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QUANTUM MECHANICS

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QUANTUM MECHANICS
Second Edition

V.K. Thankappan
Deparfment oj Physics
UnivtrsityojCalicllf, Kerala
India

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PREFACE TO THE SECOND EDITION
This second edition differs from the first edition mainly in the addition of a
chapter on the Interpretational Problem. Even before the printing of the frrst
edition, there was criticism from some quarters that the account of this problem
included in the introductory chapter is too sketchy and brief to be of much use to
the students. The new chapter, it is hoped, will remove the shortcoming. In
addition to a detailed description of the Copenhagen and the Ensemble Interpretations, this chapter also contains a brief account of the Hidden-Variable Theories
(which are by-products of the interpretational problem) and the associated
developments like the Neumann's and Bell's theorems. The important role
played by the Einstein-Podolsky-Rosen Paradox in defining and delineating the
interpretational problem is emphasized. Since the proper time to worry over the
interpretational aspect is after mastering the mathematical fonnalism, the chapter
is placed at the end of the book.
Minor additions include the topics of Density Matrix (Chapter 3) and Charge
Conjugation (Chapter 10). The new edition thus differs from the old one only in
some additions, but no deletions, of material.
It is nearly two years since the revision was completed. Consequently. an
account of certain later developments like the Greenbetger-Home-ZeilingerMermin experiment [Mennin N.D. Physics Today 36 no 4, p. 38 (1985») could not
be included in Chapter 12. It would, however, be of interest to note that the
arguments against the EPR experiment presented in Section 12.4 could be

extended to the case of the GHZ-Mermin thought-experiment also. For, the
quantum mechanically incorrect assumption that a state vector chosen as the
eigenvector of a producl of observables is a common eigenvector of the individual
(component) observables, is involved in this experiment as well.
Several persons have been kind enough to send their critical comments on the
book as well as suggestions for improvement. The author is thankful to all of
them. and. in particular. to A.W. Joshi and S. Singh. The author is also thankful
to P. Gopalakrishna Nambi for permitting to quote. in Chapter 12, from his Ph.D
thesis and 10 Ravi K. Menon for the usc of some material from his Ph.D work in
this chapter.
January 1993

V.K. THANKAPPAN

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PREFACE TO THE FIRST EDITION
This book is intended to serve as a text book for physics students at the M.Sc.
and M. Phil (Pre-Ph.D.) degree levels. It is based, with the exception of Chapter
I. on a course on quantum mechanics and quantum field theory that the author
taught for many years, starting with 1967, at Kurukshetra University and later at
the University of Calicut. At both the Universities the course is covered Over a
period of one year (or two semesters) at the final year M.Sc. level. Also at both

places, a less formal course, consisting of the developments of the pre-quantum
mechanics period (1900-1924) together with some elementary applications of
SchrOdinger's wave equation, is offered during the first year. A fairly good
knowledge of classical mechanics. the special theory of relativity, classical electrodynamics and mathematical physics (courses on these topics are standard at
most universities) is needed at various stages of the book. The mathematics of
linear vector spaces and of matrices, which play somewhat an all-pervasive role
in this book. are included in the book, the former as part of the text (Chapter 2)
and the latter as an Appendix.
Topics covered in this book. with a few exceptions, are the ones usually found
in a book on quantum mechanics at this level such as the well known books by
L. l. Schiff and by A. Messiah. However, the presentation is based on the view
that quantum mechanics is a branch of theoretical physics on the same footing as
classical mechanics or classical electrodynamics. As a result, neither accounts of
the travails of the pioneers of quantum theory in arriving at the various milestones
of the theory nor descriptions of the many experiments that helped them along the
way, are induded (though references to the original papers are given). Instead,
the empha'iis is on the ba'iic principles, the calculational techniques and the inner
consistency and beauty of the theory. Applications to particular problems are
taken up only to illustrate a principle or technique under discussion. Also, the
Hilbert space fonnalism, which provides a unified view of the different fonnulations of nonrelativistic quantum mechanics, is adopted. In particular, SchrOdinger's and Heisenberg's fonnulations appear merely as different representations,
analogous respectively to the Hamilton-Jacobi theory and the Hamilton's
formalism in classical mechanics. Problems are included with a view to supplementing the text.
From ill) early days, quantum mechanics hm; hccn bedevilled by a controversy
among its founders regarding what has come to be known as the Interpretational
Prohlem. Judging from the number of papers and books still appearing on this
topic. the controversy is far from settled. While this problem does not affect either
the mathematical framework of quantum mechanics or its practical applications,

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I···
VIII

PREFACE

a teacher of quantum mechanics cannot afford to be ignorant of it It is with a
view to giving an awareness of this problem to the teacher of this book that
Chapter 1 is included (students are advised to read this chapter only at the end, or
at least after Chapter 4). The chapter is divided into two parts: The first part is a
discussion of the two main contestants in the arena of interpretation-the Statistical (or, Ensemble) and the Copenhagen. In the second part, the path-integral
formalism (which is not considered in any detail in this book) is used to show the
connection between the 'If-function of quantum mechanics on the one hand and
the Lagrangian function L and the action integral S of classical mechanics on the
other. This too has a bearing on the interpretational problem. For, the interpretational problem is, at least partly, due to the proclivity of the Copenhagen school
to identify 'If with the particle (as indicated by the notion, held by the advocates
of this school, that observing a particle at a point leads to a "collapse" of the
'If-function to that point!). But the relationship between S and 'If suggests that, just
as S in classical mechanics, 'If in quantum mechanics is a function that characterises the paths of the particle and that its appearance in the dynamical equation
of motion need be no more mysterious than the appearance of S or L in the
classical equations of motion.
The approach adopted in this book as well as its level presumes that the course
will be taught by a theoretical physiCist. The level might be a little beyond that
currently followed in some Universities in this country, especially those with few
theorists. However, it is well to remember in this connection that, during the last
three decades, quantum theory has grown (in the form of quantum field theory)
much beyond the developments of the 1920's. As such, a quantum mechanics
course at the graduate level can hardly claim to meet the modem needs of the
student if it does not take him or her at least to the threshOld of quantum field
theory.

In a book of this size, it is difficult to reserve one symbol for one quantity. Care
is taken so that the use of the same symbol for different quantities does not lead
to any confusion.
This book was written under the University Grants Commission's scheme of
preparing University level books. Financial assistance under this scheme is
gratefully acknowledged. The author is also thankful to the,National Book Trust,
India, for subsidising the publication of the book.
Since the book had to be written in the midst of rather heavy teaching assignments and since the assistance of a Fellow could be obtained only for a short
period of three months, the completion of the book was inordinately delayed.
Further delay in the publication of the book was caused in the process of fulfilling
certain formalities.
The author is indebted to Dr. S. Ramamurthy and Dr. K.K. Gupta for a
thorough reading of the manuscript and for making many valuable suggestions.
He is also thankful to the members of the Physics Department, Calicut University,
for their help and cooperation in preparing the typescript

MarCh 1985

V.K.T~P~

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CONTENTS
v

Preface to the Second Edition

vii


Preface to the First Edition
Chapter 1. INTRODUCTION

1

1.1 The Conceptual Aspect 1
1.2 The Mathematical Aspect 9
Chapter 2. LINEAR VECfOR SPACES

19

2.1 Vectors 19
2.2 Operators 31
2.3 Bra and Ket Notation for Vectors 51
2.4 Representation Theory 52
Co-ordinate and Momentum Representation 59

Chapter 3. THE BASIC PRINCIPLES

63

3.1 The Fundamental Postulates 63
3.2 The Uncertainty Principle 75
3.3 Density Matrix 84
Chapter 4. QUANTUM DYNAMICS

87

4.1 The Equations of Motion 87
The SchrOdinger Picture 88

The Heisenberg Picture 94
The Interaction Picture 97

4.2 Illustrative Applications 98
The Linear Hannonic Oscillator 98
The Hydrogen Atom J JJ

Chapter 5. THEORY OF ANGULAR MOMENWM
5.1
5.2
5.3
5.4

The Definition 120
Eigenvalues and Eigenvectors 122
MatFix Representation 126
Orbital Angular Momentum 129

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120


x

QUANTUM MECHANICS

5.5 Addition of Angular Momenta 138
Oebsch-Gordon Coefficients 138
Racah Coefficients 148

The 9j-Symbols 154

5.6 Angular Momentum and Rotations 159
5.7 Spherical Tensors 173
5.8 Consequences of Quantization 179
Chapter 6. INV ARIANCE PRINCIPLES AND CONSERVATION
LAWS

181

6.1 Symmetry and Conservation Laws 182
6.2 The Space-Time Symmetries 183
Displacement in Space: ConselVation of Linear Momentum 184
Displacement in Time: ConselVation of Energy 187
Rotations in Space: ConselVation of Angular Momentum 188
Space Inversion: Parity 188
Time Reversal Invariance 191

Chapter 7. THEORY OF SCATIERING

196

7.1 Preliminaries 196
7.2 Method of Partial Waves 201
7.3 The Born Approximation 224
Chapter 8. APPROXIMATION METHODS

237

8.1 The WKB Approximation 237

8.2 The Variational Method 256
Bound States (Ritz Method) 256
Scbwinger's Method for Phase Shifts 263

8.3 Stationary Perturbation Theory 267
Nondegenerate Case 270
Degenerate Case 274

8.4 Time-Dependent Perturbation Theory 284
Constant Perturbation 287
Harmonic Perturbation 293
Coulomb Excitation 300

8.5 Sudden and Adiabatic Approximations 304
Sudden Approximation 304
Adiabatic Approximation 308

Chapter 9. IDENTICAL PARTICLES
9.1 The Identity of Particles 319
9.2 Spins and Statistics 324
9.3 Illustrative Examples 325
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319


CONTENTS

Chapter 10. RELATIVISTIC WAVE EQUATIONS


322

10.1 Introduction 332
10.2 The First Order Wave Equations 336
The Dirac Equation 338
The Weyl Equations 374

10.3 The Second Order Wave Equations 377
The Klein-Gordon Equation 378
Wave Equation of the Photon 390

lOA Charge Conjugation 384

Chapter 11. ELEMENTS OF FIELD QUANTIZATION
11.1
11.2
11.3
llA

390

Introduction 390
Lagrangian Field Theory 390
Non-Relativistic Fields 398
Relativistic Fields 403
The Klein-Gordm Field 405
The Dirac Field 412
The Electromagnetic Field 418

11.5 Interdcting Fields 425


Chapter 12. THE INTERPRET ATIONAL PROBLEM

445

12.1 The EPR Paradox 445
12.2 The Copenhagen Interpretation 448
12.3 The Ensemble Interpretation 454
1204 Explanations of the EPR Paradox 459
12.5 The Hidden-Variable Theories 463
Appendix A. MATRICES

472

Definition 472
Matrix Algebra 473
Important Scalar Numbers Associated with a Square Matrix 476
Special Matrices 479
Matrix Transformations 481
Solution of Linear Algebraic Equations 482
Eigenvalues and Eigenvectors 484
Diagonalizability of a Matrix 488
Bilinear, Quadratic and Hermitian Forms 490
Infinite Matrices 491
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QUANTUM MECHANICS

xii


Appendix B. ANTll...INEAR OPERATORS

494

Appendix C. FOURIER SERIES AND FOURIER TRANSFORMS

500

Fourier Series 500
Fourier Transforms 504
Appendix D. DIRAC DELTA FUNCTION

509

Appendix E. SPECIAL FUNCTIONS

512

Hermite Polynomials 512
Laguerre Polynomials 516
Legendre Polynomials 519
Bessel Functions 522
531

Index

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CHAPTER 1

INTRODUCTION
Quantum theory, like other physical theories, has two aspects: the mathematical
and the conceptual. In the former aspect, it is a consistent and elegant theory and
has been enormously successful in explaining and predicting a large number of
atomic and subatomic phenomena. But in the latter aspect, which "inquires into
the objective world hidden behind the subjective world of sense perceptions"!, it
has been a subject of endless discussions without agreed conclusions2 , provoking
one to remark that quantum theory appears to be "so contrary to intuition that the
experts themselves still do not agree what to make of it,,3. In the following section, we give a brief account of the genesis of this conceptual problem, which has
defied a satisfactory solution (in the sense of being acceptable to all) in spite of
the best efforts of the men who have built one of the most magnificent edifices of
human thought. And in Section 1.2 is presented a preview of the salient features
of the mathematical aspect of the theory.

1.1 THE CONCEPTUAL ASPECT
In order to understand the root cause of the conceptual problem in quantum
mechanics, we have to go back to the formative years of the theory. QuaI1ltirtl
theory originated at a time when it appeared that Classical physics had at last
succeeded in neatly categorising all physical entities into two groups: matter
and radiation (or field). Matter was supposed to be composed of 'particles'
obeying the laws of Newtonian (classical) mechanics. After the initial
controversy as to whether radiation consists of 'corpuscles' or 'waves', Fresnel's
work4 on the phenomenon of diffraction seemed finally to settle the question in
favour of the latter. Maxwell's electromagnetic theory provided radiation with a
theory as elegant as the Lagrangian-Hamiltonian formulation of Newtonian
mechanics.
1.
'I

~.

3.

4.

Lande. A.• Quantum Mechanics (Pitman Publishing Corporation, New York 1951), p. 7.
See, for example, Lande, A., Born. M. and Biem, W., Phys. Today, 21, No.8, p. 55 (1968)
Ballentine, L.E. et al. Phys. Today, 24, No.4, p. 36 (1971).
Dewitt, B., Phys. Today. 23, No.9, p. 30 (1970).
See, Born, M. and Wolf, E., Principles of Optics (pergamon Press, Oxford 1970), IV Edition
pp. xxiii-xxiv.

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QUANTUM MECHA~CS

Particles and Waves in Classieal Physics
Now, a particle, according to classical physics, has the following characteristics:
PI. Besides having certain invariant attributes such as rest mass, electric
charge, etc., it occupies a finite extension of space which cannot, at the same
time, be occupied by another particle.
P2. It can transfer all, or part, of its momentum and (kinetic) energy 'instantaneously' to another particle in a collision.
P3. It has a path, or orbit, characterised by certain constants of motion such as
energy and angular momentum, and determined by the principle of least
action (Hamilton'S principle).
On the other hand, a monochromatic harmonic wave motion is characterised
by the following:
WI. A frequency v and a wavelength A, related to each other by

vA=(roIk)=v,
(1.1)
where, v is the phase velocity of the wave motion.
W2. A real (that is, not complex) function
'I'.,jr, t) = 4>(k· r - wt), referred to as the wave amplitude or wave function, that satisfies the classical wave equation,
2

a 4> = v zv z",
at
Z

'1"

(1.2)

From the linearity (for a given (0) of Eq. (1.2) follows a very important property of wave motions 5; If '1'1> '1'2> •.. represent probable wave motions, then a linear
superposition of these also represents a probable wave motion. Conversely, any
wave motion could be looked upon as a superposition of two or more other wave
motions. Mathematically,
'P(r, t) =LjCj'l'j(r, t),
(1.3)
where the c/s are (real) constants. Eq. (1.3) embodies the principle of superpo-

sition. expressed in the preceding statements. It is the basis of the phenomenon
of interference, believed in classical physics to be an exclusive characteristic of
wave motions6 •
Now, experimental and theoretical developments in the domain of microparticles during the early part of this century were such as to render the above concepts of particles and waves untenable. For one thing, it was found, as in the case
of electron diffraction (Davisson and Germer 1927f, that the principle of super5.
6.


7.

In the following. we will suppress the subscripts CI) and k. so that lV... t (r, I) is written as IV (r, I).
Classical wave theory also allows for the superposition of wave motions differing in frequencie';
(and, thus, in the case of a dispersive medium, in phase velocities). Such a superposition lear.s
to a wave packel which, unlike monochromatic wave motions, shares the particle's propelly
(PI) of being limited in extension (see Appendix C).
The experimental discovery of electron diffraction was preceded by theoretical specUlation by
Louis de Broglie (1923) that matter-particles are associated with waves whose wavelength A is
related to the particle-momentum p by A=hlp. where h is the universal constant introduced
.:arlier by Max Planck (1900).
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l',

I RljI)l;CTION

position plays an important role in the motion of particles also. For another,
radiation was found to share property P2 listed above as a characteristic of particles (Photoelectric and Compton Effects)8. It was, thus, clear that the classical
concepts of particles and waves needed modification. It is the extent and the
nature of these mOdifications that became a subject of controversy.

The Two Interpretations
There have been two basically different schools of thought in this connection.
One, led by Albert Einstein and usually referred to as the Statistical (or Ensemble)
Interpretation of quantum mechanics 9 , maintains that quantum theory deals with
statistical properties of an ensemble of identical (or, 'similarly-prepared') systems, and not with the motion of an individual system. The principle of superposition is, therefore, not in conflict with properties PI and P2, though it is not
consistent with P3. However, unlike PI, P3 is not really a defining property of
particles, but is only a statement of the dynamical law governing p:U"licles (in

classical mechanics). In place of P3, quantum theory provides a law which is
applicable only to a statistical ensemble and which, of eourse, reduces to [>3 as an
approximation when conditions for the validity of classical mechanics are satis· 'dlO .
f Ie
The other school, led by Niels Bohr and known as the Copenhagen I nterpreunion, advocates radical departure from classical concepts and not just their
m;)dification. According to this school, the laws of quantum mechanics, and in
particular the principle of superposition, refer to the motion of indi vidual system s.
:s.JCh a viewpoint, of course, cannot be reconciled with the classical concept of
particles as embodied in Pl. The concept of 'wave particle duality' is, thercfore,
:niroduced according to which there arc neiUlCr particles nor waves, but only (in
c1a~sicalterminology) particle-like behaviour and wave-like behaviour, one and
the same physical entity bcing capable of both. A more detailed account of this
interpretation is given in Chapter 12; the reader is also referred to the book by
Jammerll and the article by StappI2.
8. It was iII explaining the photoelectric effect that Albert Einstein (1905) reintroduced the concept
of light corpuscles. originally due to Isaac Newton, in the fonn of light quanta which were later
named photons by G.N. Lewis (1926). Priorto this, Max Planck (1900) had introduced the idea
that exchange of energy het ween matter and radiation could take place only in units of hv. v
heing the frequency of the radiation ..
9. For a comparatively recent exposition of the Statistical Interpretation, see, L. E. Ballentine, Revs
Mod. Phys. 42, 357 (1970).
10. Thankappan, V.K. and Gopalakrishna Nambi, P. Found. Phys. 10,217 (1980); Gopalakrishna
Nambi, P. The Interpretational Problem in Quantum Mechanics (ph. D Thesis: Universily of
Cali cut, 1986), Chapter 5.
11. Jammer, M., The Conceptual Development of Quantum Mechanics (McGraw-lIiIl, New Yo k,
1966), Chapter 7.
12. Stapp, II.P., Amer. J. Phys. 40,1098 (1972).

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QUANTUM MECHANICS

4

The Tossing of Coins
It should be emphasized that the dispute between the two schools is not one that
could be settled by experiments. For, experiments in the domain of microparticles
invariably involve large number of identical systems, and when applied to large
numbers, both the interpretations yield the same result. Besides, even if it were
possible to make observations on a single isolated particle, the results could not
be taken as a contradiction of the Copenhagen Interpretation 13 • The example of
the tossing of coins might serve to illustrate this. The law governing the outcome
of tossings of identical coins is contained in the following statement: "The
probability for a coin to fall with head up is one half". According to the Statistical
Interpretation, this statement means that the ratio of the number of tosses resulting
in head up to the total number would be one half if the latter is large enough, the
ratio being nearer to the fraction half the larger the number of tosses. In any single
toss, either the head will be up or it will be down, irrespective of whether somebody is there to observe this fact or not. However, the application of the law
would be meaningless in this case since it is incapable of predicting the outcome
of a single toss. This incapability might stem from an ignorance of the factors
(parameters) that govern, and the way they influence, the motion of the coin. One
cannot, therefore, rule out the possibility of a future theory which is capable of
predicting the outcome of a single toss, and from which the above-mentioned
statistical law could be deduced (see Chapter 12, Section 5).
The Copenhagen Interpretation, on the other hand, insists that the law is
applicable to the case of a single toss, but that it is the statement that the coin falls
with either head-up or head-down that is meaningless. When no observer is
present, one can only say that the coin falls with partially (in this case, half)
head-up and partially head-down. If an observation is made, of course, it will be

found that the coin is either fully head-up or fully head-down but the act of
observation (that is, the interaction between the observer and the coin) is held
responsible for changing the coin from a half head-up state to a fully head-up state
(or a fully head-down state). Agreement with observation is, thus, achieved, but
at a heavy price. For, the coin now is not the classical coin which was capabk of
falling only with head-up or with head-down but not both ways at the same time.
Also, the role of the observer is changed from that of a spectator to an active
participant who influences the outcome of an observation. Since the law is presumed to govern the outcome of an individual tossing, it follows that the search
for a more fundamental theory is neither warranted nor likely to be fruitful.

A Thought Experiment
At this stage, one might wonder why one has to invent such a complicated scheme
of explanation as the Copenhagen Interpretation when the Statistical Interpre13.

According to the Statistical Interpretation, quantum mechanics does not have anything to say
about the outcome of observations on a single particle.
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5

INTRODUCTION

tation is able to account for the observed facts without doing any violence to the
classical concept of the coin. Unfortunately, phenomena in the world of microparticles are somewhat more complicated than the tossings of coins. The complication involved is best illustrated through the following thought-experiment.
Imagine a fixed screen W with two holes A and B (see Fig. 1.1). In front of this

x

w


X
x/l.

.................................... 0

G

Fig. 1.1. The double slit interference experiment.

screen is an election gun G which shoots out electrons, having the same energy,
uniformly in all directions. Behind W is another ~creen X on which the arrival of
the individual electrons Can be observed. We first close B and observe the electrons arriving on X for a certain interval of time. We plot the number of electrons
versus the point of arrival on X (the screen X will be assumed to be
one-dimensional) and obtain. say, the curve fA shown in Fig. 1.2. Next we close

Fig. 1.2. The distribution of particles in the double slit interference experiment when only slit A is
open (/.). when only slit B is open (I.) and when both A and B are open (I•• ). I represcn;s
the sum of IA and lB'
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QUANTUM MECHANIC~

6

A and open B and make observation for the same interval of time, obtaining 1.I1C
curve lB' We now repeat the experiment keeping both A and B open. We should

expect to get the curve I which is the sum of IAandlB , but get the curve lAB instead.

This curve is found to fit the formula
lAB (x) =1 'l'A(X) + 'l'B(X)

12,

(1.4)

with

(1.5)

where 'l'A(X) and 'l'B(X) are complex functions of x.
Apparently, our expectation that an electron going through A should not be
knowing whether B is closed or open, is not fulfilled. Could it be that every
electron speads out like a wave motion after leaving the gun, goes through both
the holes and again localises itself on arriving at X? Eqs. (1.4) and (1.5) support
such a possibility since these are identical (except for the complex character of 'l'A
and 'l'B) with the equations relating amplitudes and intensities of a wave motion.
In order to test this, we set up a device near A to observe all the electrons passing
through A, before they reach X. We will assume that the electrons arriving on X
that are not registered by the device have come through B. We find that the
electrons coming through A are, indeed, whole electrons. But, to our surprise, we
find that the curves corresponding to the electrons coming through A and B
respectively are exactly similar to IA and IB , implying that the distribution of
electrons on X is now represented not by the curve lAB' but by the Curve I. This
shows that electrons are particles conforming to the definition PI, at least whenever We make an observation on them.
Let us summarise below the main results of the experiment:
El. The number of electrons arriving at a point x on the screen X through A
depends on whether B IS closed or open. The total number of electrons
arriving on X through A is, however, independent of B14.

£2. Observations affect the outcome of experiments.
The results of the electron experiment are easily accommodated in the
Copenhagen Interpretation. The basic law governing the electrons in this case is
contained in the statement that the probability for an electron that has arrived on
X to have come through one of the holes, say A, is P and through the other hole is
(1 - P); where 0 ~ P ~ 1. Since this law governs the motion of each and every
electron, when both the holes are open and when no observations are made to see
through which hole the electrons are passing, it should be presumed that every
electron passes, in a wave-like fashion, through both the holes. Alternatively, one
14.

This follows from the relation [see Eq. (1.32)],

LJA8 (X)dx;

11

IVA (X)+1V8(X)l'iU

=L 1IVA (x) I' dx + L

11V8(X) I' dx

= LJ.
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LI'HRODUCTIOK

could take the view that, as far as the distribution fAB is concerned, the question as
to whether a particular electron has come through one or both holes, is not a
meaningful one for physics as no experiment can answer the question without
affecting the distribution JAB' For any experiment designed to answer the question
reveals the electron to be a partiele capable of passing through only one hole, but
,hen the distributon is also changed from the one corresponding to classical waves
(JAB) to one corresponding to classical panicles (I), justifying the hypothesis that
the act of observation tmnsforms the electron from a wave-like object extended in
space to a particle-like object localised in space. The dichotomy on the part of the
electron is easily understood if we realize that particles and waves are mercly
complementary aspects of one and the same physical entity l5, anyone experimen t
being capable of revealing only one of the aspects and not bothl6.
Thus, the Copenhagen Interpretation docs not appear so far-fetched when
viewed in the context of the peculiar phenomena obtaining in the world
of III icroparticles. However, it denies objective reality to physical phenomena,
and prohibits physics from being concerned with happenings in between obser"ations. The question, how is it that the act of observation at one location causes
an electron, that is supposed to be spread over an extended space, to shrink to this
location?, is dubbed as unphysical. The interpretation, thus, leaves one with an
impression that quantum theory is mysterious as no other physical theory is.
-;'l1ose who find it difficult to be at home with this positivist philosophy underlyi Ilg the Copenhagen Interpretation, will find the Statistical Interpretation morc
attractive. Let us see how this interpretation copes with the results of the clectror
~xperimcnt.

A.ccording to the Statistical Interpretation, the probability law stated carlier a,
governing the motion of electrons, is a statistical one and is applicable only wh' n
J large enough number of 'similarly-prepared' electrons are involved. The dIStribution of electrons coming through, say hole A, on the screen X being the resull
of a statistical law, need not be the same when the screen W has only hole A on it
as when both A and B are there, just as the distribution of head-up states in the
tossings of coins with only one side is different from the distribution of head-up
,tates in the tossings of coins with two sides. Let us elaborate this point: The

1istribution of electrons coming through hole A on X, is a result of the momentum
ransfer taking place between the electrons and the screen WatA. The expectation
.hat this momentum transfer, and hence the distribution, are unaffected by the
Iddition of another hole B on W is based on the presumption that a screen with
wo holes is merely a superposition of two independent screens with one hole
:1ch. The experimental result shows that the presumption is not justified. The
5.

D.

The Principle of Complementarity, which seeks to harmonize the mutually exclusive notions of
particles and waves, was proposed by Neils Bohr(I928). A detailed account of the principle i,
given in the reference quoted in footnote II as well as chapter 12.
This limitation on the part of expcrinlents is enshrined in the Uncertainty Principle proposed by
Wemer lIeisenberg (1927), which puts a limit on the precision with which complcmenlar:'
variables such as position and momentum of a particle can be measured,

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8

QUANTUM MECHAA1CS

fact that the momentum transfer at the hole A when both A and B are open is
different from the momentum transfer when only A is open, could also be understood on the basis of the quantization of the momentum transfer resulting from the
periodicity in space of the holes in the fonner case (W. Duane 1923)17.
Thus, experimental result E 1 is easily understood on the basis of the Statistical
Interpretation. As for E2, one should distinguish between the two ways in which
observations affect the outcome of experiments. One is that observations on

electrons coming through hole A affect their distribution on the screen X. This
could be understood as due to the fact that the momentum transfer involved in the
act of observation is not negligible compared with the momentum of the electrons
themselves. The other is that observations on electrons coming through hole A
affect (apparently) also the distribution of electrons coming through hole B. In
order to accommodate this fact within the framework of the Statistical Interpretation, one has to assume that the statistical correlation that exists between two
paths (of the electrons), one passing through A and the other through B, is such
that it can be destroyed by disturbing only one of the paths. In fact, a correlation
represented by the linear superposition of two functions 'l'A and 'l'B as in Eq. (1.3),
whose phases are proportional to the classical actions associated with the paths,
satisfies such a condition 1o • For, as is known from the classical theory of waves,
the correlation can be destroyed by introducing a random fluctuation in the phase
of O'le of the functions. So in order to understand the experimental result, one has
to assume that observations on the electrons always introduce such a random
variation in the action associated with the path of the electronsl8 •
The 'Mystery' in Quantum Mechanics
Thus, in the course of understanding E2, we are led to introducing a (complex)
function which, in certain aspects such as the applicability of the principle of
superposition, resembles a wave amplitude l9 • This is the really new element in
quantum mechanics; it represents an aspect of microworld phenomena quite
foreign to classical statistical processes such as the tossings of coins. But whereas
the Copenhagen school regards these functions as incompatible with the classical
17.

18.
19.

The period would be the distance d between the holes. According to Duane's hypothesis the
momentum transfer between the screen Wand the electron, when both A and B are open, has to
be an integral multiple of (h/d), h being the Planck's constant. This relationship is identical with

the de Broglie relation,p = hf).. (see footnote 7) if we recognise the wavelength A. as a periodicity
in space. Duane's hypothesis is an extension, to the case of the linear momentum, of the earlier
hypotheses of Max Planck (footnote 8) and of Neils Bohr (1913) on the relationship between
the quantization of energy and periodicity 't in time [energy = integral multiple of (hl't)l and
quantization of angular momentum and periodicity 21t in angles [angular momentum = integral
multiple of (h/21t)l, respectively.
This is nothing but the Uncertainty Principle.
Erwin Schrodinger (1926) was the first to introduce these functions and to derive an equation of
motion (the Schrodinger equation) for them. The physical interpretation of these functions as
probability amplitudes which are related to the probability of fmding the particles at a space
point in the same way as wave amplitudes are related to wave intensities, is due to Max Bom
(1926).
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INTRODUCTION

9

concept of particles, and invests them with a certain amount of physical reality,
thereby endowing quantum mechanics with an aura of mystery, the Statistical
Interpretation makes a distinction between these functions and the physical
entities involved. The physical entities are the electrons or other microparticles
(conforming to definition PI), but the functions are mathematical entities
characterising the paths of the microparticles just as the action in classical
mechanics is a mathematical function characterising the classical paths of
particles. The functions, thus, determine the dynamical law governing the motion
of microparticles. This law is, admittedly, new and different from the dynamical
law in classical mechanics. BOt, the!}, it is not the first time in physics that a set
of rules (theory) found to be adequate for a time, proved to be inadequate in the

light of new and more accurate experimental facts. Also, the fact that quantum
mechanics does not provide an explanation to the dynamical law or laws (such as
the principle of superposition) underlying it, does not justify alleging any special
mystery on its part, since such mysteries are parts of every physical theory. For
example, classical mechanics does not explain why the path of a particle is
governed by Hamilton's principle, eletromagnetic theory does not offer an
explanation for Coulomb's or Faraday's laws and the theory of relativity does not
say why the velocity of light in vacuum is the same in all inertial frames. Thus,
from the viewpoint of the Statistical Interpretation, quantum mechanics is no
more mysterious than other physical theories are. It certainly represents an
improvement over classical mechanics since it is able to explain HamilLOn'~;
principle, but an explanation of the fundamental laws underlying quantum
mechanics themselves need be expected only in a theory which is more fundamental than quantum mechanics.
It should be clear from the foregoing discussion that the choice between the
Copenhagen and the Statistical Interpretations could be one of individual taste
only. Anyway, the mathematical formalism of quantum mechanics is independent of these interpretations.

1.2 THE MATHEMATICAL ASPECT
One or the other branch of mathematics plays a dominant role in the formulation
of every physical theory. Thus, classical mechanics and electromagnetic theory
rely heavily on differential and vector calculus, while tensors playa dominant role
in the formulation of the general theory of relativity. In the case of quantum
mechanics, it is the mathematics of the infinite-dimensional linear vector spaces
(the Hilbert space) that play this role. In this section, we will show how the basic
laws of quantum mechanics 20 make this branch of mathematics the most appropriate language for the formulation of quantum mechanics.
20.

In Ihe fonn originally proposed by Feynman, R.P. [Revs. Mod. Phys. 20. 367 (1948); also,
Feynman. R.P. and Hibbs. A.R.o Quantum Mechanics and Path integrals (McGraw-Hill, New
York 1965)] and latcrmodified by V.K. Thankappan and P. Gopalakrishna Namhi lO• lhe basic

laws of non-relativistic quantum mechanics were discovered during Ihe period 1900-1924
Ihrough Ihe efforts of many physicists, and a consistent Iheory incorporating Ihese laws were
fonnulated during the period 1925-1926 mainly by Erwin Schrodinger (1926) in Ihe fonn of
Wave Muha"ics and by Werner Heisenberg, Max Born and Pascal Jordan (1925-1926) in Ihe
fonn of Matrix Mecha"ics.
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QUANTUM MECHANICS

iO

Now, in classical mechanics the motion of a particle is governed by the
Principle ofI~ast Action (Hamilton's Principle). According to this principle, the
path of a particle between two locations A and Q in space is such that the action S
(Q,IQ : A, IA ) defined by,
S (Q, IQ : A, IA) = J,'Q
Ldl
A

= IAQ pdq -

I;Q H dl,
A

(1.6)

is a minim urn, where L is the Lagrangian, p the momentum and H the Hamiltonhm
of the particle, and IA and tQ are, respectively, the time of departure from A and the
time of arrival at Q. Thus, the path between A and Q is detennined by the variational equation,

(1.7)
=0.

as

We will call the path defined by Eq. (1.7) the classical palh and will denote it by
(Xc and the action corresponding to it by Sc(Q, IQ : A, IA)'
As we have already mentioned, experiments in the domain of microparticles
have shown that the paths of these particles are not governed by the principle of
least action. However, the results of these experiments are consistent with, indeed
suggestive of, the following postulates which could be regarded as the quantum
mechanical laws of motion applicable to microparticles:
Q1. Associated with every path (X of a particle21 from location A to location Q in
space, is a complex function cjl",(Q, IQ : A, IJ given by,
cjl", = a", exp [(il1i)Sal,

(1.8)

where
S",(Q, IQ: A,

IJ =I:Qa L

dl

= IJ pdq -

J,'Q
Hdl.
a

(1.9)

I"" here, has the same meaning as IA in Eq. (1.6) except that it could be

different for the different paths (x. Also 22 , h = h(21t.
Q2. The probability amplilude for a particle to go from A (at some time) to Q
at time IQ is '!fA (Q, IQ)' where,
'!fA (Q, IQ) = L",cjl",(Q, IQ : A, IJ.

(1.1 0)

Q2a. Only those paths contribute to the summation in Eq. (1.10) that differ from
(Xc by less than 1i/2 in action. That is

M", == (S", - Sc) < (1iI2).

(1.1 Oa)

Q3. If A, B, C, '" are locations corresponding to similarly prepared states23 of a
particle in an experimental set up, the number of particles arriving at a point
of a observation, Q, at time IQ from the above locations, is proportional to
1'P(Q, IQ) 12, where,
(1.11)
21.
22.
23.

We assume that the spin of the particle is zero.
The one-letter notation for (h/21t) was first introduced by P.A.M. Dirac (1926), in the form" h".

For this reason, 11 is also called Dirac' s constant.
This phrase stands for 'elements, or members, of an ensemble'.
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!I

INTRODLJCTJOl'\

the cA's being numbers (in general, complex) to be chosen such that
JI\fl12d 3 r Q =1,

O.lla)

where d 3 r Q represents an element of volume containing the point Q.
If (X is a path between A and Q, and ~ a path between Band Q, then, as a consequence of condition (I . lOa), we will have,
(1.12a)
I (Sa - Sp) I-MAE < (N2),

where
(Xc

and

~c

being the classical paths between A and Q and between Band Q,

respectively. Also, corresponding to every path 'a' between A and
tributes to 'I'A ), there will be a path' b' between Band Q such that

I Sa-Sb I

Q (that con-

= ASAB'

Eq. (1.I2b) enables us to say that the phase difference between

0.12b)
\jIA

and

\jiB

is the

quantity (I\"SAJ/tz) whereas inequality (1.10a), from which inequality (1.12<1) folInws, ensures thauhe phase difference is sueh a definite quantity. Now, a definite
rhase ditTcrence between \jIA and \jiB is the condition for A and B to be coherent
.,e,urces (or, similarly-prepared states) from the viewpoint of Q. We will, there!'ore, refer to inequality (1.lOa) as the coherency condition.
Postulate Q3 incorporates the principle of superposition referred to in Sectioll
I.J (Eq. (1.3». However, unlike Ci and \jIi in (1.3), CA and \jIA in Eq. (1.11) are
complex quantities. Therefore, it is not possible to interpret 'I'A and \fl in (l.ll'

'IS

representing wave motions in the physical space24. Also, the principle of sU£' ..
position will conflict with property P I of particles (see, p. 2), if applied to the case
of a single particle. But there is no experimental basis for invalidating PI; on the

contrary, experiments confinn the continued validity of PI by verifying, for
example, that all electrons have the same spin, (rest) mass and electric charge both
before and after being scattered by, say, a crystal. Therefore, the principle of
superposition should be interpreted as applying to the statistical behaviour of a
large number (ensemble) of identical systems. In fact, the terms 'probability
amplitude' and 'number of particles' emphasize this statistical character of the
postulates. However, the really new element in the theory is not its statistical
Character, but the law for combining probabilities. Whereas in the classical statistics, probabilities for independent events are added to obtain the probability for
the combined event (If P A(Q) and P seQ) are, respectively, the probabilities for the
arrival of a particle al Q from A and from B, then, the probability PAB(Q) for the
arrival of a particle at Q from either A or B is given by P AB(Q) = P A(Q) + P Il(Q»
in the new thcory, this is not always so. In particular, whenever criterion (1.12a)
24.

111 classical wave theory also, complex amplitudes are employed sometimes, purely for the sake
of calculational convenience. Care is, theil, taken in the computation of physically significant
quamities sllch as the intensity (,f the wave motion, to separate out the contribution due to the
imaginary pan of the amplitlldes. In quantum mechanics the IVA'S are perforce complex. [sec
Eq. (,1.15 h)1
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