Topics in Advanced Quantum Mechanics

by Barry R. Holstein

Overview

This graduate-level text is based on a course in advanced quantum mechanics, taught many times at the University of Massachusetts, Amherst. Topics include propagator methods, scattering theory, charged particle interactions, alternate approximate methods, and Klein-Gordon and Dirac equations. Problems appear in the flow of the discussion, rather than at the end of chapters. 1992 edition.

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ISBN-13:	9780486315898
Publisher:	Dover Publications
Publication date:	08/21/2013
Series:	Dover Books on Physics
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Pages:	448
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About the Author

Barry R. Holstein is Professor of Physics at the University of Massachusetts, Amherst.

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CHAPTER 1

PROPAGATOR METHODS

I.1 BASIC QUANTUM MECHANICS

The fundamental problem of quantum mechanics is to determine the time development of quantum states. That is, given a state vector | ψ(0)> at time t = 0, what is the state at a later time t – | ψ(t)>? The answer is provided by the Schrödinger equation

(1.1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where H is the Hamiltonian operator. Usually one sees this equation expressed in terms of the coordinate space projection of the state vector — i.e. the wavefunction ψ (x, t) where

(1.2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The time-evolution of the wavefunction is then given by

(1.3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In order to evaluate the matrix element on the right we can insert a complete set of co-ordinate states

(1.4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

yielding

(1.5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Finally we need to interpret the operator matrix element [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. In general, the Hamiltonian H can be written in terms of kinetic and potential energy components as

(1.6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Here [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] so

(1.7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In order to represent the kinetic energy piece we can insert a complete set of momentum states such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then

(1.8) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

yielding

(1.9) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since <x|p> is simply a plane wave we have

(1.10) = eipx

we have

(1.11) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Substitution back into Eq. 1.3 yields

(1.12) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which is the usual version of the Schrodinger equation, where

(1.13) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

provides the representation of the operator H in coordinate space. For a free particle this reduces to the simple form

(1.14) H0(x) = -1/2m [partial derivative]2/[partial derivative]x2.

Time Development Operator

An alternative formulation of this problem is in terms of the time development operator Û (t, t') defined via

(1.15) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

with the boundary condition

(1.16) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

For the case of a free particle, obeying

(1.17) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

the solution for Û(0) (t, 0) is

(1.18) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

(1.19) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is the usual theta function. For example, if

(1.20) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

we find

(1.21) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Although one could straightforwardly evaluate this power series, it is easier to note the identity [Bl 68]

(1.22) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then using

(1.23) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

we find

(1.24) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We note that

(1.25) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which obviously exhibits the canonical spreading experienced by such a wavepacket.

We can equivalently perform the above calculation in momentum space, where the time development operator has the simple form

(1.26) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If

(1.27) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

we have

(1.28) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then

(1.29) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We can return to coordinate space via

(1.30) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which agrees precisely with Eq. 1.24 found via coordinate space methods.

PROBLEM I.1.1

Wave Packet Spreading: A Paradox

It was demonstrated above using the identity

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

that a Gaussian wavepacket

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

evolves in time via

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

[xi] = 1 + i t/2mσ2

Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

σ2(t) = σ2 + t2/4m2σ2

i) Show that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

so that the wavepacket remains normalized to unity but has a width

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which evolves with time. This is simply the usual "spreading" of a quantum mechanical wave packet.

ii) Derive the time evolution of the Gaussian wavepacket without exploiting the identity by using a power series expansion

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

iii) Now suppose that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where N is a normalization constant. Although this functional form may look a bit strange, a little thought should convince one that the wavefunction and all its derivatives are continuous at any point on the real line. However, it is easy to see that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

vanishes for all time if |x| ≥ a since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence, this type of wavepacket apparently does not undergo spreading. Is this assertion correct? If not, where have we made an error in our analysis and what does the actual time evolved wavefunction look like [HoS 72]?

I.2 THE PROPAGATOR

One can evaluate the co-ordinate space matrix element of the time development operator by transforming to momentum space and back again.

(2.1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

DF is usually called the "propagator," as it gives the amplitude for a particle produced at position x at time 0 to "propagate" to position x' at time t.

Just as a check we can verify that this form of the propagator does indeed generate the time development of the freely moving Gaussian wavefunction

(2.2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

in complete agreement with expression derived in sect. I.1

Path Integrals and the Propagator

Before going further, it is useful to note an alternative way by which the propagator can be calculated — the Feynman path integral [FeH 65]

(2.3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where the notation is that the integral represents a sum over all paths x(t) connecting the initial and final spacetime points — x, 0 and x', t respectively. For each path there is a weighting factor given by exp iS/h where S = ∫ dtL[x(t)] is the classical action associated with that path. The path integration can be carried out by dividing the time interval 0 - t into n slices of width ε. This provides a set of times ti spaced a distance e apart between the values 0 and t. At each time ti we select a point xi. A path is constructed by connecting all possible xi points so selected by straight lines as shown in Figure I.1 and the path integral is written (setting h = 1) as

(2.4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where A is a normalization constant which defines the measure — note that there is one factor of A for each straight line segment. In the limit as ε [right arrow] 0 we can evaluate the action for each line segment in the infinitesimal approximation. For the free particle we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The integrations may be performed sequentially

(2.5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

yielding

(2.6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The constant A may be determined by use of the completeness condition

(2.7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If we pick t = ε <<<1 then

(2.8) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since ε is small, the exponential will rapidly oscillate and thereby wash out the integral unless x [congruent to] x'. Thus, we can write

(2.9) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence in order to have the correct behavior as ε [right arrow] 0 we must pick

(2.10) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

so that the free propagator becomes, using t = nε

(2.11) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

in complete agreement with the expression derived via more conventional means (cf. Eq. 2.1).

The reason that the propagator can be written as a path integral can be understood by using the completeness relation

(2.12) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

For later use, we shall give the derivation here for the general case involving interaction with a potential V(x). Starting with the definition

(2.13) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and breaking the time interval tf - ti (assumed to be positive) into n discrete steps of size

(2.14) ε = tf - ti/n

we can write

(2.15) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In the limit of large n the time slices become infinitesimal and

(2.16) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Introducing a complete set of momentum states, we have

(2.17) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and, taking the continuum limit, we find the path integral prescription

(2.18) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

(2.19) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is the classical action.

Classical Connection

Perhaps the most peculiar and fascinating aspect of this prescription is that all paths connecting the spacetime endpoints must be included in the summation. This appears to be in total contradiction with the classical mechanics result that a particle traverses a well-defined trajectory. The resolution of this apparent paradox may be found by explicitly restoring the dependence on h and noting that the path integral prescription is given by

(2.20) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Classical physics results as h [right arrow] 0, and in this limit a slight change in the path x(t) produces a huge change in phase and hence little or no contribution to the path summation except for trajectories [bar.x](t) for which the action is stationary — i.e., Hamilton's principle

(2.21) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In order to find such a path we take

(2.22) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

integrate by parts and use the feature that the endpoints of the path are fixed, i.e., δx(0)= δx(t)= 0. Then

(2.23) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

so that the trajectory which satisfies the stationary phase condition for arbitrary δx(t') must obey

(2.24) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which is just the classical mechanics prescription for the motion of a freely moving particle, i.e., [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. In the limit h [right arrow] 0 the classical trajectory represents the only path contributing to the path integral and the paradox is resolved.

One can also get a feel for the meaning of the propagator by noting that since

(2.25) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

if we take

(2.26) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

so that at t = 0 the particle is located precisely at the origin, then

(2.27) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

That is to say, D(0)F (x, t; 0, 0) is just the Schrödinger wavefunction of a freely moving particle which started at the origin at time zero. If we look at a specific location x0, t0 we would say classically that if a particle is observed at this point then it must have momentum

(2.28) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and energy

(2.29) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Examining the variation of the phase of the wavefunction in the vicinity of x0, t0, we find

(2.30) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus in the vicinity of this point we can write

(2.31) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

so that both the wavelength associated with the particle

(2.32) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and the corresponding frequency

(2.33) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

are given by the usual quantum mechanical relations.

Finally, the probability that the particle is located between x and x + dx at time t is

(2.34) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and is independent of x. All momenta then are equally likely at t = 0, as would be expected from the momentum space representation of the co-ordinate space wavefunction

(2.35) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and the momentum density is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. We conclude that all our intuitive notions are satisfied by the propagator, Eq. 2.27.

Frequency Space Representation

Before moving on to the more interesting case of motion in the presence of a potential, it is important to note that the time development operator is often used in Fourier transform or frequency space form rather than in its time representation. Before examining this result, however, it is useful to prove a simple mathematical identity. Consider the integral

(2.36) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(Continues…)

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by .
Copyright © 1992 Barry R. Holstein.
Excerpted by permission of Dover Publications, Inc..
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Table of Contents

Acknowledgments,
Prologue,
I. Propagator Methods,
1 Basic Quantum Mechanics,
2 The Propagator,
3 Harmonic Oscillator Propagator,
4 Time-dependent Perturbation Theory,
5 Propagator as the Green's Function,
6 Functional Techniques,
II. Scattering Theory,
1 Basic Formalism,
2 The Optical Theorem,
3 Path Integral Approach,
4 Two-Body Scattering,
5 Two-Particle Scattering Cross Section,
7 Scattering Matrix,
8 Partial Wave Expansion,
III. Charged Particle Interactions,
1 Charged Particle Lagrangian,
2 Review of Maxwell Equations and Gauge Invariance,
3 The Bohm–Aharonov Effect,
4 The Maxwell Field Lagrangian,
5 Quantization of the Radiation Field,
6 The Vacuum Energy,
IV. Charged Particle Interactions: Applications,
1 Radiative Decay: Formal,
2 Radiative Decay: Intuitive,
3 Angular Distribution of Radiative Decay,
4 Line Shape Problem: Wigner–Weisskopf Approach,
5 Compton Scattering via Feynman Diagrams,
6 Resonant Scattering,
7 Line Shape via Feynman Diagrams,
8 The Lamb Shift,
9 Dispersion Relations,
10 Effective Lagrangians,
11 Complex Energy and Effective Lagrangians,
V. Alternate Approximate Methods,
1 WKP Approximation,
2 Semiclassical Propagator,
3 The Adiabatic Approximation,
4 Berry's Phase,
VI. The Klein-Gordon Equation,
1 Derivation and Covariance,
2 Klein's Paradox and Zitterbewegung,
3 The Coulomb Solution: Mesonic Atoms,
VII. The Dirac Equation,
1 Derivation and Covariance,
2 Bilinear Forms,
3 Nonrelativistic Reduction,
4 Coulomb Solution,
5 Plane Wave Solutions,
6 Negative Energy Solutions and Antiparticles,
8 Dirac Propagator,
9 Covariant Perturbation Theory,
10 Electromagnetic Interactions,
VIII. Advanced Topics,
1 Radiative Corrections,
2 Spinless Particles: Electromagnetic Interactions,
3 Path Integrals and Quantum Field Theory,
4 Pion Exchange and Strong Interactions,
Epilogue,
Notation,
References,
Index,