎟⎟⎟ = ⎜⎜

⎟,
⎜
⎟
cn ⎟⎟ ⎜⎜ n α ⎟⎟⎟
⎟⎟ ⎜⎜
⎟⎟
⎟⎟
 ⎟⎠ ⎜⎜⎝

⎠
(1.44)
and the matrix representation for its corresponding bra state, α = ∑ n c *n , is
α 
(c,
*
1
, cn* , 
)=(
8
n=1
α 1 , ,
)
αn ,  .
(1.45)
Inner product of two states α and β :
β α =
(
β 1 , ,
⎛
⎞
⎜⎜ 1 α ⎟⎟
⎟⎟
⎜⎜
⎟⎟
⎜⎜

⎟⎟ =
βn ,  ⎜
βn n α .
⎜⎜ n α ⎟⎟ ∑
⎟⎟ n=1
⎜⎜
⎟⎟
⎜⎜⎝
⎟⎠

)
(1.46)
Outer product:
⎛
⎞
⎜⎜ 1 β ⎟⎟
⎟⎟
⎜⎜
⎟⎟
⎜⎜

⎟⎟
β α =⎜
⎜⎜ n β ⎟⎟
⎟⎟
⎜⎜
⎟⎟
⎜⎜⎝
⎟⎠

(
α 1 , ,
αn , 
)
⎛
⎜⎜ 1 β α 1
⎜
= ⎜⎜⎜ 2 β α 1
⎜⎜
⎜⎜⎝

1β α2
2β α2

⎞
 ⎟⎟
⎟⎟
⎟
 ⎟⎟⎟
⎟⎟
 ⎟⎠
(1.47)
Transpose, AT , complex conjugate, A* , and Hermitian adjoint, A† = (A* )T , of A:
T
*
†
Amn
= Anm ; Amn
= (Amn )*; Amn
= (Anm )*
(1.48)
From the definition, one can derive that
(AB)† = B †A† .
(1.49)
Eq. (1.49) indicates that if A and B are Hermitian, their product AB is Hermitian only if A and B
commute, i.e.,
[A,B] ≡ AB − BA = 0 ,
where [A,B] is commutator. On the other hand, the anti-commutator,
{A,B } ≡ AB + BA ,
(1.50)
(1.51)
is Hermitian if both A and B are Hermitian operators.
1.6 Eigenvalues and Eigenvectors of Hermitian operators
The following are crucial properties of Hermitian operators in a finite dimensional Hilbert space:
1) An Hermitian operator A can be diagonalized by a unitary transformation: U †AU , with
U †U = I .
2) In the diagonalized form the diagonal matrix elements, a j with j = 1, 2,...,n , are
eigenvalues of A. They are real numbers, but not necessary to be all different; instead,
some of they might be equal, which is called degenerate.
3) The basis vectors, a1 , a 2 , !, an , that diagonalize A are called its eigenvectors:
n
A a j = a j a j ; hence, A = ∑ a j a j a j .
j =1
9
(1.52)
4) Eigenvalues are obtained by solving the secular equation:
det(A − λI ) = 0 ,
(1.53)
and then eigenvectors are obtained by solving the matrix equation (1.52).
5) Eigenvectors with different eigenvalues are orthogonal. Those belong to a degenerate
subspace will not be orthogonal automatically, but can be always made orthogonal.
6) If two Hermitian operators A and B commute, then they can be diagonalized
simultaneously by the same set of eigenvectors. Observables A and B are compatible
when they commute; otherwise they are incompatible.
7) More generally, if A(i ) ( i = 1, 2,...,m ) is a set of m commuting Hermitian operators,
these operators can be diagonalized simultaneously with eigenvalues {a (ij ) } , then we use
to designate the common eigenvectors of these operators, which satisfy
a (1)
,a (2)
,...,a (m)
j
j
j
A(i ) a (1)
,a (2)
,...,a (m)
= a (ij ) a (1)
,a (2)
,...,a (m)
j
j
j
j
j
j
(1.54)
Note that the above statements are derived for finite-dimensional complex vector space, and
it is non-trivial to establish them for infinite dimensions, especially for a Hilbert space spanned
by a continuous (nondenumerable) basis. Some of them are false for certain conditions;
nevertheless, we shall assume they are correct in this introductory class.
1.7 Change of Basis and Unitary Transformations
Theorem: Given two sets of orthonormal and complete basis kets,
{ a } and { b } , there exists
j
j
a unitary operator U such that
bj =U a j .
(1.55)
U †U =UU † = 1 .
(1.56)
Here the operator U satisfies that
The proof is straightforward by explicitly constructing U :
U = ∑ bj a j ,
(1.57)
j
then using orthonormality of
{ a } and { b } , one can derive that
j
j
U †U = ∑ ∑ ai bi bj a j = ∑ ∑ ai δij a j = ∑ ai ai = 1 .
(1.58)
⇒ det(U ) = 1 .
(1.59)
i
j
i
10
j
i
Here we used the properties (1-3) of determinant:
1) det(I n ) = 1 .
2) det(AB) = det(A)det(B) , for square matrices A and B .
3) det(AT ) = det(A); det(A* ) = [det(A)]*; det(A† ) = [det(A)]* .
4) det(A−1 ) =
1
if det(A) ≠ 0 .
det(A)
Comparisons:
Inverse operator: A−1A = AA−1 = I .
Unitary operator: U †U =UU † = I
⇒
U † =U −1 .
Identity operator: I α = α
(1.60)
(the short hand of the unit operator is simply 1.)
Unitary operator: U α = β ⇒ α U † = β ⇒ β β = α α , i.e.,
the norm of a vector is conserved after unitary transformation.
Now consider a change of basis from
{ a } to { b } by U = ∑ b
j
j
j
j
a j . Here
{ a } and
j
{ b } are eigenvectors of operators A and B, respectively:
j
A a j = a j a j ; B bj = bj bj .
B is not a diagonal matrix by basis
U a j = bj
{ a } , while it is diagonalized by basis { b } .
j
⇔ U † bj = a j
j
⇔
′ ≡ bm B ′ bn = bn δmn ;
Bmn
Eq. (1.63) indicates that using
(1.61)
bj U = a j
⇔
bj = a j U † .
′ = am U †BU an .
Bmn
(1.62)
(1.63)
{ a } as basis B is not diagonal, but it can be diagonalized by the
j
unitary transformation B ′ =U †BU . Here the transformation matrix elements
U mn = am U an = am bn ,
(1.64)
i.e., the columns of U form an orthonormal basis which are eigenvectors of B.
det(B ′) = det(U †BU ) = det(B) .
11
(1.65)
If another operator C is also diagonalized by the same unitary operator U : C ′ =U †CU , then
B and C commute, i.e., [B,C ] = 0 .
The trace of an operator A is defined as the sum of diagonal elements:
Tr(A) ≡ ∑ n A n .
(1.66)
Tr(AB) = Tr(BA) .
(1.67)
Tr(A′) = Tr(U †AU ) = Tr(A) ,
(1.68)
n
An important property:
Then one can verify that
which means that an operator’s trace is independent of representation (basis).
The expectation value of observable A for a system in the quantum state ψ :
A ≡ ψ A ψ = Tr(APψ ) ,
(1.69)
where the projector operator Pψ = ψ ψ .
1.8 Time Evolution and Functions of Operators
Let’s consider a unitary operator, namely, the time-evolution operator
⎡ (t −t ) ⎤
0
U(t,t0 ) = exp ⎢⎢−i
H ⎥⎥ ,
(1.70)

⎢⎣
⎥⎦
which can be obtained by formally solving the time-dependent Schrödinger Equation:
∂
i ψ(r,t) = H ψ(r,t) ⇒ ψ(r,t) =U(t,t0 )ψ(r,t0 ) .
(1.71)
∂t
Here the operator U(t,t0 ) is a function of Hamiltonian operator H . We define a function of
operator as
∞
f (A) = ∑ anAn ,
(1.72)
n=0
where An = AAA . A−1 , or
1
, is the reverse operator: A−1A = AA−1 = I , while
A
eA = I + A +
If A is diagonalized by its eigenvectors,
A2 A3
+
+
2!
3!
12
(1.73)
⎛
⎜⎜
⎜⎜
⎜
A = ⎜⎜
⎜⎜
⎜⎜
⎜⎝
λ1
0
0
0
λ2
0
0
0
λ3



 ⎞⎟⎟
⎟⎟
 ⎟⎟⎟
⎟⎟ ,
 ⎟⎟
⎟⎟
 ⎟⎠
(1.74)
then
⎛
⎜⎜
⎜⎜
⎜
n
A = ⎜⎜
⎜⎜
⎜⎜
⎜⎝
λ1
0
0
0
λ2
0
0
0
λ3



n
⎛ n
⎜⎜ λ1 0 0
 ⎞⎟⎟
⎟⎟
⎜⎜
 ⎟⎟⎟
⎜ 0 λn2 0
⎟⎟ = ⎜⎜⎜
 ⎟⎟
⎜⎜ 0 0 λn3
⎟⎟
⎜⎜
⎜⎝ 
 ⎟⎠


⎞
 ⎟⎟
⎟⎟
 ⎟⎟⎟
⎟⎟ .
 ⎟⎟⎟
⎟⎟
 ⎟⎠
(1.75)
⎞⎟
⎟⎟
⎟⎟
⎟⎟ ,
⎟⎟
⎟⎟
⎟⎟⎠
(1.76)
In particular, A0 = I . Eq. (1.73) leads to
⎛
⎜⎜
⎜⎜
⎜
A
e = ⎜⎜
⎜⎜
⎜⎜
⎜⎝
α1
0
0
0
α2
0
0
0
α3



 ⎞⎟⎟ ⎛⎜
⎟⎟ ⎜
 ⎟⎟⎟ ⎜⎜⎜
⎟=⎜
 ⎟⎟⎟ ⎜⎜⎜
⎟⎟ ⎜
⎜
 ⎟⎠ ⎝
λ
e1 0 0
λ
0 e2 0
λ
0 0 e3







1
λ
(λn )m = e n .
m=0 m !
∞
where αn = ∑
Then a change of basis if A is not diagonal: A′ =U †AU . One can show that,
U †AnU =U †AUU †AUU † U †AU = (A′)n ,
(1.77)
U † f (A)U = f (A′)
(1.78)
Hence,
Another example of function of operator:
1
= (1 − A)−1 = I + A + A2 + A3 + 
1−A
In contrast to Eq. (1.73) for e A ,
(1.79)
1
might diverge, i.e., it is not always well defined as above.
1−A
Two operators A and B, in general,
e A+B ≠ e Ae B .
(1.80)
Only when [A,B] = 0 , e A+B = e Ae B . In particular, the time-evolution operator U(t) with t0 = 0
13
⎡ t
⎤
⎛ t ⎞
⎛ t ⎞
U(t) = exp ⎢−i (T +V )⎥ ≠ exp ⎜⎜⎜−i T ⎟⎟⎟ exp ⎜⎜⎜−i V ⎟⎟⎟ ,
⎢ 
⎥
⎝  ⎟⎠
⎝  ⎟⎠
⎣
⎦
1 2
p̂ and potential operator V =V(r,t) .
where the kinetic energy operator T =
2m
For a time-independent Hamiltonian H, i.e., V =V(r) ,
(1.81)
H φn = En φn .
(1.82)
ψ(0) = ∑ cn φn .
(1.83)
At t = 0 ,
n
Then the solutions ψ(t) to time-dependent Schrödinger equation are
ψ(t) =U(t) ψ(0) = e
t
−i H
!
∑ cn φn = ∑ e
n
t
−i En
!
n
cn φn .
(1.84)
Derivative of operators with parameters.
For example, A(α) = exp(αB) , where A and B are operators, while α is a number (parameter).
Then derivative with respect to this parameter is:
⎡ A(α + Δα)− A(α) ⎤
dA(α)
⎥.
≡ lim Δα→0 ⎢
⎢
⎥
dα
Δα
⎣
⎦
(1.85)
d αB
d
1
1
1
e
=
(αB)n = ∑ (nαn−1 )B n = ∑
(m + 1)αmB m+1
∑
dα
dα n=0 n !
n=1 n !
m=0 m + 1!
1 m m
= B∑
α B = Be αB = e αBB.
m=0 m !
(1.86)
( )
One can verify that
d ⎡⎢A(α)B(α)⎤⎥ dA(α)
dB(α)
⎣
⎦=
.
B(α) + A(α)
dα
dα
dα
(1.87)
d αA αB
e e
= Ae αAe αB +e αAe αBB.
dα
(1.88)
For example,
(
)
1.9 Quantum Measurements
The statistical (probabilistic) character of quantum mechanics is irreducible – NO hidden
variables which behave in a deterministic manner.
The basic postulates of quantum mechanics:
14
1) The state of a system is represented by a vector ψ(t) in a Hilbert space.
2) The physically meaningful entities of classical mechanics, such as momentum, energy,
position, etc, are represented by Hermitian operators.
3) If the particle is in a state ψ , measurement of the variable (corresponding to) A will
yield one of the eigenvalues a j with probability
P(a j ) = a j ψ
2
,
(1.89)
and the state of the system will change from ψ to a j .
4) The state vector ψ(t) obeys the Schrödinger equation
i
d
ψ(t) = H ψ(t)
dt
(1.90)
Postulate III describes the basic principle of quantum measurement, which is the foundation
of quantum interpretation. While the mathematical structure of quantum mechanics is extremely
successful, its interpretation remains controversial. In this class we adopt the standard
Copenhagen interpretation formulated by Niels Bohr and Werner Heisenberg: Nature is
intrinsic probabilistic and a measurement causes an instantaneously stochastic collapse of wave
function to an eigenstate of the observable measured.
We’ll discuss quantum interpretation in full depth when studying quantum entanglements
and decoherence next semester. Here we only briefly illustrate the principles in QM
measurement proposed by Bohr and Heisenberg.
For a quantum state ψ , a measurement of observable A whose eigenvalues and eigenvector
are {a j } and
{ a } , “collapses”
p(a j ) = a j ψ
j
2
ψ = ∑ c j a j to a j randomly with a probability of
j
. Therefore, (1) any single measurement gives an eigenvalue of A; (2) when ψ
is an eigenstate of A, there is no wave function collapse, and any measurement of A gives the
same value; (3) in general, ψ is not an eigenstate of A, the expectation value can be obtained
by averaging measured data of a quantum ensemble:
ψ A ψ = ∑ a j p(a j ) .
j
15
(1.91)
Compatible Observables. A and B are compatible if [A, B] = 0, then they can be
diagonalized by a common basis eigenvectors denoted by
{ a ,b } . A state
j
j
ψ is written as a
linear combination of this basis: ψ = ∑ c j a j ,bj . A ψ = ∑ c ja j a j ,bj , while
j
ψ
Measurement A
= a j ,bj
→→→→→
j
Measurement B
a j ,bj
→→→→→
Measurement A
a j ,bj ,
→→→→→
(1.92)
if there are no degeneracy. When A has degenerate states (B is non-degenerate) with degeneracy
n for the kets a,b(i ) , i = 1, 2,n , then
Measurement A
n
= ∑ ca(i ) a,b(i )
→→→→→
ψ
i=1
Measurement B
a,b(i )
→→→→→
Measurement A
a,b(i ) . (1.93)
→→→→→
We conclude that whether or not there are degeneracy, A measurements and B measurements do
not interfere, i.e., measuring A will not mess up measuring B; thus we call A and B compatible.
Incompatible Observables. If [A,B] ≠ 0 then they can not be diagonalized by a common
basis. The measurements of A and B for an arbitrary state ψ :
ψ
Measurement A
= aj
→→→→→
Measurement B
bk
→→→→→
Measurement A
ai .
→→→→→
(1.94)
Here the A and B measurements are incompatible, and they interfere: the complete determination
of observable A makes the following B measurements totally probabilistic − the uncertainty
principle.
Except when ψ is an eigenstate of A, there is non-zero dispersion of A, i.e., the deviation
of individual measurements from the average result, the expectation value A . The expectation
value of operator (ΔA)2 is known as dispersion (variance, mean square deviation) of A, where
ΔA ≡ A − A ,
(1.95)
2
(ΔA)2 = A2 − A .
For example, for the z;+ state of a spin-half system, Sz z;+ =
16
(1.96)
!
z;+ , we obtain
2
(ΔSz )2 = 0,
2
(ΔSx )2 = Sx2 − Sx
while
= !2 / 4 .
(1.97)
Experimentally, Sz measurements are “sharp” whereas Sx are “fuzzy”.
Using the Schwarz inequality,
αα β β ≥ αβ
2
,
(1.98)
setting α = ΔA ψ and β = ΔB ψ , one can derive the general uncertainty relation:
(ΔA)2 (ΔB)2 ≥
1
4
⎡A,B ⎤
⎢⎣
⎥⎦
2
.
(1.99)
1.10 Continuous Spectra
We have discussed the discrete basis, now we generalize it for the continuous spectra. For a
continuous basis, such as
Ξ ξ =ξ ξ ,
(1.100)
compared with A a j = a j a j . We do the following two replacements:
δij → δ(ξ − ξ ′) and
N
∑ →∫
∞
−∞
j =1
dξ
(1.101)
Therefore,
ai a j = δij
N
∑a
j =1
j
→
aj = 1 →
N
ψ = ∑ aj aj ψ
→
j =1
2
p(a j ) = a j ψ ;
N
ξ ξ ′ = δ(ξ − ξ ′)
∫
ξ ξ dξ = 1
ψ =
∫
2
j =1
ai A a j = a j δij
→
17
(1.103)
ξ ξ ψ dξ
∑ p(a j ) = 1 → p(ξ) = ξ ψ ;
(1.102)
∫ p(ξ)dξ = 1
ξ ′ Ξ ξ = ξδ(ξ − ξ ′)
(1.104)
(1.105)
(1.106)
1.11 Position and Momentum
Now we consider the Hilbert space with (one-dimensional) position eigenkets as a basis,
X x =x x .
(1.107)
Then an arbitrary vector ψ in this Hilbert space can be written as
∞
∫
ψ =
x x ψ dx , and ψ(x) ≡ x ψ .
−∞
(1.108)
The inner product of two states,
α ψ =∫
∞
α x x ψ dx = ∫
−∞
∞
−∞
α*(x)ψ(x)dx .
(1.109)
The normalization condition:
⎧⎪
1
⎪
ψ ψ = ∫ ψ (x)ψ(x)dx = ⎨
−∞
⎪⎪ δ-function
⎪⎩
∞
*
(bound, discrete states)
(unbound, continuous states)
(1.110)
The definition of δ -function:
⎧⎪ 0 (x ≠ x ′)
⎪
′
δ(x − x ) = ⎨
;
⎪⎪ ∞ (x = x ′)
⎪⎩
∫
b
a
δ(x − x ′)dx = 1, a < x ′ < b.
(1.111)
= δ *(x ′ − x) = δ(x ′ − x) ,
(1.112)
The properties of δ -function:
δ(x − x ′) = x x ′ = x ′ x
∫
∞
−∞
*
δ(x − x ′)f (x)dx = f (x ′)
(1.113)
Define the derivative of δ -function:
δ ′(x − x ′) ≡
∫
∞
−∞
dδ(x − x ′)
dx
δ ′(x − x ′)f (x)dx = −
df (x)
| ,
dx x=x ′
(1.114)
(1.115)
therefore,
δ ′(x − x ′) = −δ(x − x ′)
δ (x − x ′) ≡
(n)
d
,
dx ′
d n δ(x − x ′)
dn
n
′
= (−1) δ(x − x ) n .
dx n
dx ′
18
(1.116)
(1.117)
Finally, a very important expression for δ -function:
δ(x − x ′) =
1 ∞ ik(x−x ′)
e
dk
2π ∫−∞
(1.118)
This can be derived from the Fourier transformation (analysis):
f (k) =
1
∫
2π
∞
−∞
e −ikx f (x)dx ,
(1.119)
and its inverse transformation
f (x ′) =
1
∫
2π
∞
−∞
′
e ikx f (k)dk .
(1.120)
Feeding Eq. (1.119) into Eq. (1.120), we obtain
f (x ′) =
1 ∞ ⎛⎜ ∞ ik(x ′−x ) ⎞⎟
e
dk ⎟⎟ f (x)dx .
⎜
⎠
2π ∫−∞ ⎜⎝ ∫−∞
(1.121)
Comparing Eq. (1.121) with Eq. (1.113), we reach Eq. (1.118).
Operators in continuous basis
Any Hermitian operator that corresponds to a classical observable is a function of position and
momentum: A = A(r̂, p̂) . In comparison, classically A = A(r, p) .
A ket state ψ in the Hilbert space spanned by the continuous basis x :
ψ(x) = x ψ
(1.122)
Then for an operator A (not limited by Hermitian operators), A ψ = const. φ in x representation:
x Aψ = x φ
⇔ Aψ(x) = φ(x) .
(1.123)
For example, the derivative operator
D≡
d
dψ(x)
; Dψ(x) =
, or
dx
dx
x D ψ =
d x ψ
dx
.
(1.124)
Question: is D a Hermitian operator? If so, then Dxx ′ = Dx*′x . But
Dxx ′ = x D x ′ =
while
d
dδ(x − x ′)
x x′ =
,
dx
dx
19
(1.125)
D *x ′x = x ′ D x
*
=
d
x′ x
dx
*
=
dδ(x ′ − x)
dδ(x − x ′)
=−
= −Dxx ′
dx
dx
(1.126)
Therefore D is not Hermitian. We then define (the wave vector operator)
K ≡ −iD = −i
d
,
dx
(1.127)
and it is straightforward to verify that K xx ′ = K x*′x . Q: Is K a Hermitian operator? Not yet…
The momentum operator (in x -representation):
P = K = −i
d
dx
(1.128)
in three-dimensional real space,
p̂ = k̂ = −i∇
(1.129)
Position and momentum operators, a summary
X x =x x ; P p =p p
∞
∫
−∞
∫
x x dx = 1;
∞
−∞
(1.130)
p p dp = 1
(1.131)
ψ(x) = x ψ ; ψ(p) = p ψ
φ ψ =∫
∞
−∞
⎧
⎪
⎪
⎪
⎪
φ X ψ =⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎩
∫
∞
−∞
*
∞
−∞
⎧
⎪
⎪
⎪
⎪
φ P ψ =⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎩
φ*(x)xψ(x)dx
dψ(p)
∫−∞ φ (p)(i) dp dp
∞
φ ψ =∫
φ*(x)ψ(x)dx ;
(1.132)
;
φ*(p)ψ(p)dp
∫
∞
−∞
φ*(p)pψ(p)dp
dψ(x)
∫−∞ φ (x)(−i) dx dx
∞
(1.133)
(1.134)
*
Change of basis: Fourier transformation [Eq. (1.119)] and the inverse Fourier transformation
[Eq. (1.120)] suggest that (set k = p /  )
ψ(p) =
ψ(x) =
1
∞
∫
2π
−∞
1
∫
2π
e −ipx/ ψ(x)dx ,
(1.135)
e ipx/ ψ(p)dp .
(1.136)
∞
−∞
Since
ψ(p) = p ψ = ∫
∞
−∞
20
p x x ψ dx ,
(1.137)
comparing with Eq. (1.135), we obtain the wave function for an eigenvector of P in x-representation:
ψp (x) = x p =
1
2π
e ipx/ .
(1.138)
e −ipx/ .
(1.139)
Similarly,
ψx (p) = p x =
1
2π
1.12 Momentum as a Generator of Space Translation
We define a space translation (similar to time evolution) by a unitary operator
T(Δx) x = x + Δx .
(1.140)
T −1(Δx) = T †(Δx) = T(−Δx) .
(1.141)
T(Δx)ψ(x) = x T(Δx) ψ = x − Δx ψ = ψ(x − Δx) .
(1.142)
Obviously, T(0) = I , and
Then
Consider an infinitesimal translation by Δx → 0 ,
ψ(x − Δx) = ψ(x)−
dψ(x)
Δx +O(Δx)2 ,
dx
(1.143)
dψ(x)
Δx ,
dx
(1.144)
T(Δx)ψ(x) = ψ(x)−
therefore
T(Δx) = 1 − Δx
d
P
= 1 −i Δx .
dx

(1.145)
Assuming Δx = x / N , with N → ∞ , we find that
N
⎛
⎞
P
T(x) = limN →∞ T (x / N ) = limN →∞ ⎜⎜⎜1 −(i x) / N ⎟⎟⎟ = e −iPx/ .
⎟⎠

⎝
N
(1.146)
Here, the momentum operator P is the generator operator for space translation, whereas the
Hamiltonian operator H is the generator operator for time evolution. Later on we’ll prove that the
angular momentum operator J is the generator operator for rotation.
We can easily generalize from one-dimension to three-dimension:
T(x) = e −iPx/
⇒
21
T(r) = e −ip̂⋅r/
(1.147)
In addition,
r r′ = δ 3(r − r′);
p p′ = δ 3(p − p′) ,
(1.148)
where the 3D Dirac delta-function:
δ 3(r − r′) = δ(x − x ′)δ(y − y ′)δ(z − z ′) .
(1.149)
Matrix elements for p̂ and change of basis:
φ p̂ ψ = ∫ φ*(r)(−i∇)ψ(r)d 3r
ψ(p) =
1
3
(2π)
∫e
ψr (p) = p r =
−ip⋅r/
ψ(r)d 3r; ψ(r) =
1
3
(2π)
1
3
(2π)
e −ip⋅r/ ; ψp (r) = r p =
(1.150)
∫e
ip⋅r/
ψ(p)d 3p
(1.151)
e ip⋅r/
(1.152)
1
3
(2π)
1.13 Canonical (Fundamental) Commutators
First, let’s prove that
⎡X,P ⎤ = ih
⎣⎢
⎦⎥
(1.153)
T †(Δx)XT(Δx) = X + Δx .
(1.154)
Eq. (1.140) indicates that
Obviously, [X,T(Δx)] ≠ 0 ; therefore, [X,P ] ≠ 0 . Eq. (1.154) is true for any value of Δx ;
specifically, when Δx → 0 , T(Δx) = 1 − iPΔx /  and T †(Δx) = 1 + iPΔx /  , then
T †(Δx)XT(Δx) = (1 + iPΔx / )(X − iXPΔx / )
.
i
= X + (XP − PX )(− )Δx +O[(Δx)2 ]

(1.155)
Comparing this with Eq. (1.154), we obtain [X,P ] = XP − PX = i .
Since any Hermitian operator A corresponding to a classical observable can be written as a
function of operator X and P, we often end up with evaluating [P,A(X )] and [X,B(P)] . The
following commutator relations will be used:
[A,A] = 0; [A,B] = −[B,A]; [A,c] = 0
[A + B,C ] = [A,C ] + [B,C ]
[A,BC ] = [A,B]C + B[A,C ]
[A,[B,C ]] + [B,[C,A]] + [C,[A,B]] = 0
22
(1.156)
(1.157)
(1.158)
(1.159)
Using (1.158), one can derive
[X,P 2 ] = [X,P ]P + P[X,P ] = 2iP
[X,P n ] = [X,P n−1 ]P + P n−1[X,P ] = niP n−1 = i
[X,B(P)] = i
Similarly,
(1.160)
dP n
dP
dB(P)
dP
dX n
dX
dA(X )
[P,A(X )] = −i
dX
[P, X n ] = −niX n−1 = −i
(1.161)
(1.162)
(1.163)
(1.164)
Commutators in 3D. The following are the canonical (fundamental) communication
relations, with i, j = 1, 2, 3 corresponding to x,y,z directions, respectively:
[X i , X j ] = 0; [Pi ,Pj ] = 0; [X i ,Pj ] = iδij
(1.165)
which form the cornerstone of quantum mechanics.
Proof: (1) T(r1 )T(r2 ) = T(r2 )T(r1 ) ⇒ [Pi ,Pj ] = 0 . Then (2) using the translation in
momentum space, one can readily derive: T(p1 )T(p2 ) = T(p2 )T(p1 ) ⇒ [X i , X j ] = 0 .
(3) In 3D, Eq. (1.155) becomes
Then
T †(Δr)r̂T(Δr) = (1 + ip̂ ⋅ Δr / )[r̂ −ir̂(p̂ ⋅ Δr) / ]
i
= r̂ + (− )[r̂(p̂ ⋅ Δr)−(p̂ ⋅ Δr)r̂]+O[(Δr)2 ]

(1.166)
r̂(p̂ ⋅ Δr)−(p̂ ⋅ Δr)r̂ = iΔr .
(1.167)
For Δr in the direction of rj , Eq. (1.167) becomes
r̂(Pj Δrj )−(Pj Δrj )r̂ = iΔrj ,
(1.168)
which leads to [X i ,Pj ] = iδij .
Relation with classical Poisson brackets. In 1925 P. A. M. Dirac postulated that
{A,B}PB →
1
[Â, B̂] ,
i
(1.169)
where the classical Poisson brackets (PB) are defined for functions of generalized momenta p
and coordinates q as
23
⎛ ∂A ∂B ∂A ∂B ⎞⎟
⎟⎟ .
{A(p,q),B(q, p)}PB ≡ ∑ ⎜⎜⎜
−
s ⎜
⎝ ∂qs ∂ps ∂ps ∂qs ⎟⎟⎠
(1.170)
For example, in classical mechanics, we have
{x i , p j }PB = δij ,
(1.171)
corresponding to [X i ,Pj ] = iδij . Later on we’ll discuss that this fundamental commutator is
responsible for (the first) quantization.
24