quantum mechanics classical results modern systems and visualized examples jun 2006

Quantum Mechanics
Classical Results, Modern Systems, and
Visualized Examples


This page intentionally left blank


Quantum
Mechanics
Classical Results, Modern
Systems, and Visualized
Examples
Second Edition

Richard W. Robinett
Pennsylvania State University

1


3

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British Library Cataloguing in Publication Data
Data available
Library of Congress Cataloging in Publication Data
Robinett, Richard W. (Richard Wallace)
Quantum mechanics : classical results, modern systems, and
visualized examples / Richard W. Robinett.—2nd ed.
p. cm.
ISBN-13: 978–0–19–853097–8 (alk. paper)

ISBN-10: 0–19–853097–8 (alk. paper)
1. Quantum theory. I. Title.
QC174.12.R6 2006
530.12—dc22
2006000424
Typeset by Newgen Imaging Systems (P) Ltd., Chennai, India
Printed in Great Britain
on acid-free paper by
Biddles Ltd, King’s Lynn, Norfolk
ISBN 0–19–853097–8
10 9 8 7 6 5 4 3 2 1

978–0–19–853097–8


Preface to the Second Edition
One of the hallmarks of science is the continual quest to refine and expand one’s
understanding and vision of the universe, seeking not only new answers to old
questions, but also proactively searching out new avenues of inquiry based on
past experience. In much the same way, teachers of science (including textbook
authors) can and should explore the pedagogy of their disciplines in a scientific
way, maintaining and streamlining what has been documented to work, but also
improving, updating, and expanding their educational materials in response to
new knowledge in their fields, in basic, applied, and educational research. For that
reason, I am very pleased to have been given the opportunity to produce a Second
Edition of this textbook on quantum mechanics at the advanced undergraduate
level.
The First Edition of Quantum Mechanics had a number of novel features,
so it may be useful to first review some aspects of that work, in the context
of this Second Edition. The descriptive subtitle of the text, Classical Results,

Modern Systems, and Visualized Examples, was, and still is, intended to suggest a
number of the inter-related approaches to the teaching and learning of quantum
mechanics which have been adopted here.
• Many of the expected familiar topics and examples (the Classical Results)
found in standard quantum texts are indeed present in both editions, but we
also continue to focus extensively on the classical–quantum connection as one
of the best ways to help students learn the subject. Topics such as momentumspace probability distributions, time-dependent wave packet solutions, and the
correspondence principle limit of large quantum numbers can all help students
use their existing intuition to make contact with new quantum ideas; classical
wave physics continues to be emphasized as well, with its own separate chapter,
for the same reason. Additional examples of quantum wave packet solutions
have been included in this new Edition, as well as a self-contained discussion
of the Wigner quasi-probability (phase-space) distribution, designed to help
make contact with related ideas in statistical mechanics, classical mechanics,
and even quantum optics.
• An even larger number of examples of the application of quantum mechanics to Modern Systems is provided, including discussions of experimental
realizations of quantum phenomena which have only appeared since the First
Edition. Advances in such areas as materials science and laser trapping/cooling


vi

PREFACE TO THE SECOND EDITION

have meant a large number of quantum systems which have historically been
only considered as “textbook” examples have become physically realizable. For
example, the “quantum bouncer”, once discussed only in pedagogical journals, has been explored experimentally in the Quantum states of neutrons in
the Earth’s gravitational field.É The production of atomic wave packets which
exhibit the classical periodicity of Keplerian orbitsÊ is another example of a
Classical Result which has become a Modern System.

The ability to manipulate nature at the extremes of small distance (nanoand even atomic-level) and low temperatures (as with Bose–Einstein condensates) implies that a knowledge of quantum mechanics is increasingly
important in modern physical science, and a number of new discussions of
applications have been added to both the text and to the Problems, including
ones on such topics as expanding/interfering Bose–Einstein condensates, the
quantum Hall effect, and quantum wave packet revivals, all in the context of
familiar textbook level examples.
• We continue to emphasize the use of Visualized Examples (with 200 figures
included) to reinforce students’ conceptual understanding of the basic ideas
and to enhance their mathematical facility in solving problems. This includes
not only pictorial representations of stationary state wavefunctions and timedependent wave packets, but also real data. The graphical representation of
such information often provides the map of the meeting ground of the sometimes arcane formalism of a theorist, the observations of an experimentalist,
and the rest of the scientific community; the ability to “follow such maps” is
an important part of a physics education.
Motivated in this Edition (even more than before) by results appearing from
Physics Education Research (PER), we still stress concepts which PER studies have indicated can pose difficulties for many students, such as notions of
probability, reading potential energy diagrams, and the time-development of
eigenstates and wave packets.
As with any textbook revision, the opportunity to streamline the presentation
and pedagogy, based on feedback from actual classroom use, is one of the most
important aspects of a new Edition, and I have taken this opportunity to remove
some topics (moving them, however, to an accompanying Web site) and adding
new ones. New sections on The Wigner Quasi-Probability Distribution (and many
related problems), an Infinite Array of δ-functions: Periodic Potentials and the
Dirac Comb, Time-Dependent Perturbation Theory, and Timescales in Bound State
É The title of a paper by V. V. Nesvizhevsky et al. (2002). Nature 415, 297.
Ê See Yeazell et al. (1989).


PREFACE TO THE SECOND EDITION vii

Systems: Classical Period and Quantum Revival Times reflect suggestions from
various sources on (hopefully) useful new additions. A number of new in-text
Examples and end-of-chapter Problems have been added for similar reasons, as
well as an expanded set of Appendices, on dimensions and mathematical methods.
An exciting new feature of the Second Edition is the development of a Web
siteË in support of the textbook, for use by both students and instructors, linked
from the Oxford University PressÌ web page for this text. Students will find
many additional (extended) homework problems in the form of Worksheets on
both formal and applied topics, such as “slow light”, femtosecond chemistry, and
quantum wave packet revivals. Additional material in the form of Supplementary
Chapters on such topics as neutrino oscillations, quantum Monte Carlo approximation methods, supersymmetry in quantum mechanics, periodic orbit theory
of quantum billiards, and quantum chaos are available.
For instructors, copies of a complete Solutions Manual for the textbook, as
well as Worksheet Solutions, will be provided on a more secure portion of the site,
in addition to copies of the Transparencies for the figures in the text. An 85-page
Guide to the Pedagogical Literature on Quantum Mechanics is also available there,
surveying articles from The American Journal of Physics, The European Journal
of Physics, and The Journal of Chemical Education from their earliest issues,
through the publication date of this text (with periodic updates planned.) In
addition, a quantum mechanics assessment test (the so-called Quantum Mechanics Visualization Instrument or QMVI) is available at the Instructors site, along
with detailed information on its development and sample results from earlier
educational studies. Given my long-term interest in the science, as well as the
pedagogy, of quantum mechanics, I trust that this site will continually grow in
both size and coverage as new and updated materials are added. Information on
accessing the Instructors area of the Web site is available through the publisher
at the Oxford University Press Web site describing this text.
I am very grateful to all those from whom I have had help in learning quantum
mechanics over the years, including faculty and fellow students in my undergraduate, graduate, and postdoctoral days, current faculty colleagues (here at
Penn State and elsewhere), my own undergraduate students over the years, and
numerous authors of textbooks and both research and pedagogical articles, many

of whom I have never met, but to whom I owe much. I would like to thank all
those who helped very directly in the production of the Second Edition of this
text, specifically including those who provided useful suggestions for improvement or who found corrections, namely, J. Banavar, A. Bernacchi, B. Chasan,
Ë See robinett.phys.psu.edu/qm
Ì See www.oup.co.uk


viii

PREFACE TO THE SECOND EDITION

J. Edmonds, M. Cole, C. Patton, and J. Yeazell. I have truly enjoyed recent collaborations with both M. Belloni and M. A. Doncheski on pedagogical issues
related to quantum theory, and some of our recent work has found its way into
the Second Edition (including the cover) and I thank them for their insights, and
patience.
No work done in a professional context can be separated from one’s personal
life (nor should it be) and so I want to thank my family for all of their help
and understanding over my entire career, including during the production of
this new Edition. The First Edition of this text was thoroughly proof-read by my
mother-in-law (Nancy Malone) who graciously tried to teach me the proper use
of the English language; her recent passing has saddened us all. My own mother
(Betty Robinett) has been, and continues to be, the single most important role
model in my life—both personal and professional—and I am deeply indebted
to her far more than I can ever convey. Finally, to my wife (Sarah) and children
(James and Katherine), I give thanks everyday for the richness and joy they bring
to my life.
Richard Robinett
December, 2005
State College, PA

Contents

Part I The Quantum Paradigm
1

1
3

How this Book Approaches Quantum Mechanics

3

1.2

Essential Relativity

1.3

Quantum Physics:

1.4

Semiclassical Model of the Hydrogen Atom

17

1.5

Dimensional Analysis

21

1.6

2

A First Look at Quantum Physics
1.1

Questions and Problems

23

8
as a Fundamental Constant

10

Classical Waves

34

2.1

The Classical Wave Equation

34

2.2

Wave Packets and Periodic Solutions

36

2.2.1

General Wave Packet Solutions

36

2.2.2

Fourier Series

38

2.3

Fourier Transforms

43

2.4

Inverting the Fourier transform: the Dirac δ-function

46

2.5

Dispersion and Tunneling

51

2.5.1

Velocities for Wave Packets

51

2.5.2

Dispersion

53

2.5.3

Tunneling

56

2.6

3

Questions and Problems

57

The Schrödinger Wave Equation

65

3.1

The Schrödinger Equation

65

3.2

Plane Waves and Wave Packet Solutions

67

3.2.1

Plane Waves and Wave Packets

67

3.2.2

The Gaussian Wave Packet

70

3.3

“Bouncing” Wave Packets

75

3.4

Numerical Calculation of Wave Packets

77

3.5

Questions and Problems

79


x

CONTENTS

4

Interpreting the Schrödinger Equation

84

4.1

84

Introduction to Probability
4.1.1

Discrete Probability Distributions

84

4.1.2

Continuous Probability Distributions

87

4.2

Probability Interpretation of the Schrödinger Wavefunction

91

4.3

Average Values

96

4.3.1

Average Values of Position

96

4.3.2

Average Values of Momentum

4.3.3

Average Values of Other Operators

98
100

4.4

Real Average Values and Hermitian Operators

102

4.5

The Physical Interpretation of φ(p)

104

4.6

Energy Eigenstates, Stationary States, and the Hamiltonian Operator

107

4.7

The Schrödinger Equation in Momentum Space

111

4.7.1

Transforming the Schrödinger Equation Into Momentum
Space

4.7.2
4.8

Uniformly Accelerating Particle

Commutators

111
114
116

4.9

5

The Wigner Quasi-Probability Distribution

118

4.10

Questions and Problems

121

The Infinite Well: Physical Aspects
5.1

134

The Infinite Well in Classical Mechanics: Classical Probability
Distributions
Stationary States for the Infinite Well

137

5.2.1

5.2

134
137

5.2.2

Position-Space Wavefunctions for the Standard Infinite Well
Expectation Values and Momentum-Space Wavefunctions for

the Standard Infinite Well

5.2.3

The Symmetric Infinite Well

140
144

5.3

The Asymmetric Infinite Well

146

5.4

Time-Dependence of General Solutions

151

5.4.1

Two-State Systems

151

5.4.2

Wave Packets in the Infinite Well

154

5.4.3

Wave Packets Versus Stationary States

157

5.5

6

Questions and Problems

157

The Infinite Well: Formal Aspects

166

6.1

Dirac Bracket Notation

166

6.2

Eigenvalues of Hermitian Operators

167

6.3

Orthogonality of Energy Eigenfunctions

168

6.4

Expansions in Eigenstates

171


CONTENTS xi
6.5

175

Parity

181

6.7

Simultaneous Eigenfunctions

183

6.8

7

Expansion Postulate and Time-Dependence

6.6

Questions and Problems

185

Many Particles in the Infinite Well: The Role of Spin and
Indistinguishability

192

7.1

The Exclusion Principle

192

7.2

One-Dimensional Systems

193

7.3

Three-Dimensional Infinite Well

195

7.4

Applications

198

7.4.1

Conduction Electrons in a Metal

198

7.4.2

Neutrons and Protons in Atomic Nuclei

200

7.4.3

White Dwarf and Neutron Stars

200

7.5

8

Questions and Problems

206

Other One-Dimensional Potentials

210

8.1

210

Singular Potentials
8.1.1

Continuity of ψ (x)

210

8.1.2

Single δ-function Potential

212

8.1.3

Twin δ-function Potential

213

8.1.4

Infinite Array of δ-functions: Periodic Potentials and the Dirac
Comb

216

The Finite Well

221

8.2.1

Formal Solutions

221

8.2.2

8.2

Physical Implications and the Large x Behavior of
Wavefunctions

8.3

Applications to Three-Dimensional Problems

225
230

8.3.1
8.4

9

The Schrödinger Equation in Three Dimensions

230

8.3.2

Model of the Deuteron

231

Questions and Problems

234

The Harmonic Oscillator

239

9.1

The Importance of the Simple Harmonic Oscillator

239

9.2

Solutions for the SHO

243

9.2.1

Differential Equation Approach

243

9.2.2

Properties of the Solutions

247

9.3

Experimental Realizations of the SHO

249

9.4

Classical Limits and Probability Distributions

251


xii

CONTENTS
9.5

Unstable Equilibrium: Classical and Quantum Distributions

254

9.6

Questions and Problems

255

10 Alternative Methods of Solution and Approximation
Methods

260

10.1

Numerical Integration

261

10.2

The Variational or Rayleigh–Ritz Method

266

10.3

The WKB method

273

10.3.1 WKB Wavefunctions

274

10.3.2 WKB Quantized Energy Levels

277

10.4

Matrix Methods

278

10.5

Perturbation Theory

286

10.5.1 Nondegenerate States

286

10.5.2 Degenerate Perturbation Theory

293

10.5.3 Time-Dependent Perturbation Theory

295

Questions and Problems

299

10.6

11 Scattering
11.1

307

Scattering in One-Dimensional Systems

307

11.1.1 Bound and Unbound States

307

11.1.2 Plane Wave Solutions

310

11.2

Scattering from a Step Potential

310

11.3

Scattering from the Finite Square Well

315

11.3.1 Attractive Well

315

11.3.2 Repulsive Barrier

321
321

11.4.2 Scanning Tunneling Microscopy

324

11.4.3 α-Particle Decay of Nuclei

325

11.4.4 Nuclear Fusion Reactions
11.5

319

Applications of Quantum Tunneling
11.4.1 Field Emission

11.4

328

Questions and Problems

330

12 More Formal Topics

333

12.1

Hermitian Operators

333

12.2

Quantum Mechanics, Linear Algebra, and Vector Spaces

337

12.3

Commutators

341

12.4

Uncertainty Principles

343

12.5

Time-Dependence and Conservation Laws in Quantum Mechanics

346

12.6

Propagators

352

12.6.1 General Case and Free Particles

352

12.6.2 Propagator and Wave Packets for the Harmonic Oscillator

353


CONTENTS xiii
12.7

Timescales in Bound State Systems: Classical Period and Quantum
Revival Times

12.8

357

Questions and Problems

360

13 Operator and Factorization Methods for the Schrödinger
370
Equation

13.1

Factorization Methods

370

13.2

Factorization of the Harmonic Oscillator

371

13.3

Creation and Annihilation Operators

377

13.4

Questions and Problems

380

14 Multiparticle Systems

384

14.1

Generalities

384

14.2

Separable Systems

387

14.3

Two-Body Systems

389

14.3.1 Classical Systems

390

14.3.2 Quantum Case

391

14.4

Spin Wavefunctions

394

14.5

Indistinguishable Particles

396

14.6

Questions and Problems

407

Part II The Quantum World

413

15 Two-Dimensional Quantum Mechanics
15.1

415
417

15.1.1 2D Infinite Well

418

15.1.2 2D Harmonic Oscillator

422

Central Forces and Angular Momentum

423

15.2.1 Classical Case

15.2

2D Cartesian Systems

423

15.2.2 Quantum Angular Momentum in 2D

429
429

15.3.2 Circular Infinite Well

432

15.3.3 Isotropic Harmonic Oscillator
15.4

425

Quantum Systems with Circular Symmetry
15.3.1 Free Particle

15.3

435

Questions and Problems

437

16 The Schrödinger Equation in Three Dimensions

448

16.1

Spherical Coordinates and Angular Momentum

449

16.2

Eigenfunctions of Angular Momentum

454

16.2.1 Methods of Derivation

454


xiv

CONTENTS
16.2.2 Visualization and Applications

465

Diatomic Molecules

467

16.3.1 Rigid Rotators

467

16.3.2 Molecular Energy Levels

16.3

463

16.2.3 Classical Limit of Rotational Motion

469

16.3.3 Selection Rules

472

16.4

Spin and Angular Momentum

475

16.5

Addition of Angular Momentum

482

16.6

Free Particle in Spherical Coordinates

491

16.7

Questions and Problems

492

17 The Hydrogen Atom

501

17.1

Hydrogen Atom Wavefunctions and Energies

501

17.2

The Classical Limit of the Quantum Kepler Problem

507

17.3

Other “Hydrogenic” Atoms

513

17.3.1 Rydberg Atoms

513

17.3.2 Muonic Atoms

515

Multielectron Atoms

517

17.4.1 Helium-Like Atoms

519

17.4.2 Lithium-Like Atoms

524

17.4.3 The Periodic Table

527

Questions and Problems

529

17.4

17.5

18 Gravity and Electromagnetism in Quantum Mechanics

540

18.1

Classical Gravity and Quantum Mechanics

540

18.2

Electromagnetic Fields

543

18.2.1 Classical Electric and Magnetic Fields

543

18.2.2 E and B Fields in Quantum Mechanics

548

18.3

Constant Electric Fields

550

18.4

Atoms in Electric Fields: The Stark Effect

552

18.4.1 Classical Case

552

18.4.2 Quantum Stark Effect

555

18.5

Constant Magnetic Fields

561

18.6

Atoms in Magnetic Fields

564

18.6.1 The Zeeman Effect: External B Fields

564

18.6.2 Spin-Orbit Splittings: Internal B Fields

569

18.6.3 Hyperfine Splittings: Magnetic Dipole–Dipole Interactions

574

Spins in Magnetic Fields

576

18.7.1 Measuring the Spinor Nature of the Neutron Wavefunction

576

18.7.2 Spin Resonance

578

18.7


CONTENTS xv
18.8

The Aharonov–Bohm Effect

583

18.9

Questions and Problems

586

19 Scattering in Three Dimensions

596

19.1

Classical Trajectories and Cross-Sections

597

19.2

Quantum Scattering

603

19.2.1 Cross-Section and Flux

603

19.2.2 Wave Equation for Scattering and the Born Approximation

606

19.3

Electromagnetic Scattering

612

19.4

Partial Wave Expansions

619

19.5

Scattering of Particles

624

19.5.1 Frames of Reference

625

19.5.2 Identical Particle Effects
19.6

A

635

Dimensions and MKS-type Units for Mechanics, Electricity
641
and Magnetism, and Thermal Physics
A.1

B

631

Questions and Problems

Problems

642

644

B.1

Physical Constants

644

B.2

The Greek Alphabet

646

B.3

Gaussian Probability Distribution

646

B.4

C

Physical Constants, Gaussian Integrals, and the Greek
Alphabet

Problems

648
649

C.1

D

Complex Numbers and Functions

651

Problems

653

D.1

Integrals

653

D.2

Summations and Series Expansions

658

D.3

Assorted Calculus Results

661

D.4

Real Integrals by Contour Integration

661

D.5

Plotting

664

D.6

E

Integrals, Summations, and Calculus Results

Problems

665

Special Functions

666

E.1

Trigonometric and Exponential Functions

666

E.2

Airy Functions

667


xvi

CONTENTS
E.3

668

Cylindrical Bessel Functions

669

E.5

Spherical Bessel Functions

669

E.6

Legendre Polynomials

669

E.7

Generalized Laguerre Polynomials

670

E.8

The Dirac δ-Function

671

E.9

The Euler Gamma Function

672

E.10

F

Hermite Polynomials

E.4

Problems

672
674

Vectors and Matrices

674

F.2

Group Theory

679

F.3

G

Vectors, Matrices, and Group Theory
F.1

Problems

679

Hamiltonian Formulation of Classical Mechanics

680

G.1

685

Problems

REFERENCES

687

INDEX

695


PART I

The Quantum Paradigm


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ONE

A First Look at Quantum
Physics

1.1 How This Book Approaches Quantum
Mechanics
It can easily be argued that a fully mature and complete knowledge of quantum

mechanics should include historical, axiomatic, formal mathematical, and even
philosophical background to the subject. However, for a student approaching
quantum theory for the first time in a serious way, it can be the case that an
approach utilizing his or her existing knowledge of, and intuition for, classical
physics (including mechanics, wave physics, and electricity and magnetism) as
well as emphasizing connections to experimental results can be the most productive. That, at least, is the point of view adopted in this text and can be
illustrated by a focus on the following general topics:
(1) The incorporation of a wave property description of matter into a consistent
wave equation, via the Schrödinger equation;
(2) The statistical interpretation of the Schrödinger wavefunction in terms of
a probability density (in both position- and momentum-space);
(3) The study of single-particle solutions of the Schrödinger equation, for both
time-independent energy eigenstates as well as time-dependent systems, for
many model systems, in a variety of spatial dimensions, and finally;
(4) The influence of both quantum mechanical effects and the constraints
arising from the indistinguishability of particles (and how that depends
on their spin) on the properties of multiparticle systems, and the resulting
implications for the structure of different forms of matter.
By way of example of our approach, we first note that Fig. 1.1 illustrates
an example of a precision measurement of the wave properties of ultracold
neutrons, exhibiting a Fresnel diffraction pattern arising from scattering from


4

CHAPTER 1 A FIRST LOOK AT QUANTUM PHYSICS

1500

1000

500
100 µm

Scanning slit position

Figure 1.1. Fresnel diffraction pattern obtained from scattering at a sharp edge, obtained using ultracold
neutrons by Gähler and Zeilinger (1991).

a sharp edge, nicely explained by classical optical analogies. We devote Chapter 2
to a discussion of classical wave physics and Chapter 3 to the description of
such wave effects for material particles, via the Schrödinger equation. Figure 1.2
demonstrates an interference pattern using electron beams, built up “electron by
electron,” with the obvious fringes resulting only from a large number of individual measurements. The important statistical aspect of quantum mechanics,
simply illustrated by this experiment, is discussed in Chapter 4 and beyond.
It can be argued that much of the early success of quantum theory can be traced
to the fact that many exactly soluble quantum models are surprisingly coincident with naturally occurring physical systems, such as the hydrogen atom and
the rotational/vibrational states of molecules and such systems are, of course,
discussed here. The standing wave patterns obtained from scanning tunneling microscopy of “electron waves” in a circular corral geometry constructed
from arrays of iron atoms on a copper surface, seen in Fig. 1.3, reminds us of
the continuing progress in such areas as materials science and atom trapping
in developing artificial systems (and devices) for which quantum mechanics
is applicable. In that context, many exemplary quantum mechanical models,
which have historically been considered as only textbook idealizations, have also
recently found experimental realizations. Examples include “designer” potential
wells approximating square and parabolic shapes made using molecular beam
techniques, as well as magnetic or optical traps. The solution of the Schrödinger
equation, in a wide variety of standard (and not-so-standard) one-, two-, and
three-dimensional applications, is therefore emphasized here, in Chapters 5, 8, 9,

1.1 HOW THIS BOOK APPROACHES QUANTUM MECHANICS 5

Figure 1.2. Interference patterns obtained by using an electron microscope showing the fringes being
“built up” from an increasingly large number of measurements of individual events. From Merli, Missiroli,
and Pozzi (1976). (Photo reproduced by permission of the American Institute of Physics.)

and 15–17. In parallel to these examples, more formal aspects of quantum theory
are outlined in Chapters 7, 10, 12, 13, and 14.
The quantum in quantum mechanics is often associated with the discrete
energy levels observed in bound-state systems, most famously for atomic systems
such as the hydrogen atom, which we discuss in Chapter 17, emphasizing that this
is the quantum version of the classical Kepler problem. We also show, in Fig. 1.4,
experimental measurements leading to a map of the momentum-space probability density for the 1S state of hydrogen and the emphasis on momentum-space
methods suggested by this result is stressed throughout the text. The influence of
additional“real-life”effects, such as gravity and electromagnetism, on atomic and
other systems are then discussed in Chapter 18. We note that the data in Fig. 1.4


6

CHAPTER 1 A FIRST LOOK AT QUANTUM PHYSICS

Figure 1.3. Standing wave patterns obtained using scanning tunneling microscopy from a circular “corral”
of radius ∼70 Å, constructed from 48 iron atoms on a copper surface. (Photo courtesy of IBM Almaden.)

Differential cross section

1.0

H(1s)
1200 eV

0.8

800 eV
400 eV

0.6

(1+q2)–4

0.4

0.2

0

0.2

0.4

0.5
0.8
1.0
Momentum q (a.u.)

1.2

1.4

Figure 1.4. Electron probability density obtained by scattering with three different energy probes, compared
with the theoretically calculated momentum-space probability density for the hydrogen-atom ground state,
from Lohmann and Weigold (1981). The data are plotted again the scaled momentum in atomic units (a.u.),
q = a0 p/ .

was obtained via scattering processes, and the importance of scattering methods
in quantum mechanics is emphasized in both one-dimension (Chapter 11) and
three-dimensions (Chapter 19). The fact that spin-1/2 particles must satisfy the
Pauli principle has profound implications for the way that matter can arrange


Rn

Xe

Kr

Ar

Ne

He

1.1 HOW THIS BOOK APPROACHES QUANTUM MECHANICS 7

30

40
20

20

10

0

0

10

20

30

40

50

60

70

80

90

Polarizability (10–24 cm3) (dashed)

Ionizational potential (eV) (solid)

60

0
100

Nuclear charge (Z)

Figure 1.5. Plots of the ionization energy (solid) and atomic polarizability (dashed) versus nuclear charge,
showing the shell structure characterized by the noble gas atoms, arising from the filling of atomic energy
levels as mandated by the Pauli principle for spin-1/2 electrons.

itself, as shown in the highly correlated values of physical parameters shown in
Fig. 1.5 for atoms of increasing size and complexity. While it is illustrated here in
a numerical way, this should also be reminiscent of the familiar periodic table of
the elements, and the Pauli principle has similar implications for nuclear structure. We discuss the role of spin in multiparticle systems described by quantum
mechanics in Chapters 7, 14, and 17.
We remind the reader that similar dramatic manifestations of quantum phenomena (including all of the effects mentioned above) are still being discovered,
as illustrated in Fig. 1.6. In a justly famous experiment,É two highly localized
and well-separated samples of sodium atoms are cooled to sufficiently low temperatures so that they are in the ground states of their respective potential wells
(produced by laser trapping.) The trapping potential is removed and the resulting coherent Bose–Einstein condensates are allowed to expand and overlap,
exhibiting the quantum interference shown in Fig. 1.6 (the solid curve, showing
É From the paper entitled Observation of interference between two Bose condensates by Andrews et al.
(1997).


8

CHAPTER 1 A FIRST LOOK AT QUANTUM PHYSICS

Absorpition (%)

60

30

0

200

0

400
Position (µm)

Figure 1.6. Data (from Andrews et al. (1997)) illustrating the interference of two Bose condensates as
they expand and overlap (solid curve), compared to a single expanding Bose condensate (dotted curve).

regular absorption variations across the central overlap region), while no such
interference is observed for a single expanding quantum sample (dotted data.)
Many of the salient features of this experiment can be understood using relatively
simple ideas outlined in Chapters 3, 4, and 9.
The ability to use the concepts and mathematical techniques of quantum
mechanics to confront the wide array of experimental realizations that have
come to characterize modern physical science will be one of the focuses of this
text. Before proceeding, however, we reserve the remainder of this chapter for
brief reviews of some of the essential aspects of both relativity and standard
results from quantum theory.

1.2 Essential Relativity

While we will consider nonrelativistic quantum mechanics almost exclusively,
it is useful to briefly review some of the rudiments of special relativity and the
fundamental role played by the speed of light, c.
For a free particle of rest mass m moving at speed v, the total energy (E),
momentum (p), and kinetic energy (T ) can be written in the relativistically
correct forms
E = γ mc 2 ,

and

T ≡ E − mc 2 = (γ − 1)mc 2

1

p = γ mv,

= 1−

(1.1)

where
γ ≡

1 − v 2 /c 2

v2
c2

−1/2

(1.2)