Download Quantum Mechanics II SS 2014

Quantum Mechanics II
SS 2014
Roser Valentı́
Institut für Theoretische Physik
Universität Frankfurt
Contents
1 Scattering theory
1.1 Importance of collision experiments . . . . . . . . . . . . . . . . .
1.2 Elastic scattering . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Scattering cross section . . . . . . . . . . . . . . . . . . . . . . . .
1.4 Calculation of the scattering cross section . . . . . . . . . . . . . .
1.4.1 Asymptotic form of stationary scattering states.
Scattering amplitude . . . . . . . . . . . . . . . . . . . . .
1.4.2 Relation between σ(θ, φ) and f (θ, φ) . . . . . . . . . . . .
1.5 Partial wave analysis . . . . . . . . . . . . . . . . . . . . . . . . .
1.5.1 Example: Quantum hard-sphere scattering . . . . . . . . .
1.5.2 Phase shift . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5.3 Phase shift for the 3-dimensional case . . . . . . . . . . . .
1.6 The Born approximation . . . . . . . . . . . . . . . . . . . . . . .
1.6.1 Integral form of the Schrödinger equation . . . . . . . . . .
1.6.2 Solving the Green’s function . . . . . . . . . . . . . . . . .
1.6.3 The first Born approximation . . . . . . . . . . . . . . . .
1.6.4 Example: Yukawa potential . . . . . . . . . . . . . . . . .
1.6.5 The Born series . . . . . . . . . . . . . . . . . . . . . . . .
1.7 Collisions with absorption . . . . . . . . . . . . . . . . . . . . . .
1.8 Formal scattering theory . . . . . . . . . . . . . . . . . . . . . . .
1.8.1 Propagator theory and the Lippmann-Schwinger equation .
1.8.2 Meaning of the Lippmann-Schwinger equation . . . . . . .
1.8.3 Position and momentum representation . . . . . . . . . . .
1.8.4 Retarded and advanced Green’s functions . . . . . . . . .
1.8.5 Interpretation of the Lippmann-Schwinger equation and its
sion in terms of Feynmann’s diagrams . . . . . . . . . . . .
2 Theory of angular momentum
2.1 Importance of angular momentum . . . . . . . . . . . . . .
2.1.1 Orbital angular momentum in Quantum Mechanics
2.1.2 Spin . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Generalization: angular momentum J~ . . . . . . . . . . . .
2.2.1 Special properties of J = S = 1/2 . . . . . . . . . .
2.2.2 Non-relativistic description of a spin-1/2 particle . .
2.3 Angular momentum and rotations . . . . . . . . . . . . . .
-3
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
1
1
1
2
3
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
expan. . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
4
5
6
10
11
12
13
13
14
17
19
21
23
25
25
27
28
29
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. 31
.
.
.
.
.
.
.
33
33
33
34
35
38
39
41
-2
CONTENTS
2.4
2.5
2.6
2.7
2.8
2.9
2.3.1 Definition of rotation . . . . . . . . . . . . . . . . . . . . . . .
2.3.2 Orthogonal group . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.3 Infinitesimal rotations . . . . . . . . . . . . . . . . . . . . . .
2.3.4 Rotation operators in state space . . . . . . . . . . . . . . . .
2.3.5 Rotation operators in terms of angular momentum observables
2.3.6 System of several spinless particles . . . . . . . . . . . . . . .
Rotation of observables . . . . . . . . . . . . . . . . . . . . . . . . . .
Rotation invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.1 Invariance of physical laws . . . . . . . . . . . . . . . . . . . .
Rotation operators for a spin 1/2 particle . . . . . . . . . . . . . . . .
2.6.1 Unitary unimodular group SU(2) . . . . . . . . . . . . . . . .
Addition of angular momenta . . . . . . . . . . . . . . . . . . . . . .
2.7.1 Addition of two spins 1/2 . . . . . . . . . . . . . . . . . . . .
Formal theory of angular-momentum addition . . . . . . . . . . . . .
2.8.1 Clebsch-Gordan coefficients . . . . . . . . . . . . . . . . . . .
2.8.2 Recursion relations for the Clebsch-Gordan coefficients . . . .
2.8.3 Clebsch-Gordan coefficients and rotation matrices . . . . . . .
Bell’s inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3 Symmetry in quantum mechanics
3.1 Symmetries, conservation laws, degeneracies . . . . . .
3.1.1 Symmetries in classical physics . . . . . . . . .
3.1.2 Symmetries in quantum mechanics . . . . . . .
3.2 Discrete symmetries: parity . . . . . . . . . . . . . . .
3.2.1 Wavefunctions under parity . . . . . . . . . . .
3.2.2 Example: double-well potential . . . . . . . . .
3.3 Discrete symmetry: lattice translation . . . . . . . . .
3.3.1 Tight-binding approximation . . . . . . . . . . .
3.3.2 Bloch’s theorem . . . . . . . . . . . . . . . . . .
3.4 Discrete symmetry: time-reversal . . . . . . . . . . . .
3.4.1 Antiunitary transformation . . . . . . . . . . .
3.4.2 Time-reversal operator . . . . . . . . . . . . . .
3.4.3 Transformation of operators under time reversal
3.4.4 Wavefunctions under time reversal . . . . . . .
3.4.5 Kramers degeneracy . . . . . . . . . . . . . . .
4 Approximation methods: perturbation theory
4.1 Non-degenerate time-independent perturbation theory .
4.1.1 Two-state Hamiltonian . . . . . . . . . . . . . .
4.1.2 General perturbation expansion . . . . . . . . .
4.1.3 Renormalization of the wave-function . . . . . .
4.1.4 Example . . . . . . . . . . . . . . . . . . . . . .
4.2 Degenerate time-independent perturbation theory . . .
4.2.1 First-order approximation . . . . . . . . . . . .
4.2.2 Second-order approximation (λ2 ) . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
41
42
42
43
45
46
47
48
49
50
52
53
55
57
59
61
62
63
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
65
65
65
66
69
73
75
79
81
82
84
85
86
88
89
90
.
.
.
.
.
.
.
.
93
93
94
95
99
100
102
103
104
-1
CONTENTS
4.3 Example I: linear Stark effect . . . . . . . . . .
4.4 Example II: fine structure . . . . . . . . . . . .
4.5 Time-dependent Hamiltonian: the Dirac picture
4.5.1 Dirac picture / Interaction picture . . .
4.6 Time-dependent perturbation theory . . . . . .
4.6.1 Operator formalism: Dyson series . . . .
4.6.2 Concept of transition probability . . . .
4.6.3 Perturbation expansion . . . . . . . . . .
4.6.4 Example: A constant perturbation . . .
4.7 Fermi’s golden rule . . . . . . . . . . . . . . . .
4.8 Example: harmonic perturbation . . . . . . . .
4.9 Example: interaction of a bound electron with a
. . . . . . . . . . . . .
. . . . . . . . . . . . .
. . . . . . . . . . . . .
. . . . . . . . . . . . .
. . . . . . . . . . . . .
. . . . . . . . . . . . .
. . . . . . . . . . . . .
. . . . . . . . . . . . .
. . . . . . . . . . . . .
. . . . . . . . . . . . .
. . . . . . . . . . . . .
classical radiation field
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
104
106
109
110
112
112
113
114
114
115
118
118
5 Many-particle systems. Second quantization
5.1 Identical particles and permutation theory . . . . .
5.2 Totally symmetric and totally antisymmetric states
5.3 Bosons . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.1 Creation and annihilation operators . . . . .
5.3.2 Particle-number operator . . . . . . . . . . .
5.3.3 Single- and many-particle operators . . . . .
5.4 Fermions . . . . . . . . . . . . . . . . . . . . . . . .
5.4.1 Creation-annihilation operators . . . . . . .
5.4.2 Single- and many-particle operators . . . . .
5.5 Momentum representation . . . . . . . . . . . . . .
5.5.1 Hamiltonian in second-quantized form . . .
5.6 Spin-1/2 fermions . . . . . . . . . . . . . . . . . . .
5.6.1 The Fermi sphere, excitations . . . . . . . .
5.6.2 Electron gas with Coulomb repulsion . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
121
121
123
123
124
126
126
128
129
130
131
131
133
133
134
6 Relativistic quantum mechanics
6.1 The Klein-Gordon equation . . . . . . . . . . . . .
6.1.1 Derivation . . . . . . . . . . . . . . . . . . .
6.1.2 The continuity equation . . . . . . . . . . .
6.1.3 Free solutions of the Klein-Gordon equation
6.2 Dirac equation . . . . . . . . . . . . . . . . . . . .
6.2.1 Derivation of the Dirac equation . . . . . . .
6.2.2 Continuity equation . . . . . . . . . . . . . .
6.2.3 Properties of the Dirac matrices αk and β .
6.2.4 Covariant form of the Dirac equation . . . .
6.2.5 Non-relativistic limit of the Dirac equation .
6.2.6 Physical interpretation of the solution to the
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
Dirac equation
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
137
137
137
141
142
143
143
144
145
146
147
152
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
0
CONTENTS
Literature
1) J. J. Sakurai: ” Modern Quantum Mechanics”, Addison-Wesley Publishing Company 1994.
2) J. J. Sakurai and J. J. Napolitano: ” Modern Quantum Mechanics”, Pearson Ed.
Limited 2014.
3) W. Nolting: ”Grundkurs Theoretische Physik 5/2. Quantenmechanik-Methoden
und Anwendungen”, Springer 2004.
4) F. Schwabl: ”Advanced Quantum Mechanics”, Springer 2000.
5) C. Cohen-Tannoudji, B. Diu, F. Laloke: ”Quantum Mechanics II”, Wiley-VCH
2005.
6) M. Tinkham: ”Group Theory and Quantum Mechanics”, McGraw-Hill, Inc. 1964.
7) L. H. Ryder: ”Quantum Field Theory”, Cambridge University Press, Inc. 1985
Acknowledgements I would like to thank Kateryna Foyevtsova for her help in typing
the present manuscript.
Chapter 1
Scattering theory
1.1
Importance of collision experiments
Many experiments in physics ( atomic and nuclear physics, solid state physics and high
energy physics) consist of directing a beam of particles (1) onto a target of particles
(2) (see Fig. 5.4), and studying the resulting collisions. This is possible since the various particles constituting the final state of the system (state after the collision) are
detected and their characteristics (direction of emissiom, energy, etc.) are measured.
Aim of such experiments: Determine the
interactions that occur between the various
particles entering the collision.
The collisions give rise to reactions:
(1) + (2) → (3) + (4) + (5) + . . . .
In quantum mechanics, we can only speak
of probabilities of the possible states coming out of the collision. Among the reactions possible under given conditions, scattering reactions are defined as those in
which the final state and the initial state
are composed of the same particles (1) and
(2). In an elastic scattering reaction, none
of the particles’ internal states change during the collision.
1.2
Figure 1.1: The collision experiment
Elastic scattering
Let us focus on the elastic scattering of the incident particles (1) by the target particles
(2).
→ In classical mechanics, we would determine the deviations in the incident particles’
trajectories (1) due to the forces exerted by particles (2).
1
2
CHAPTER 1. SCATTERING THEORY
→ In quantum mechanins, we study the evolution of the wave function associated with
the incident particles under the influence of their interactions with the target particles,
i.e., “scattering” of particles (1) by particles (2).
In order to analyze the scattering process, we will consider the following simplifying
hypotheses:
1. we assume that particles (1) and (2) have no spin;
2. we do not consider the internal structure of particles (1) and (2); this assumption
excludes therefore inelastic scattering phenomena, where part of the kinetic energy
of (1) is absorbed in the final state by the internal degrees of freedom of (1) and
(2);
3. we assume that the target is thin enough to neglect multiple scattering processes,
i.e., processes during which a particular incident particle is scattered several times
before leaving the target;
4. we neglect coherence between waves scattered by the different particles of the target;
this is justified if the spread of the wave packet associated with (1) is small compared
to the average distance of particles in (2). An example that we are neglecting is
the coherent scattering by a crystal (Bragg diffraction). We concentrate on the
scattering of particle (1) of the beam by particle (2) of the target;
5. we assume that the interactions between particles (1) and (2) can be described by
a potential energy:
V (~r1 − ~r2 ) = V (~r).
In the center of mass reference,
1
1
1
+
=
m
m1 m2
→
m : relative mass,
the problem reduces then to the study of a single particle with mass m scattered by the
potential V (~r).
1.3
Scattering cross section
To introduce the concept of scattering cross section, we count the particles scattered in a
certain direction in space. The process is illustrated in Fig. 1.2, and the following notation
is used:
Oz: direction of the incident particle of
mass m;
dn: number of particles scattered per unit
time into the solid angle dΩ about the direction (θ, φ);
Figure 1.2
3
1.4. CALCULATION OF THE SCATTERING CROSS SECTION
where θ and φ is the angular dependence
in spherical coordinates.
Fi : flux of incident particles; it is the number of particles which traverse per unit
time a unit surface ⊥ to Oz in the region where z takes on very large negative values;
V (~r): potential localized around the origin O;
D: detector situated at a distance from O, which is large compared to the linear dimensions of the potential’s zone of influence, measures the number dn of particles scattered
per unit time into solid angle dΩ.
The fraction of scattered particles should be proportional to the flux of incident particles,
therefore:
dn
is proportional to Fi :
dΩ
dn = Fi σ(θ, φ)dΩ
→
σ(θ, φ): coefficient of proportionality.
(1.1)
Dimensions:
[dn] = T −1
[Fi ] = (L2 T )−1
[σ(θ, φ)] = (L2 )
→ dimensions of a surface
σ(θ, φ) is the differential scattering cross section in the direction (θ, φ). Units:
1 barn = 10−24 cm2
→ The number of particles per unit time which reach the
detector is equal to the number of particles which would
cross a surface dσ = σ(θ, φ)dΩ placed ⊥ to Oz in the incident beam.
Differential
cross
σ(θ, φ) = dσ/dΩ
The total scattering cross section σ:
σ=
Z
section
dΩσ(θ, φ).
Important: The concept of cross section is not limited to the case of elastic scattering.
1.4
Calculation of the scattering cross section
In quantum mechanics, in order to describe the scattering of a given particle by the
potential V (~r), it is necessary to study the time evolution of the wave packet representing
the state of the particle. We have that
4
CHAPTER 1. SCATTERING THEORY
1. the wavefunctions are known at t → −∞ for Oz < 0 far from V (~r) and
2. the evolution of the wave packet can be obtained if it is expressed as a superposition
of stationary states (because V (~r) 6= V (~r, t)),
therefore, we study the eigenvalue equation of the Hamiltonian:
H = H0 + V (~r)
→
H0 =
~2
P
:
2m
describes the particle’s kinetic energy
Stationary scattering states Ψ(~r) are defined as follows.
Ψ(~r, t) = Ψ(~r)e−iEt/~
→
evolution of a particle in the potential V (~r),
where Ψ(~r) is the solution of the eigenvalue equation:
~2
∆ + V (~r) Ψ(~r) = EΨ(~r).
−
2m
(1.2)
We assume that V (~r) decreases faster than 1/r as r approaches ∞ (this hypotheses
excludes the Coulomb potential). Then, if we write the kinetic energy of the incident
particle, E, as
~2 k 2
E=
2m
and the potential, V (~r), as
~2
V (~r) =
U(~r),
2m
eq. (1.2) becomes
[∆ + k 2 − U(~r)]Ψ(~r) = 0.
(1.3)
Boundary conditions: conditions that must be imposed on the solutions of eq. (1.3) if
they are to be used in the description of a scattering process.
1.4.1
Asymptotic form of stationary scattering states.
Scattering amplitude
The wavefunction Ψ~k (~r) representing
the stationary scattering state associated with a given energy, E =
~2 k 2 /2µ, is a superposition of the
plane wave eikz and the scattered wave.
For large r, the scattered wave approches an asymptotic function whose
radial dependence in a given direcikr
tion (θ, φ) is of the form e r (spherical wave).
It is worth noticing
that:
1.4. CALCULATION OF THE SCATTERING CROSS SECTION
5
• it is an outgoing wave which has the same energy as the incident wave;
• the factor 1r results from the fact that we have three spacial directions, and |Ψ|2
must go like r12 to conserve probability:
(∆ + k 2 )eikr
(∆ + k 2 )
is not zero, but
eikr
=0
r
for r ≥ r0 .
where r0 is the range of the potential.
For r → ∞, Ψ~k (~r) therefore behaves as
Ψ~k (~r) ∼ eikz + fk (θ, φ)
eikr
,
r
(1.4)
f (θ, φ): scattering amplitude, depends on the potential V (~r).
Analogous is the scattering of waves in classical mechanics, f.i., water waves encountering
a rock.
1.4.2
Relation between σ(θ, φ) and f (θ, φ)
We define dV as the volume of incident beam that passes through an
area dσ in the time dt with velocity
v, dV = dσ vdt
The whole problem for a scattering process reduces to
determine the scattering amplitude, fk (θ, φ), whose
squared value gives the probability of scattering in a
given direction (θ, φ) and, as we will see, is related to
the differential cross section.
Remembering that
Ψincident = Aeikz ,
and
eikr
,
Ψscattered = A f (θ, φ)
r
the probability that the incident particle, travelling at speed v, passes through the infinitesimal area dσ in time dt,
dP = |Ψincident |2 dV = |A|2(vdt)dσ
6
CHAPTER 1. SCATTERING THEORY
is equal to the probability that the particle scatters into the corresponding solid angle dΩ
(see Fig. 1.2),
|A|2
dP = |Ψscattered |2 dV = 2 |f |2 (vdt)r 2 dΩ .
| {z }
r
dV
It follows that:
dσ = σ(θ, φ)dΩ = |f |2 dΩ,
σ(θ, φ) = |f (θ, φ)|2
→
differential cross section.
The differential cross section (which is what the experimentalist measures) is equal to the
absolute square of the scattering amplitude (which is obtained by solving the Schrödinger
equation). This equation relates directly observation with theory. There are two important techniques for calculating the scattering amplitude:
• partial wave analysis;
• Born approximation.
1.5
Partial wave analysis
~ of the
In the special case of a central potential, V (r), the orbital angular momentum, L,
particle is a constant of motion. Therefore, there exist stationary states with well-defined
~ 2 and Lz . And, as we found in
angular momentum, i.e., eigenstates common to H, L
QM I, the Schrödinger equation of a spherically symmetrical potential V (r) has separable
solutions of the form:
Ψ(r, θ, φ) = R(r)Ylm (θ, φ),
where Ylm (θ, φ) are spherical harmonics and u(r) = rR(r) satisfies the radial equation:
~2 l(l + 1)
~2 d 2 u
u = Eu.
+ V (r) +
−
2m dr 2
2m r 2
where the term proportional to 1/r 2 is the centrifugal potential energy.
We consider first the asymptotics of this equation since in section 1.4 we reasoned how
the wavefunction of the particle should behave at large distances away from the influence
of the potential.
For r → ∞ (scattered region), V (r) → 0 and the centrifugal term goes to 0. Then
d2 u
≈ −k 2 u,
dr 2
has the general solution:
ikr
−ikr
u(r) = Ce
| {z } + De
| {z } .
outgoing
incoming
7
1.5. PARTIAL WAVE ANALYSIS
An incoming wave to describe the scattered wave makes no sense and therefore D = 0.
Then, at very large r,
R(r) =
u(r)
eikr
∼
r
r
(valid for kr ≫ 1)
what in optics is called as the radiation zone.
Since we assume that the potential is localized, in the intermediate region the equation is:
d2 u l(l + 1)
−
u = −k 2 u
dr 2
r2
(V can be ignored but not the centrifugal term)
Please, note that this cannot be applied to the Coulomb
potential since 1/r goes to zero slowlier than 1/r 2 as
r → ∞ and the centrifugal term does not dominate this region. In this sense, the Coulomb potential is not localized, and partial wave analysis is inappropriate. Other potentials, like quantum hard-sphere
scattering or a finite depth potential, fulfill this condition.
In the intermediate region the solution of the equation:
d2 u l(l + 1)
−
u = −k 2 u
dr 2
r2
The centrifugal term
is a linear combination
u(r) = Arjl (kr) + Brnl (kr)
of the spherical Bessel functions (see Arfken,mathematical Methods):
l
sin x
1 d
l
,
jl (x) = (−x)
x dx
x
l
cos x
1 d
l
nl (x) = −(−x)
,
x dx
x
8
CHAPTER 1. SCATTERING THEORY
and
sin x
cos x
, n0 (x) = −
.
x
x
Since none of them represents an outgoing (or incoming) wave and we should have the
correct asymptotics for the scattering process, we consider spherical Hankel functions
which are the linear combinations of j0 (x) and n0 (x):
j0 (x) =
(1)
hl (x) ≡ jl (x) + inl (x),
(2)
hl (x) ≡ jl (x) − inl (x),
(1)
h0 = −i
(1)
hl
(2)
hl
eix
,
x
e−ix
(2)
h0 = i
,
x
1 ix
i
(1)
e ,
h1 = − 2 −
x
x
i
1 −ix
(2)
hl =
e ,
−
x2 x
)
→ x1 (−i)l+1 eix
for x >> 1.
→ x1 (i)l+1 e−ix
For outgoing waves, we should choose the solution (boundary conditions):
(1)
R(r) ∼ hl (kr),
which gives the correct asymptotics and the wavefunction outside the scattering region
(V (r) = 0) is:












X
(1)
m
ikz
cl,m hl (kr)Yl (θ, φ) .
Ψ(r, θ, φ) = A
e
+
|{z}




l,m

incident plane wave


|
{z
}


scattered wave
We assume that the potential is spherically symmetric, and Ψ is independent of φ, which
means that m = 0 (Ylm ∼ eimφ ). Then,
r
2l + 1
Pl (cos θ),
Yl0 (θ, φ) =
4π
Pl : the lth Legendre polynomial.
We redefine the expansion coefficients as:
cl,0 ≡ il+1 k
p
4π(2l + 1)al .
9
1.5. PARTIAL WAVE ANALYSIS
Then
(
ikz
Ψ(r, θ) = A e
+k
∞
X
l+1
i
(1)
1)al hl (kr)Pl (cos θ)
(2l +
l=0
)
,
al : lth partial wave amplitude.
For r → ∞,
(1)
hl (kr) ∼ (−i)l+1
then
eikr
;
kr
eikr
ikz
,
Ψ(r, θ) ≈ A e + f (θ)
r
and
f (θ) =
P∞
l=0 (2l
+ 1)al Pl (cos θ)
→
scattering amplitude, f (θ), in terms
of the partial wave amplitude, al
These results confirm more rigorously the general structure of the wavefunctions postulated previously.
The differential cross section σ(θ) is then:
σ(θ) = |f (θ)|2 =
XX
l
(2l + 1)(2l′ + 1)a∗l al′ Pl (cos θ)Pl′ (cos θ),
l′
and the total cross section: σ = 4π
P∞
l=0 (2l
+ 1)|al |2 ,
where we used the orthogonality of the Legendre polynomials:
Z
1
−1
Pl (x)Pl′ (x)dx =
2
δll′ .
2l + 1
What to do next? : All that remains is to determine the partial wave amplitudes, al , for
the potential in question. This is accomplished by solving the Schrödinger equation in
the interior region (where V (r) is distinctly non-zero) and matching this to the exterior
solution using the appropriate boundary conditions.
Now, a problem with notation arises since we have:
• spherical coordinates for the scattered wave;
• cartesian coordinates for the incident wave.
10
CHAPTER 1. SCATTERING THEORY
Let’s rewrite the wavefunction in a more consistent notation: eikz satisfies the Schrödinger
equation with V = 0; on the other hand, the general solution to the Schrödinger equation
with V = 0 can be written as:
X
[Al,m jl (kr) + Bl,m nl (kr)] Ylm (θ, φ).
l,m
Therefore, let us express eikz in this way:
• z = r cos θ has no φ dependence, only m = 0 occurs for Ylm ;
• eikz is finite at z = 0 therefore nl (kr) is not allowed since it diverges at r = 0.
Therefore,
ikz
e
=
∞
X
il (2l + 1)jl (kr)Pl (cos θ)
l=0
→
Rayleigh’s formula
Then, the wave function in the exterior region of the potential can be expressed entirely
in terms of r and θ:
i
h
P∞ l
(1)
Ψ(r, θ) = A l=0 i (2l + 1) jl (kr) + i k al hl (kr) Pl (cos θ).
1.5.1
Example: Quantum hard-sphere scattering
We consider the following potential:
∞ for r ≤ b;
V (r) =
0 for r > b.
And we want to calculate the scattering cross section for elastic
scattering under the influence of this potential.
Boundary conditions:
Ψ(b, θ) = 0
The potential is central, therefore we consider the partial wave
analysis. Then,
Ψ(b, θ) = 0 =
∞
X
l=0
h
i
(1)
il (2l + 1) jl (kb) + i k al hl (kb) Pl (cos θ) = 0,
al = i
jl (kb)
(1)
khl (kb)
,
and the total cross section is:
∞
jl (kb) 2
4π X
.
(2l + 1) (1)
σ= 2
k l=0
h (kb) 1.5. PARTIAL WAVE ANALYSIS
11
What do we learn out of that?
First, let’s consider the low-energy scattering limit: kb ≪ 1.
In this limit, the wavelength is much greater than the radius of the sphere:
l 2
jl (x)
2 l!
jl (x)
2l l!xl /(2l + 1)!
i
jl (x)
=
≃ −i
≃ −i
=
x2l+1 ,
(1)
−l−1
l
j
(x)
+
in
(x)
n
(x)
−(2l)!x
/2
l!
2l
+
1
(2l)!
l
l
l
hl (x)
and hence,
l 4
∞
2 l!
4π X 1
σ≈ 2
(kb)4l+2 .
k
2l + 1 (2l)!
l=0
Since kb ≪ 1, the higher powers are negligible, and only l = 0 will be significant. This
means that the differential cross section for low-energy hard-sphere scattering is independent of θ, as in the classical case:
σ = 4πb2 .
Remarks:
• the scattering cross section is four times the geometrical cross section;
• σ is the total surface area of the sphere.
This “larger effective size” of σ is characteristic of long-wavelength scattering (this is also
in optics); the waves “feel” their way around the whole sphere, whereas classical particles
only see the head-on cross section.
1.5.2
Phase shift
Phase shift is a very useful variable in scattering theory since it often reduces the complexity of the calculations of scattering processes.
In order to introduce this concept in scattering theory,
let’s consider a one-dimensional scattering of a particle from a localized potential V (x) on the half line
x < 0:
V =
V (x) for − a < x < 0;
∞
for x = 0.
(bounded on the right
by an infinite well)
Incident wave coming from −∞:
Ψi (x) = Aeikx (x < −a),
reflected wave:
Ψr (x) = Be−ikx (x < −a).
Independently of what happens in the interaction region (−a < x < 0), the amplitude of
the reflected wave has got to be the same as that of the incident wave (conservation of
probability), but it need not have the same phase.
12
CHAPTER 1. SCATTERING THEORY
If there were no potential at all, V (x) = 0 (only the well ∞ at x = 0), then B = −A,
we would have total reflection, since the total wavefunction must vanish at the origin
(Ψ(0) = 0) and:
Ψ0 (x) = A eikx − e−ikx .
For V (x) 6= 0,
Ψ(x) = A eikx − ei(2δ−kx) for x < −a.
The whole theory of scattering reduces to the problem of calculating the phase shift δ for
a specific potential.
The advantage of working with phase shift (as opposed to calculating the complex amplitude B) is that it makes use of the physics:
• conservation of probability: all the potential can do is to shift the phase of the
reflected wave;
• simplifies mathematics extremely.
1.5.3
Phase shift for the 3-dimensional case
The incident plane wave, Aeikz , carries no angular momentum (m = 0) but includes all
values of the total angular momentum (l = 0, 1, 2, . . .).
In a spherically symmetrical potential, angular momentum is conserved. Each partial
wave (labelled with l) scatters independently with no change of amplitude, only in phase.
When V (r) = 0, Ψ0 = Aeikz and the lth partial wave is:
(l)
Ψ0 = Ail (2l + 1)jl (kr)Pl (cos θ);
where we used the Rayleigh formula
(eikz =
∞
X
il (2l + 1)jl (kr)Pl (cos θ)).
l=0
For x = kr ≫ 1,
jl (x) =
i
1 h (1)
1 (2)
hl (x) + hl (x) ≈
(−i)l+1 eix + il+1 e−ix ,
2
2x
and the lth partial wave:
(l)
Ψ0 ≈ A
(2l + 1) ikr
e − (−1)l e−ikr Pl (cos θ).
2ikr
For V (r) = 0 the first term in the square brackets represents the outgoing wave and the
second term the incoming wave. When V (r) 6= 0 the first term picks up a phase shift δl :
Ψ(l) ≈ A
(2)
The hl
(2l + 1) i(kr+2δl )
e
− (−1)l e−ikr Pl (cos θ).
2ikr
(1.5)
component in eikz is phase shifted 2δl and emerges as an outgoing spherical wave.
1.6. THE BORN APPROXIMATION
13
Connection between phase shift δl , partial wave amplitude al and scattering
cross section σ:
We had as general wavefunction in the exterior region:
Ψ(r, θ) = A
∞
X
l=0
h
i
(1)
i (2l + 1) jl (kr) + ikal hl (kr) Pl (cos θ).
l
we consider the asymptotic behavior of eq. (1.6):
2l + 1 ikr
(2l + 1) ikr
(l)
l −ikr
Ψ ≈A
e − (−1) e
+
Pl (cos θ).
al e
2ikr
r
(1.6)
(1.7)
Comparing eq. (1.5) and eq. (1.7):
al =
It follows that:
and
1
1
e2iδl − 1 = eiδl sin δl .
2ik
k
∞
1X
(2l + 1)eiδl sin δl Pl (cos θ),
f (θ) =
k l=0
∞
4π X
(2l + 1) sin2 δl .
σ= 2
k l=0
The advantage of working with phase shifts (as opposed to partial wave amplitudes) is
that they are easier to interpret physically. δl exploits conservation of angular momentum
and reduces solving a complex quantity al to a simple real one δl .
1.6
The Born approximation
We consider now the Born approximation which is suitable for scattering processes where
the scattering potential is weak. First we introduce the integral form of the Schrödinger
equation.
1.6.1
Integral form of the Schrödinger equation
The stationary Schrödinger equation,
−
~2 2
∇ Ψ + V Ψ = EΨ,
2m
can be written (analogous to the Helmholtz equation) as:
∇2 + k 2 Ψ = Q,
14
CHAPTER 1. SCATTERING THEORY
where
k≡
√
2mE
,
~
and Q ≡
2mV
Ψ.
~2
Note that it has only the form of the Helmholtz equation since Q depends on Ψ.
We remind about the concept of a Green’s function, G(~r):
G(~r) is the solution of the Helmholtz equation,
∇2 + k 2 G(~r) = δ (3) (~r),
(1.8)
when the source is a delta function. It can be shown that Ψ can be expressed in terms of
G as an integral:
Ψ(~r) =
Z
G(~r − ~r0 )Q(~r0 ) d3r0 ,
which satisfies the Schrödinger equation:
Z
2
2
2
∇ + k Ψ(~r) =
∇ + k 2 G(~r − ~r0 ) Q(~r0 ) d3 r0
Z
=
δ (3) (~r − ~r0 ) Q(~r0 ) d3~r0 = Q(~r),
The Green’s function for the Helmholtz equation, G(~r − ~r0 ), is the “response” to a delta
function “source”.
1.6.2
Solving the Green’s function
The Fourier transform of G(~r),
1
G(~r) =
(2π)3/2
Z
ei~s·~r g(~s) d3s,
(1.9)
transforms the differential equation Eq. 1.8 into an algebraic equation. g(~s) can be now
obtained by multiplying both sides of Eq. (1.9) by (∇2 + k 2 ):
Z
2
1
2
2
∇ + k G(~r) =
∇ + k 2 ei~s·~r g(~s) d3s.
3/2
(2π)
Then, using that
and
∇2 ei~s·~r = −s2 ei~s·~r ,
Z
1
(3)
ei~s·~r d3 s,
δ (~r) =
(2π)3
∇2 + k 2 G(~r) = δ (3) (~r),
15
1.6. THE BORN APPROXIMATION
we arrive at:
1
(2π)3/2
Z
2
2
i~s·~
r
(−s + k ) e
1
g(~s) d s =
(2π)3
3
Z
e~s·~r d3 s,
and therefore:
g(~s) =
1
.
(2π)3/2 (k 2 − s2 )
We now introduce g(~s) into eq. (1.9):
1
G(~r) =
(2π)3
Z
ei~s·~r
(k 2
1
d3 s.
− s2 )
and solve the integral for G(~r). We choose spherical coordinates (s, θ, φ), with the polar axis along ~r as shown in the
figure. With this choice of the coordinate frame,
Coordinate frame
~s · ~r = s r cos θ,
Z
d3 s = dφ s2 sin θ dθ ds,
Z
dφ = 2π,
π
isr cos θ
e
0
Thus,
1 2
G(~r) =
(2π)2 r
Z
0
π
2 sin (sr)
eisr cos θ =
.
sin θ dθ = −
isr 0
sr
∞
1
s sin (sr)
ds
=
k 2 − s2
4π 2 r
We consider the relation:
sin x =
Z
+∞
−∞
s sin (sr)
ds .
k 2 − s2
(1.10)
eix − e−ix
,
2i
and we rewrite eq. (1.10) as:
Z +∞
Z +∞
s eisr
i
s e−isr
ds −
ds
G(~r) =
8π 2 r
−∞ (s − k)(s + k)
−∞ (s − k)(s + k)
i
(I1 − I2 ).
=
8π 2 r
(1.11)
We can now solve the integrals I1 and I2 by using the Cauchy’s integral formula:
I
f (z)
2πif (z0 ) for z0 within the contour;
(z → complex variable)
dz =
0
otherwise.
z − z0
16
CHAPTER 1. SCATTERING THEORY
For Eq. 1.11 the integration is along the real
axis, and it passes right over the pole singularity at ±k. One can decide different ways
to skirt the poles. We choose to go over the
pole at −k and under the one at +k (some
other conventions will give different Green’s
Skirting the poles in the contour integral
functions, but they are all equally valid). For
each integral in eq. (1.11), we have to “close
the contour” in such a way that the semicircle
at ∞ doesn’t contribute. Thus, in the integral I1 ,
Z +∞
s eisr
I1 =
ds,
−∞ (s − k)(s + k)
eisr goes to zero when s has a large positive imaginary part:
ei(Re (s)+i Im (s))r ,
and therefore we close the integral from above. The contour encloses only the singularity
at s = +k and:
I1 =
For the integral I2 ,
I isr 1
se
s eisr
= iπeikr .
ds = 2πi
s+k s−k
s + k s=k
I2 =
Z
+∞
−∞
−isr
s e−isr
ds,
(s − k)(s + k)
the factor e
goes to zero when s has a large negative imaginary part, and therefore we
close from below :
and the contour encloses the singularity at s = −k. Since it goes around in the clockwise
direction, we pick up a (−) sign:
−isr I −irs se
1
se
= −iπeikr .
I2 = −
ds = −2πi
s−k s+k
s − k s=−k
17
1.6. THE BORN APPROXIMATION
Then,
G(~r) =
eikr
i ikr
ikr
−
.
iπe
−
−iπe
=
8π 2 r
4πr
(1.12)
This is the expression for the Green’s function for the Helmholtz equation.
We can add to G(~r) any function, G0 (~r), that satisfies the homogeneous Helmholtz equation,
∇2 + k 2 G0 (~r) = 0,
and the result, (G + G0), will still satisfy the initial equation, which reflects the ambiguity
of how to skirt the poles: a different choice of how to skirt the poles amounts to picking
a different G0 (~r).
With the explicit expression for G(~r), (eq. (1.12)), the general solution of the Schrödinger
equation,
√
2m
2mE
2
2
∇ + k Ψ = Q, with Q ≡ 2 V Ψ and k ≡
,
~
~
in then:
m
Ψ(~r) = Ψ0 (~r) −
2π~2
Z
eik|~r−~r0|
V (~r0 )Ψ(~r0 )d3 r0
|~r − ~r0 |
(1.13)
where Ψ0 satisfies the free-particle Schrödinger equation. This equation is the Schrödinger
equation in integral form.
1.6.3
The first Born approximation
We will now make use of the previous equation to derive the Born approximation.
We assume that V (~r0 ) is localized around ~r0 = 0 and calculate Ψ(~r) at points far away
from the scattering center. Since |~r| >> |~r0 |,
~r · ~r0
2
2
2
2
∼
,
|~r − ~r0 | = r + r0 − 2~r · ~r0 = r 1 − 2 2
r
18
CHAPTER 1. SCATTERING THEORY
and hence
with r̂ =
~r
and where we used:
r
|~r − ~r0 | ∼
= r − r̂ · ~r0 .
x=
Then
r
1−2
2~r · ~r0
.
r2
~r · ~r0 √
x
1
−
x
≃
1
−
=
r2
2
(for x << 1).
We define:
~k ≡ kr̂ ;
then,
~
eik|~r−~r0| ∼
= eikr e−ik·~r0
and therefore,
eik|~r−~r0 | ∼ eikr −i~k·~r0
e
.
=
|~r − ~r0 |
r
In a scattering process the incident wave is:
Ψ0 (~r) = Aeikz ,
which is the solution of the free-particle Schrödinger equation.
For large r and using the integral form of the Schrödinger equation, the total wavefunction
outside the region of the potential is:
Z
m eikr
~
ikz
∼
Ψ(~r) = Ae −
e−ik·~r0 V (~r0 )Ψ(~r0 )d3 r0 ,
2
2π~ r
which can be compared to the equation:
eikr
ikz
Ψ(r, θ, φ) ≈ A e + f (θ, φ)
r
so that,
m
f (θ, φ) = −
2π~2 A
This result is exact.
Z
(large r),
~
e−ik·~r0 V (~r0 )Ψ(~r0 )d3 r0 .
Now, we invoke the Born approximation which corresponds to a weak potential approximation. In the Born approximation, one assumes that the incoming plane wave is not
substantially altered by the potential. Then:
~′
Ψ(~r0 ) ≈ Ψ0 (~r0 ) = Aeikz0 = Aeik ·~r0 ,
with ~k ′ ≡ kẑ .
19
1.6. THE BORN APPROXIMATION
i.e we assume that the wavevector keeps the direction of ~z, while the norm |~k ′ | = k is the
norm of the incident wavevector since we have energy conservation.
This would be the exact wave function, if V were zero. Otherwise, it is a weak potential
approximation.
In the Born approximation, f (θ, φ) then becomes:
Z
m
~′ ~
∼
ei(k −k)·~r0 V (~r0 )d3 r0 .
f (θ, φ) = −
2
2π~
While the partial wave analysis is useful when the incident particle has low energy (since
only the first few terms of the series contribute significantly), the Born approximation applies when the potential is weak compared to the incident energy or, in other words, when
the deflection is small.
• for the low energy scattering (long wavelengths), the exponential factor is essentially
constant over the scattering region, and the
Born approximation simplifies to:
Z
m
V (~r0 )d3 r0 .
f (θ, φ) ≃ −
2π~2
• for a spherically symmetric potential, V (~r) =
V (r), the Born approximation can also be reduced. In this case we have:
- ~k′ points in the incident direction - ~k points in the scattered direction - ~(~k′ − ~k) = ~q: momentum transfer θ
- 2ksin = q
2
~q ≡ ~k ′ − ~k,
and
(~k ′ − ~k) · ~r0 = qr0 cos θ0 .
We define the polar axis for ~r0 along ~q. Then,
Z
m
f (θ) ≃ −
eiqr0 cos θ0 V (r0 )r02 sin θ0 dr0 dθ0 dφ0.
2π~2
Since
we obtain:
1.6.4
Z
Z
π
2m
f (θ) ≃ − 2
~q
Z
dφ0 = 2π
and
eiqr0 cos θ0 sin θ0 dθ0 =
0
2 sin(qr0 )
,
qr0
∞
r0 V (r0 ) sin(qr0 )dr0
0
Example: Yukawa potential
(1.14)
20
CHAPTER 1. SCATTERING THEORY
The Yukawa potential,
V (r) = α2
e−µr
,
r
(1.15)
describes, in a very crude way, the binding force in an
atomic nucleus. It was introduced by Yukawa in 1938
to describe the nuclear interaction between protons and
neutrons due to the pion exchange. In 1940 Yukawa got
the Nobel Prize for predicting the existence of pions.
V0
1
Often this potential is written with α2 =
and corµ
µ
responds to the range of the potential. V (r) goes to zero
very rapidly for r >> 1/µ.
We first want to understand where the expression Eq. 1.15 comes from. Let us consider
the electrostatic potential about a charged point particle:
e
Φ=
4πǫ0 r
which fulfills the Laplace equation
∇2 Φ = 0
Φ describes the potential for a force mediated by massless particles, the photons and it
has long-ranged behavior.
We consider now a relativistic particle with mass m, momentum p and energy E, then:
E 2 = p2 c2 + m2 c4
This equation can be converted into a wave equation by using the correspondence principle:
2
∂
∂
2∂ Φ
2 4
2 2 2
−~
.
= (m c − ~ c ∇ )Φ
E → i~ , px → −i~
∂t2
∂t
∂x
Φ is the potential field (pion field in our case).
In the stationary case,
which reduces to:
m2 c2
2
∇ − 2
Φ = 0,
~
∇2 Φ = 0,
for the massless case.
For a point source with spherical symmetry:
∇2 Φ →
1 d2
(rΦ),
r dr 2
and the differential equation is then:
m2 c2
d2
(rΦ)
=
rΦ,
dr 2
~2
21
1.6. THE BORN APPROXIMATION
with the solution:
e−µr
,
r
~
1
is the range of the potential. This potential
where α is the coupling strength and =
µ
mc
has the form of a screened Coulomb interaction.
Φ = α2
We assume now that α is sufficiently small for the Born approximation Eq. 1.14 to be
valid. The scattering amplitude f (θ) is then:
2mα2
f (θ) ≃ − 2
~q
Z
∞
0
e−rµ sin(qr)dr = −
to perform this integral we used:
Z ∞
Im[
e−µr eiqr dr] = −Im
0
we have that
2mα2
~2 (µ2 + q 2 )
q
1
= 2
−µ + iq
µ + q2
(1.16)
(1.17)
θ
= 2k 2 (1 − cosθ)
2
Then, in the first Born approximation, the differential cross section for scattering by a
Yukawa potential is given by:
q 2 = 4k 2 sin2
2mα2 2
1
) 2
(1.18)
2
2
~
[µ + 2k (1 − cosθ)]2
→ Note that as µ → 0, the Yukawa potential is reduced to the Coulomb potential,
provided that α2 is fixed, f.i. α2 = ZZ ′ e2 in the limiting process. Then the differential
cross section in the first Born approximation becomes:
σ(θ) = (
σ(θ) =
(2m)2 (ZZ ′ e2 )2
1
4
4
~
16k sin4 (θ/2)
(1.19)
which is the Rutherford scattering cross section that can be obtained classically.
The term
2mα2
2mα2
=
−
−
.
2 2
m2 c2 + ~2 q 2
~2 ( m~2c + q 2 )
can be interpreted as a function of the propagator term, which forr c = 1 and ~ = 1 is:
q2
1
+ m2
It arises from the exchange of a virtual boson (pion) whose rest mass is m.
1.6.5
The Born series
22
CHAPTER 1. SCATTERING THEORY
In order to understand the Born approximation, one may compare it
with the impulse approximation in
classical scattering theory. In the impulse approximation, it is assumed
that the particle after scattering continues undeflected. The delivered
transverse momentum is calculated
as:
Z
I = F⊥ dt.
θ∼
= tan−1 (I/p) for small θ
p: incident momentum
In the zeroth-order Born approximation, the incident plane wave passes by with no modifications; in the first-order Born approximation, the model is improved by introducing
the first-order correction. This idea can be iterated to generate a series of higher-order
corrections, which should converge to the exact value.
Let us recall the integral form of the Schrödinger equation derived earlier:
Z
m eikr
Ψ(~r) = Ψ(~r0 ) + g(~r − ~r0 )V (~r0 )Ψ(~r0 )d3 r0 , with g(~r) = − 2π~
,
(1.20)
2 r
Ψ0 : incident wave;
g(~r): the Green’s function (multiplied by the factor 2m/~2 ), g(~r) =
V : scattering potential.
2G(~r)m
;
~2
Schematically, Eq. (1.20) can be written as:
Ψ = Ψ0 +
Z
gV Ψ.
Now, we take this expression for Ψ and plug it into the integral sign of its own:
Z
Z Z
Ψ = Ψ0 + gV Ψ0 +
gV gV Ψ.
In this way, we generate a formal series:
Z
Z Z
Z Z Z
Ψ = Ψ0 + gV Ψ0 +
gV gV Ψ0 +
gV gV gV Ψ0 + . . . .
(1.21)
1.7. COLLISIONS WITH ABSORPTION
23
In each integrand of Eq. (1.21), only the incident wavefunction, Ψ0 , appears together
with more and more powers of gV . Diagrammatically, it can be represented as:
zeroth order:
first order:
second order:
Ψ0 is untouched by the potential;
Ψ0 is ”kicked”’ once and then ”propagates” out in some new direction;
Ψ0 is kicked, propagates to a new location, is kicked again, and then propagates
out, and so on.
In the context of the Born series, the Green’s function is called the propagator (the
solution we found for the Yukawa potential was g in momentum space). It describes how
the disturbance propagates between one interaction and the next one.
The Born series was the original source for Feynman’s formulation of relativistic quantum
mechanics which is expressed in terms of vertex factors, V , and propagators, g, connected
together in Feynman’s diagrams.
1.7
Collisions with absorption
Up to now we considered elastic scattering of particles. Collisions between particles can
also be inelastic and lead to other reactions (creation or destruction of particles, etc.),
particularly, if the energy of the incident particles is high. During such processes, some
particles ”disappear”: they are said to be ”absorbed”.
In the partial wave method, the effect of the potential is to multiply the wavefunction by
e2iδl . For elastic scattering, δl is real since:
• the amplitude of the outgoing wave is equal to that of the incoming wave;
• the probability is conserved during the scattering;
• the total number of particles is constant.
This suggests to consider the absorption phenomena by giving the δl an imaginary part,
we define ηl :
ηl = e2iδl , with |ηl | < 1.
e2i(Reδl +iImδl ) = e2iReδl e−2Imδl
and
|ηl |2 = |e−2Imδl |2
24
CHAPTER 1. SCATTERING THEORY
Then, we can define the absorption cross section as:
σabs
∞
π X
(2l + 1)[1 − |ηl |2 ].
= 2
k
l=0
If |ηl |2 = 1 there is no absorption.
We had that the elastic cross section was (now expressed in terms of ηl ):
σel =
∞
π X
(2l + 1)|1 − ηl |2 ,
k 2 l=0
therefore, the total cross section will be:
∞
σtot = σel + σabs
2π X
= 2
(2l + 1)[1 − Re ηl ].
k l=0
(1.22)
The total cross section is the number of particles which, per unit time, participate in one
or another of the possible reactions, divided by the incident flux.
Optical Theorem:
For θ = 0, we are considering the scattering in the forward direction.
f (0) =
∞
X
(2l + 1)al
l=0
The imaginary part of the elastic scattering amplitude in the forward direction is:
Imf (0) =
∞
X
(2l + 1)Imal
l=0
with
al =
1
(ηl − 1)
2ik
Then,
∞
Im fk (0) =
1X
1 − Re ηl
(2l + 1)
k
2
(1.23)
l=0
Comparing Eq. 1.22 and Eq. 1.23 we have:
σtot =
4π
Im fk (0).
k
which is the optical theorem.
It is the interference in the forward direction between the incident plane wave and the
scattered wave which accounts for the attenuation of the transmitted beam due to the
scattering of particles in all directions of space.
25
1.8. FORMAL SCATTERING THEORY
1.8
Formal scattering theory
So far we considered the scattering problem in the position representation. We represented
the incoming particle by a plane wave, and, since this is an eigenstate of the free-particle
hamiltonian,
p2
Ψ = EΨ,
2m
and the collision potential is time independent, we were able to reduce the problem to the
stationary case.
We will consider now a more abstract representation, which is more general that what we
did up to now.
1.8.1
Propagator theory and the Lippmann-Schwinger equation
As we have seen, the scattering problem consists of finding the solution of the Schrödinger
equation with particular boundary conditions for r → ∞, namely, an incoming wave and
an angle dependent scattered wave, Ψ = Ψin +Ψscatt , for r → ∞. The problem is separated
into:
1. a general solution of the Schrödinger equation with open boundary conditions;
2. a particular solution which specifies the appropriate boundary conditions.
This can be achieved systematically with the propagator theory.
General solution
As we are interested in the stationary problem, we consider the Schrödinger equation
with fixed E. The resolvent operator (≡ Green’s operator ≡ inverse operator) of the
Schrödinger equation is defined as the solution of the following Schrödinger equation:
(E − H)Ĝ = 1
←
representation-free form
(1.24)
H = H0 + V ;
~2
P
: the free Hamiltonian;
H0 = 2m
V : scattering potential;
1: unit operator in Hilbert space.
Note: we will come to the known equation for the Ĝ with the δ function once we choose
a representation.
For the free system, the Green’s operator is Ĝ0 :
(E − H0 )Ĝ0 = 1.
26
CHAPTER 1. SCATTERING THEORY
The full Green’s operator, Ĝ, can be expressed in terms of Ĝ0 by multiplying eq. (1.24)
from the left with Ĝ0 :
Ĝ0 (E − H)Ĝ = Ĝ0 1,
Ĝ0 E Ĝ − Ĝ0 H Ĝ = Ĝ0 ,
Ĝ0 E Ĝ − Ĝ0 (H0 + V )Ĝ = Ĝ0 ,
Ĝ0 (E − H0 ) Ĝ − Ĝ0 V Ĝ = Ĝ0 ,
{z
}
|
1
Ĝ = Ĝ0 + Ĝ0 V Ĝ .
Particular solution
Let’s now find the particular solution, |Ψi, for the scattering boundary problem:
(a) (E − H0 ) |Ψi = V |Ψi
(b) (E − H0 ) |Φ0 i = 0 →
solution of the free problem
By substracting (b) from (a), we get:
(E − H0 ) |Ψi − (E − H0 ) |Φ0 i = V |Ψi ,
which is then to be multiplied by Ĝ0 from the left side:
Ĝ0 (E − H0 ) (|Ψi − |Φ0 i) = Ĝ0 V |Ψi ,
{z
}
|
1
|Ψi = |Φ0 i + Ĝ0 V |Ψi .
Formally,
1
Ĝ0 =
(E − H0 )
(E − H0 )Ĝ0 = 1 .
(1.25)
(1.26)
Finally, combining eq. (1.25) and (1.26), we arrive at:
|Ψi = |Φ0 i +
1
V |Ψi ,
E − H0
which is called the Lippmann-Schwinger equation.
Remarks:
• This equation is very similar to the one describing the time evolution of a state in
the context of time-dependent perturbation theory. The only difference is that here
we consider the stationary case only. Both are called Lippmann-Schwinger equation;
• the free solution, |Φ0 i, accounts for the incoming plane wave Ψin ;
27
1.8. FORMAL SCATTERING THEORY
• the equations
Ĝ = Ĝ0 + Ĝ0 V Ĝ
and
|Ψi = |Φ0 i +
1
V |Ψi
E − H0
differ from each other in that in the latter the boundary conditions are implemented
via |Φ0 i.
Both equations generate a perturbation series for Ĝ and |Ψi, respectively, via iterations:
Ĝ = Ĝ0 + Ĝ0 V Ĝ0 + Ĝ0 V Ĝ0 V Ĝ0 + . . .
= Ĝ0 + Ĝ0 T Ĝ0
= Ĝ0 S Ĝ0 ,
T : = V + V Ĝ0 V + V Ĝ0 V Ĝ0 V + . . .
S : = Ĝ−1
0 +T
= E − H0 + T
→
→
T -matrix
S-matrix
Since the matrix elements of S are transition probabilities between incoming and outgoing states, S has a direct relation to quantities that are measured in scattering theory,
e.g., scattering amplitude and scattering cross section. With boundary conditions, the
wavefunctions can be expressed as:
|Ψi = |Φi + Ĝ0 V |Φ0 i + Ĝ0 V Ĝ0 V |Φ0 i + . . .
= |Φ0 i + Ĝ0 T |Φ0 i .
1.8.2
Meaning of the Lippmann-Schwinger equation
In the Lippmann-Schwinger equation,
|Ψi = |Φ0 i +
1
V |Ψi ,
E − H0
|Ψi reduces to |Φ0 i as V vanishes. However, without prescriptions for dealing with a sin1
gular operator E−H
, it has no meaning. Then, in order to make the Lippmann-Schwinger
0
equation more workable, one uses the trick of making E slightly complex :
(±) Ψ
= |Φ0 i +
1
V
E−H0 ±iη
(±) Ψ
.
This is the conventional Lippman-Schwinger equation. In a moment, we will discuss the
meaning of ±.
28
CHAPTER 1. SCATTERING THEORY
1.8.3
Position and momentum representation
Position representation
The Lippmann-Schwinger equation is a ket equation independent of a particular representation. We now consider the position basis by multiplying it with h~r| from the left:
Z
(±) 1
h~r Ψ
= h~r |Φ0 i + d3 r ′ h~r|
|~r ′ i h~r ′ | V Ψ(±) .
E − H0 ± iη
This is an integral equation for scattering because the unknown ket Ψ(±) appears under
an integral sign. If |Φ0 i stands for a plane wave state with momentum p~, we can write:
h~r |Φ0 i =
ei~p·~r/~
(2π~)3/2
(not normalizable and is not a vector in Hilbert space).
The normalization is such that:
Z
d3 r h~p ′ |~ri h~r|~pi = δ (3) (~p − ~p ′ ).
Momentum representation
The Lippmann-Schwinger equation in the momentum representation,
1
h~p Ψ(±) = h~p |Φ0 i +
(~p 2 /2m)
E−
|
± iη
}
{z
Ĝ0~p
h~p| V Ψ(±) ,
is derived using the following arguments:
Z
X
d3 p −i~p·~r i~p·~r ′
e
e
= δ (3) (~r − ~r ′ ),
1=
|~pi h~p| →
3
(2π)
p
~
1 = (E − H0 )Ĝ0
→
→
h~p| 1 |~p ′ i = h~p| (E − H0 )
X
p
~ ′′
|~p ′′ i h~p ′′ | Ĝ0 |~p ′ i
X
p~ 2
δp~p~ ′′ Ĝ0~p ′′ ~p ′ .
δp~p~ ′ =
E−
|{z}
2m
′′
p
~ |
{z
}
diagonal
diagonal
Solving for Ĝ0~pp~ ′ :
Ĝ0~pp~ ′ =
1
E−
(~p 2 /2m)
± iη
.
Since Ĝ0~pp~ ′ is diagonal in p-representation, it is often identified with its diagonal element:
Ĝ0~p (E) =
1
E−
(~p 2 /2m)
± iη
.
29
1.8. FORMAL SCATTERING THEORY
1.8.4
Retarded and advanced Green’s functions
Physical significance of the poles in Ĝ:
1
(±)
Ĝ0~p (E) =
E = ~ω,
p2
= ǫp
2m
E−
(±)
p2
2m
± iη
→ Ĝ0~p (~ω) =
E = ǫp are the energy eigenvalues of H!
,
1
,
~ω − ǫp ± iη
To treat the poles, e.g., in the integration over E, they are shifted in the complex plane
E away from the real axis by an infinitesimal positive or negative imaginary part ±iη in
the form:
The relevance of ±iη becomes clear by transforming the Lippmann-Schwinger equation
into the time domain:
Z
(±) (±)
h~p Ψ (t) = h~p |Φ0 (t)i + dt′ Ĝ0~p (t − t′ ) h~p| V Ψ(±) (t′ ) .
(1.27)
(±)
We Fourier transform Ĝ0~p (t − t′ ):
(±)
Ĝ0~p (t
The integral is calculated with
the method of residues,
I
f (z)
dz = 2πif (z0 ),
(z − z0 )
as:
′
Z
dω −iω(t−t′ )
e
Ĝ0~p (~ω)
2π
Z
′
dω e−iω(t−t )
.
=
2π ~ω − ǫp ± iη
−t) =
30
(±)
Ĝ0~p (t − t′ ) =
CHAPTER 1. SCATTERING THEORY
(
i
′
−iθ(t − t′ )e− ~ ǫp (t−t )
i
′
iθ(t − t′ )e− ~ ǫp (t−t )
+ iη;
− iη.
Using this result, we rewrite eq.
(1.27):
Z
(±) h~p Ψ (t) = h~p |Φ0 (t)i +
+∞
−∞
r/a
dt′ Ĝ0~p (t − t′ ) h~p| V Ψ(±) (t′ ) .
Ĝr/a is the retarded/advanced propagator because it propagates the wavefunction from t′
to t with t > t′ (retarded, causal) or t < t′ (advanced, acausal).
Position representation
Let us consider the position basis:
Ĝ(±) (~r, ~r ′ ) ≡
1
~2
h~r|
|~r ′ i ,
2m
E − H0 ± iη
or, explicitly,
′
Ĝ
(±)
1 e±ik|~r−~r |
(~r, ~r ) = −
,
4π |~r − ~r ′ |
′
with E ≡
~2 k 2
.
2m
(1.28)
Remember that we already had such an expression!
Let us derive eq. (1.28).
1
~2
~2
h~r|
|~r ′ i =
2m
E − H0 ± iη
2m
Z
3 ′
dp
Z
d3 p′′ h~r |~p ′ i × h~p ′ |
where H0 acts on h~p ′|. Now,
h~p ′ |
1
E−
′2
p
~
2m
± iη
|~p ′′ i =
′
ei~p ·~r/~
h~r |~p i =
,
(2π~)3/2
′′
′
e−i~p ·~r /~
h~p |~r i =
.
(2π~)3/2
′
E−
p
~ ′2
2m
δ (3) (~p ′ − ~p ′′ )
′2 ,
p
~
E − 2m ± iη
′
′′
1
± iη
|~p ′′ i h~p ′′ | ~r ′ i ,
(1.29)
31
1.8. FORMAL SCATTERING THEORY
The integral in eq. (1.29) thus becomes:
~2
2m
Z
′
′
d 3 p′
ei~p (~r−~r )/~
.
(2π~)3 E − p~ ′2 ± iη
2m
We write
~2 k 2
2m
E=
and set
p~ ′ ≡ ~~q.
We can now perform the integral:
Z ∞
Z 2π
Z +1
′
d(cos θ)ei|~q||~r−~r | cos θ
1
2
q
dq
dφ
(2π)3 0
k 2 − q 2 ± iη
0
−1
Z +∞
′
′ dq q eiq|~r−~r | − e−iq|~r−~r |
1
1
= − 2
8π i|~r − ~r ′ | −∞
q 2 − k 2 ∓ iη
′
1 e±ik|~r−~r |
r/a
= Ĝ0 (~r, ~r ′ ).
= −
′
4π |~r − ~r |
We used:
q = ±k
s
1±
iη
k2
≃ ±k ± iη ′ .
This is the Green’s function of the Helmholtz equation,
∇2 + k 2 Ĝ(±) (~r, ~r ′ ) = δ (3) (~r − ~r ′ ),
2m
h~r Ψ(±) = h~r |Φ0 i − 2
~
Z
′
(±) e±ik|~r−~r |
′
Ψ
d r
h~
r
|
V
,
4π|~r − ~r ′ |
3 ′
which is the equation we have already derived! Eq. 1.13
1.8.5
Interpretation of the Lippmann-Schwinger equation and
its expansion in terms of Feynmann’s diagrams
Ĝ = Ĝ0 +
Z
dr ′′ Ĝ0 (~r, ~r ′′ )V (~r ′′ )Ĝ(~r ′′ , ~r ′ )
32
CHAPTER 1. SCATTERING THEORY