Table Of ContentINTERNATIONAL SERIES IN NATURAL P H I L O S O P HY
VOLUME 82
GENERAL EDITOR! D. TER HAAR
(Full list of titles in Natural Philosophy see p. 397)
SOME OTHER TITLES OF INTEREST
ABRIKOSOV, A. A. et al
Quantum Field Theoretical Methods in
Statistical Physics
DAVYDOV, A. S.
Quantum Mechanics
KOTKIN, G. L. & SERBO, V. G.
Collection of Problems in Classical
Mechanics
LANDAU, L. D. & LIFSHITZ, E. M.
A Shorter Course of Theoretical Physics
— Volume 1: Mechanics and Electrodynamics
— Volume 2: Quantum Mechanics
TER HAAR, D.
The Old Quantum Theory
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MECHANICS:
CLASSICAL AND QUANTUM
by
T. T. TAYLOR
Professor of Physics, Loyola Marymount University,
Los Angeles, California
PERGAMON PRESS
OXFORD N E W Y O RK TORONTO
SYDNEY PARIS BRAUNSCHWEIG
U. K. Pergamon Press Ltd., Headington Hill Hall,
Oxford OX3 0BW, England
U. S. A. Pergamon Press Inc., Maxwell House, Fairview Park,
Elmsford, New York 10523, U.S.A.
CANADA Pergamon of Canada, Ltd., 207 Queen's Quay West,
Toronto 1, Canada
AUSTRALIA Pergamon Press (Aust.) Pty. Ltd., 19a Boundary Street,
Rushcutters Bay, N.S.W. 2011, Australia
FRANCE Pergamon Press SARL, 24 rue des Ecoles,
75240 Paris, Cedex 05, France
WEST GERMANY Pergamon Press GmbH, D.3300 Braunschweig,
Burgplatz 1, West Germany
Copyright © 1976 T. T. Taylor
All Rights Reserved. No part of this publication may be
reproduced, stored in a retrieval system or transmitted in
any form or by any means: electronic, electrostatic,
magnetic tape, mechanical, photocopying, recording or
otherwise, without permission in writing from the
publishers
First edition 1976
Library of Congress Cataloging in Publication Data
Taylor, Thomas Tallott.
Mechanics: classical and quantum.
(International series in natural philosophy; v. 82)
Bibliography: p.
1. Quantum theory. 2. Mechanics, Analytic.
I. Title.
QC174.12.T4 1975 530.T2 75-12794
ISBN 0-08-018063-9 h
0-08-020522-4 f
Printed in Hungary
PREFACE
EVER since the appearance in 1950 of Herbert Goldstein's deservedly
well-known text on classical mechanics, the once esoteric methods of
this branch of physics have become increasingly accessible, not only
to graduate students but also to undergraduates. Now it is quite
common for junior-year textbooks on mechanics to include chapters
on Lagrangian and Hamiltonian theory and it is generally agreed
that this material, if competently presented, is within the capa-
bilities of the students who use these books. The trend exemplified
here is part of an ever-operative process in the teaching of physics
whereby advanced topics are simplified (in the constructive sense of
that term) and placed earlier in the curriculum. Thus the efficiency of
learning is increased and time is made available for new subject matter
generated by research. Having established that the Lagrangian-
Hamiltonian approach, also known as analytical mechanics, should
be welcome at the undergraduate level, one may discuss the manner
in which it may best be introduced. If presented in the context of a
few classical applications only, students may miss much of its true
relevancy. Would it not be better utilized as a logical prelude to
quantum theory? This rhetorical question expresses the author's
position and reveals the motif of the present work.
Designed for the junior year, this book is intended to give students
a solid grounding in the principles of quantum mechanics after leading
them to these principles via the medium of analytical mechanics. In
doing this, the use of gratuitous postulates is minimized; quantum
theory, as exemplified by Schrodinger's formulation, is introduced
ix
X PREFACE
through the concept that macroscopic particle motion is to be ex-
plained by the motion of a packet of waves. This is possible if the
eikonal of the typical wave is made proportional to the classical action
function. The Hamilton-Jacobi equation then becomes the eikonal
equation and the linear wave equation corresponding thereto is shown
to be the Schrodinger equation. Once this derivation is completed,
the treatment becomes deductive and the implications of the Schrodinger
formulation are explored in the microscopic domain where phenom-
ena cannot be described successfully in classical terms. Techniques
appropriate to this domain are introduced and the power and uni-
versality of quantum theory are made evident.
Beginning as it does with the Lagrangian formulation and ending
with the hydrogenic atom and matrix mechanics, this book is ideally
suited to a two-semester sequence. However, in curricula where the
Lagrangian-Hamiltonian methodology is introduced in an earlier
course on intermediate mechanics and where some familiarity with
wave motion and with the historical background of quantum theory
can be presumed, this book could be adapted to a one-semester
presentation. In such an event, it is unlikely that the two final chapters
could be covered; the substance of these chapters would probably be
incorporated into a subsequent course, e.g. the senior course on atomic
physics. In curricula where time is divided into units smaller than
traditional semesters, still other ways of utilizing the material made
available here will suggest themselves.
The author wishes to thank Dr. James S. Albertson who first
suggested that a course of this type be presented and to express his
gratitude for the interest and the reactions of the many students who
attended the lectures now incorporated into this book. The author is
particularly indebted to Dr. Joseph R. Schwartz who audited the
entire sequence of lectures and who participated in many stimulating
conversations. Also appreciated are the hospitality of Dr. William G.
Spitzer of the University of Southern California and the many profit-
able discussions held with members, particularly with Dr. Jan Smit, of
the faculty of that university during a sabbatical leave in which a
significant part of this project was completed. The tangible encourage-
PREFACE xi
ment of the author's immediate administrator, Dr. James E. Fox-
worthy, is also gratefully acknowledged.
In the preparation of the illustrations and typescript, the author is
greatly indebted to Mr. John F. McCaffrey, to Mrs. Josephine D.
Murphy, and to his wife, Grace Bonaro Taylor, for very helpful
assistance.
Los Angeles THOMAS T. TAYLOR
CHAPTER 1
THE LAGRANGIAN FORMULATION OF
MECHANICS
1.01. The Harmonic Oscillator; a New Look at an Old
Problem
Every student is familiar with the one-dimensional harmonic os-
cillator consisting of a given mass M attached to a linear spring (whose
own mass is negligible) as in Fig. 1.01. The mass M is constrained to
M
FIG. 1.01. The one-dimensional harmonic oscillator.
move without friction along the line of action of the spring and x
denotes its displacement from equilibrium. The force exerted on M by
the spring is called F and is reckoned positive when it tends to in-
x
crease x. In terms of the elastic constant K, it is given by:
Fx Kx, (1.001)
whereupon the equation of motion becomes:
Mx + Kx = 0. (1.002)
t x = dx/dt; x = d2x/dt2; etc.
l
2 THE LAGRANGIAN FORMULATION OF MECHANICS [§1.01
Notice that this is a second-order differential equation since it involves
the second derivative of x with respect to t. It has as a general solution:
x = A sin {Qt+ 0); Q = (K/Mf. (1.003)
Here A and 0 are the familiar amplitude and phase, respectively. The
presence of these two arbitrary and independent parameters indicates
that there is a two-fold infinity of motions available to the oscillator;
the specification of A and d determines one particular motion out of
this continuum of possible motions. Other pairs of quantities, e.g. the
initial displacement x and the initial velocity x, could serve equally
0 0
well as determinative parameters in this situation and are often used
for this purpose. Equations expressing A and 0 in terms of xo and xo
and vice versa can be easily derived.
It might be supposed that nothing could be added to this simple
analysis which so quickly produces a complete description of the
motion of the oscillator. In a sense, this is true. However, another
method, which reduces (1.002) to a first-order differential equation
before finally solving it, can also be brought to bear upon this problem.
Although obviously not necessary in the present case, this method is
very valuable in the analysis of more complicated systems. It employs
the concepts of potential energy and kinetic energy and introduces
other useful ideas.
The potential energy V(x) is energy stored in the spring when the
displacement of M is increased from zero to x; it can be found by
integrating the force an external agent would have to apply to the end
of the spring to produce the required extension. The force in question,
F', is equal to — F, i.e. to Kx. Thus:
x x
V(x) = JKxdx= \Kx2. (1.004)
o
A potential energy exists and is derivable from the force whenever, as
here, the latter can be integrated to yield a single-valued function of
position.* Conversely, in such cases, the force can be obtained from
t A force having this characteristic is called a conservative force.
§ 1.01] THE HARMONIC OSCILLATOR 3
the potential energy by applying the following simple formula:
(1.005)
From this it is evident that a constant added to the potential energy
would not change the force in any way and would therefore have no
effect upon the motion of the oscillator. The privilege of adding such
a constant to the potential energy (and therefore to the total energy
also) is exercised in many situations in physics. In the present example,
the potential energy without an additive constant can be plotted as
the parabolic curve shown in Fig. 1.02.
Energy
Potential Energy, V{x)
Total Energy, f
FIG. 1.02. Harmonic oscillator potential energy curve and associated
concepts.
The expression for the kinetic energy is the well-known \Mx*\ the
total energy therefore becomes:
(1.006)
Differentiation of this with respect to time yields:
E = Mxx+Kxx = x(Mx + Kx) = 0. (1.007)
The null result follows because the original differential equation (1.002)
asserts that (Mx -f- Kx) is zero. The fact that E = 0 or that E is constant
is, of course, an expected result and E in this context is said to be a
constant of the motion. The equation expressing this constant in terms
of x and x, namely (1.006), is actually a first-order non-linear differen-
tial equation of motion; its discovery is tantamount to an integration
of equation (1.002) with E playing the part of a constant of integration*
4 THE LAGRANGIAN FORMULATION OF MECHANICS [§1.01
An expression appearing in this way is often called a first integral of the
motion.
Equation (1.006) may now be solved for x and the variables separ-
ated:
(1.008)
On rearrangement, this becomes:
(1.009)
This may be integrated to yield / as a function of x:
t = (M/Kf arc sin [x(K/2Ey] + t. (1.010)
e
The constant t, which is the second constant of integration produced
e
in this analysis, is of the same mathematical genus as E itself and
quantities like it are also classified as constants of the motion. Equa-
tion (1.010) can be solved for x:
x = (2E/KY sin Q(t-t). (1.011)
e
This is obviously equivalent to (1.003) with (2EjKf = A and -Qt = 0.
e
Notice that when t = t, the mass M passes through its neutral posi-
e
tion while moving to the right. A constant of the motion such as t,
e
which identifies the instant at which a moving body passes through
some specially designated point on its path, is called an epoch.*
The solution in the form of (1.011) involves, as it must, two arbi-
trary and independent determinative parameters. These are none other
than the constants of the motion or, if one prefers, the constants of
integration E and t. It follows that E and t, A and 0, x and x, are
e e 0 0
all different forms of the same two entities; any one of these pairs
may be obtained from any other. It is clear that two constants of the
t This term is borrowed from celestial mechanics where it usually refers to the
time of periapsidal passage of a planet or satellite.
