Table Of ContentINTERNATIONAL SERIES IN PURE AND APPLIED PHYSICS
G. P. HARNWELL, Consuttive Eprron
Apvisory Eprroria, Commrrres: E, U. Condon, George R. Harrison,
Elmer J[utehisson, K. K. Darrow
QUANTUM MECHANICS
Donated by
Mrs. Yemuna Bappu
to
The Indian Institute of Astrophysics
trom the personal collection
of
Dr. M. K. V. Bappu
INTERNATIONAL SERIES IN
PURE AND APPLIED PHYSICS
G. P. Harnwuun, Consulting Editor
Bacwea ann Goupsszr—ATOMIC ENERGY STATES
Brrraz—INTRODUCTION TO FERROMAGNETISM
Bririowx—WAVE PROPAGATION IN PERIODIC STRUCTURES
Oso¥—PIELORLECTRICITY
Crank—APPLIED X-RAYS
Cunmis—ELECTRICAL MEASUREMENTS
Daver—CRYSTAL STRUCTURE AND ITS APPLICATIONS
Epwanps—-ANALYTIC AND VECTOR MECHANICS
Hanpy an Puurin—THE PRINCIPLES OF OPTICS
Harwwe1—ELECTRICITY AND ELECTROMAGNETISM
Hagnwawt aNp Lryincooo EXPERIMENTAL ATOMIC PHYSICS
Hoveron—PRINCIPLES OF MATHEMATICAL PHYSICS
Hoamns ano DuBatwor—PHOTOELECTRIC PHENOMENA
Hunp—HIGH-FREQUENCY MEASUREMENTS
PHENOMENA IN HIGH-FREQUENCY SYSTEMS
Inoprsot, Zonet, anp Inomesont—HEAT CONDUCTION
Kewein—PRINOIPLES OF QUANTUM MECHANICS
Kannarn—KINETIC THEORY OF GASES
Kousr—THE PHYSICS OF ELECTRON TUBES
Morse—VIBRATION AND SOUND
Pavuine ano Govpenat—THE STRUCTURE OF LINE SPHCTRA
Rucwrmyer avn Kenwarn—INTRODUCTION TO MODERN PHYSICS
Rosa ano Urer—ATOMS, MOLECULES, AND QUANTA
Soure—QUANTUM MECHANICS
Sarm—TH MODURN THEORY OF SOLIDS
Statze—INTRODUCTION TO CHEMICAL PHYSICS
MICROWAVE TRANSMISSION
Sarr any Franx—ELECTROMAGNETISM
INTRODUCTION TO THEORETICAL PHYSICS
MECHANICS
Suyruz—STATIC AND DYNAMIC ELECTRICITY
Srrartox—LLECTROMAGNETIC THEORY
Wurrs—INTRODUCTION TO ATOMIC SPECTRA
‘Wirztams—MAGNETIC PHENOMENA
Dr, Lee A, DuBridge was consulting editor of the series from 1939
to 1946.
QUANTUM MECHANICS
BY
LEONARD 1, SCHIFF
Professor of Physics
Stanford University
First Hoveron
Sxconp IMpiession:
AW YORK TOROVrO LONDON
McGRAW-LILL BOOK COMPANY, INC.
1949
uA Lib.
TA ARE
g
QUANTUM MECHANICS
Copyright, 1949, by the McGraw-Hill Book Company, Ine. Printed in the
United States of America, All rights reserved. This book, or purts thereof,
gaay not be reproduced in any form without permission of the publishers.
‘THE MAPIE PRESS COMPANY, YORK, PA.
PREFACE
This volume has a threefold purpose: to explain the physical concepts
of quantum mechanics, to describe the mathematical formalism, and to
present illustrative examples of both the ideas and the methods. The
book is intended to serve as a text at the graduate level and also as a
reference book. It is assumed that the reader is reasonably familiar
with atomic structure, classical mechanics, and differential equations.
Tn addition, he should have had some contact with electromagnetic theory
and, for the latter part of the book, with the special theory of relativity.
The author believes that the analytical methods employed in the book
will satisfy most theoretical physicists even though no attempt is made
to achieve mathematical rigor. For example, there is little or no dis-
cussion of the justification for the interchange of sum, derivative, and.
integral operations, or for the usc of the 6 function. On the other hand,
the physical reasons for the nature of the results obtained are investigated.
wherever possible.
Problems are given nt the end of each chapter. They are often used
to illustrate or amplify points discussed in the text. Original theorctical
papers are referred to throughout the hook; the list is representative
rather than exhaustive. Experimental results are, for the most part,
quoted without reference, since the large amount of documentation
required for an adequate survey seems out of place in a book on theoretical
physics. Several othor books on quantum mechanics and related sub-
jects arc referred to for more detailed discussions of particular topics.
‘The scope of this volume is best outlined if the book is divided into
three parts. The first three chapters constitute an introduction to
quantum mechanies, in which the physical concepts are discussed and
the Schrédinger wave formalism is established. The detniled treatment
of the wave function (Chap. TH) may be omitted ina first rending. The
next eight chapters comprise the central part of the book. This part
presents exact solutions of the wave equation for both energy-level and
collision problems, the Heixenberg matrix formalism and transformation
theory, approximation methods, radiation theory, and some applications
to atomic sysiems. Since the first eleven chapters correspond to a
typical one-year graduate course, il seemed desirable to inelude a semi-
classical treatment of clectromagnetic radintion in the contral part of
the hook (Chap. X) even though some of the results are obtained again in
Chap. XIV. The last part of the book corresponds to a short course in
what is often called advanced quantum mechanicr, It consists of rela-
v
PREFACE
cle theory and an introduction to quantized field theory and
aectrodynamics.
all this book was written while the author was at the Uni-
ut Pennsylvania, and he gratefully acknowledges the continued
ugement of Prof. G. P. Harnwell. He is also indebted 10 Drs. E. EH.
.....ard and §, Pasternack for helpful criticism of the early chapters, to
W. Miller and L. Spruch for their careful reading of most of the manu-
script, and to Profs. F. Bloch, R. F. Christy, and W. W. Hunsen for
valuable comments on certain sections. It is a particular pleasure for
the author to thank Prof. R. Serber for many discussions of both the
conceptual and formal aspects of quantum mechanics that took place
during the Jast eleven years.
Finally, the author wishes to acknowledge his indebtedness to Prof.
J. R. Oppenheimer for his introduction to severul of the ideas and exam-
ples that appear in the book. Indeed, the writing of this book in its
present form owes much to the period from 1937 to 1840, which the
author spent in association with Professor Oppenheimer.
Lxuonann I, Scrtrr
Sranvorp, CALIFORNIA
Fane, 1948
Pre
9.
10,
CONTENTS
ACD. ee ee Pee
Cuarrer T
THE PHYSICAL RASIS OF QUANTUM MECHANICS
. Haperimental Background. 6. 0 ee
Tnadequacy of classical physics. Summury of prineipal experiments and
inferonces.
The Old Quantum Pheary. . . bee e eee eB
Bobr-Sommerfeld quantization rules, Practical difficulties. Conceptual
difficulties, Quantum-mechanieal viewpoint.
Uncertainty and Complementarity 0 eB
Uncertainty principle. Complementarity principle. Lintitati
periment.
Discussion of Measurement. ee DO
Anulysis of diffraction experiment. Discussion of diffrs
Tocatiaition experiment,
Wave Packets in Spnee and
Space packets. Time p
tion experiment.
a re P
ots. Wave f
Cuaeren IL
THE SCHRODINGER WAVE EQUATION
Denlopmient of the Wane Bquation .. . . fee WT
Traveling harmonie waves. Need for a wave entiation, ‘The owedimen=
sion) wave equation. Extension to three dimensions. Inclusion of
for
Taterpretation of the Wann Punetion. . . . . . . oe al
Stalintieal interpretation, Normalisation of yj. Peohabll
sity. Expectation value, Khrenfost'y theorem,
Energy Bigenfunction® 0 ..
Seperation of the wave equation. Significance of the separation co
stunt B. Boundary conditions at great distanes, Continuity conditions,
Boundary conditions for infinite potential energy. Energy eigenvalues
one dimension, Discrete energy levels. Continuous energy cigenvalucs.
Diserete and continuous cigenvilues in Uhees dimensions.
One-Dimensional Syuare Well Potentinl. 6. ee vb
Perlectly rigid walls, Finite: potential step. Energy levels. Pac
simplified solrtion.
y current den
in
t ITT
TWIGENEFUNCTIONS AND BLIGENVALITES
Frterpretative Postitates and Energy Kigenfunctions. we Al
Dymamieal variables as operators, Expansion in eigenfunctions, The
vi
ie
12.
13.
id.
16.
17,
18.
19.
20,
. Three-Dimensional Square Well Potential
CONTENTS
total energy operator. Normalization in a box. Orthonormality of
energy eigenfunctions. Reality of the energy eigenvalues. Expansion in
energy cigenfunctions. The closure property. Probability function and
expectation value. General solution of the Sehridinger equation,
Momentum. Eigenfunctions . . .
Form of the eigenfunetions. Box normalization, ‘The Dirac § function.
A representation of the 5 function, Normalization in terms of the 3 fno-
tion, Some properties of the 5 function, Closure. Expansion in
momentum eigenfunctions, Probability function and expectation value.
Motion of a Free Wave Packet in One Dimension, :
‘The minimum uncertainty product. Form of the minimum packet.
Momentum expansion coefficients, Change with time of a minimum
packet. Classical limit.
Caprese IV
DISCRETE EIGENVALUES: ENERGY LEVELS
Linear Harmonie Oscillator © be
Asymptotic behavior. Energylovels. Zero-point energy. Parity. Here
mite polynomials, Harmonic-oseillator wave functions. Correspondence
with classical theory, Oseillating wave packet.
Spherically Symmetric Potentials in Three Dimensions... a
Separation of the wave equation. Legendre polynomials, Sphe
harmonies, Parity. Angular momentum,
Zero angular momentum, Interior solutions for arbitrary f, Exterior
solutions for arbitrary 1. Energy levels.
The Hydrogen Atom ee
Redneod mass. Asymptotic behavior. Energy levels. Laguerre_ pol
nomials, Tlydrogen-atom wave functions. TDogeneracy. Separation i
parabolic coordinates. Energy lovels, Wave funetions.
Cuaprar V
CONTINUOUS TFIGENVALURS: COLLISION TITKORY
One-Dimensional Square Potential Barrier... .
Asymptotic behavior. Normalization. Scattering Coeffieients.
Collisions in Three Dimensions...
Scattering cross section, Relations between angles in the laborntory and
conter-of-mass systems, Relation between cross sections. Dependence
ony. Asymptotic behavior. Normalization.
Scattering by Spherically Symmeiric Potentials Lee a
Asymptotic behavior. Scattering cross section, Phase shift, Caleu-
lation of 8. Relation between signs of and V(r). Ramsuuer-Townsend
effect. Scattering hy a perfoctly rigid sphere. Scattering by a square well
potential. Resonance stattering. Angular distribution at low energies.
Scattering by a Coulomb Field . : an
Parabolic coordinates, Confluent hypergeometric function, Seattering
cross section and normalization. Solution in sphorieal coordinates. Modi-
fied Coulomb field. Classical limit for a pure Coulomb field.
48
Bt
GO
69
76
80
. 103
meecy
Ql.
22.
23,
24,
28.
. Stationary Perturbation Theory.
CONTENTS
Cuarrer VI
MATRIX FORMULATION OF QUANTUM MECHANICS
Matrix Algebra. . Lee
Matrix addition and multiplication. Null, unit, and constant matrices.
Spur, determinant, and inverse of a matrix, Hormitian end unitary
matrices. ‘Transformation and diagonalization of matrices, Matrices of
infinite rank.
Matrices in Quantum Mechanics. Lee eee
A unitary transformation matrix. Hamiltonian matrix, Dynamical
variables as Hermitian matrices. Wave functions as unitary matrices.
The z representation. A useful identity.
Equations of Motion in Matrix Form. Le eee
‘Time rate of change of matrix. Classical Lagrangian and Hamiltonian
equations of motion. Poisson brackets and commutator brackets. Quan-
tization of a classical system. Motion of a particle in an electromagnetic
field. Evaluation of commutator brackets. Velocity and acceleration of
a charged particle. The Lorenta force. Constants of the motion, The
parity operator. Energy representation.
Angular Momentum . . Lee ee
Definition of angular momentum. Choise of a representation, Relations
between matrix clements, Kigenvalues of MW. Eigenvalues of M3; the
Linateix. Connection with tho spherical harmonies, Spin angular mo-
mentum. Addition of angular momenta. Kigenvalues of (ML, + Mz)%
Car VIL
APPROXIMATION METHODS FOR STATIONARY PROBLEMS:
nid-order perturba~
tor, Degenerate
Nondegenerate case, First-order perturbation, See
tion. Normal nol y. Perturbation of an oxe
case. Removal of degeneracy in first order, Removal of degeneracy in
second order. Vinst-order Stark effect in hydrogen. Perturbed energy
levels. Ovcurrence of permanent cloelric-dipole moments.
The Born Approximation. .
Perturbation approximation,
free particle, Kxplieit e
tion treatment ofp:
well potential, Vali¢
serconed Coulomb field.
The Variation Method we : re woe
Txpeetation vw Fihe ouergy. Application to exeited stat Ground
stateal helium. Kleetron interaction energy. Variation of the parameter
Z. yan der Waals interaction. Perturbation calculation. Variation cal-
culatio
The WKB Approximation Le
Classical limit, Approximate solutions, Asymptotie nature of the solu-
tions. Solutinn near a turning point, Linear turning point. Connection
at the turning point. Asymptotic connection formulas, Energy levels
of a potential well. A quantization rule. Special houndary conditions.
function. Clroen’s function for a
Ination, Scattering eros section. Perturba-
waves. Phaso shifte. Seuttering hy a square
iy of the Born approximation. Seattering by a
ix
. 22
. 127
» 131
. 140
. 149
. 159
. 169
. IR
x CONTENTS
Cnarren VIIT
APPROXIMATION METHODS FOR TIME-DEPENDENT PROBLEMS
29, Time-Dependent Perturbation Theory . Fe 189
Expansion in unperturbed eigenfunctions. ¥ storder perturbation,
Physical interpretetion, ‘Transition probability. Scattering cross xeetion,
Harmonie perturbation. Second-order perturbation, Vffect, af first-order
transitions, Intermediate states.
30. Inelastis Collisions, 2. 2. ee re)
Expression for the cross seotion, Evaluation of the matrix element,
Differential and total cross sections. Production of a clowd-chamnber
track. Formulation of the problem. Evaluation of the k sum. Second.
order matrix element, Discussion of the cross section.
81. Adiabatic and Sudden Approvimations .... 1. . cee ee . 207
Expansion in instantaneous energy eigenfunctions. Choive of phases,
Adiabatic approximation. Connection with perturbation theary. Dine.
continuous change in H. Sudden approximation. Transient disturbance,
Disturbance of an oscillator.
Cuarrern IX
IDENTICAL PARTICLES AND SPIN
32, Identical Particles... . . ae .
Physical meaning of identity. Symmetric and antisyramcti wave f
tions. Construction from unsymmetrized functions. Dislingn viity
of identical particles. The exclusion principle. Conneetion with stat is:
cal mechanics. Collisions of identical particles.
83. Spin Angular Momentum. woe
Connection betweon spin and statictios, Spin mut
Collisions of identical particles, Electron spin fun:
atom. Spin functions for three electrons.
34. Rearrangement Collisions. 2... 0. be .. 230
Notation for rearrangement collisions, Use of the Bori approximation.
Lack of orthogonality of initial and final states. Connoetion with time
dependent perturbation theory. Exchange collisions of clerirons with
hydrogen. Differential cross section, Exchunge collisions with heliuun.
ancl cigonfunetionss.
ns. The helium
Crarrer X
SEMICLASSICAL TREATMENT OF RADIATION
85, Absorption and Induced Emission . . ee B10
Maxwell's equations. Plane electromagnetic waves, Uwn of peeturrlat ten
theory. Transition probability. Interpretation in forma of absorptions
Hostiedipale wansilions, Forbidden transitions,
%5. Spontaneous Eméaion. ok an . BAX
a energy. Dipole
radiation. Angular momentue, Dipole e caso, Conversion from elagsien]
ta quantum theory. Planck distribution formula. Line breadth.
87. Some Applications of Radiation Theory .
Selection rules for a single particle. Pola
287
‘ation of emitted radiation.
