2-- 5l- \

This graduate text introduces relativistic quantum theory, emphasizing
its important applications in condensed matter physics.
Basic theory, including special relativity, angular momentum and particles of spin zero are first reprised. The text then goes on to discuss the
Dirac equation, symmetries and operators, and free particles. Physical
consequences of solutions including hole theory and Klein's paradox are
considered. Several model problems are solved. Important applications
of quantum theory to condensed matter physics then follow. Relevant
theory for the one-electron atom is explored. The theory is then developed to describe the quantum mechanics of many electron systems,
including Hartree-Fock and density functional methods. Scattering theory, band structures, magneto-optical effects and superconductivity are
amongst other significant topics discussed. Many exercises and an extensive reference list are included.
This clear account of relativistic quantum theory will be valuable to
graduate students and researchers working in condensed matter physics
and quantum physics.

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DokOmantasyon Dalre Ba~kanhgl

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Relativistic Quantum Mechanics

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RELATIVISTIC
QUANTUM MECHANICS
WITH APPLICATIONS IN CONDENSED MATTER
AND ATOMIC PHYSICS

Paul Strange
Keele University

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CAMBRIDGE UNIVERSITY PRESS
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo
Cambridge University Press
The Edinburgh Building, Cambridge CB2 2RU, UK
Published in the United States ofAmerica by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521562713
© Cambridge University Press 1998
This publication is in copyright. Subject to statutory exception
and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without
the written permission of Cambridge University Press.
First published 1998

A catalog1le record/or this publication is available from the British Library
Library a/Congress Cataloguing in Publication data
Strange, Paul, 1956Relativistic quantum mechanics I Paul Strange.
p. cm.
Includes bibliographical references and index.

ISBN 0 521562716
1. Relativistic quantum theory. I. Title.
QCI74.24.R4S87 1998
530.12-dc21 97-18019 CIP
ISBN-13 978-0-521-56271-3 hardback
ISBN-IO 0-521-56271-6 hardback
ISBN-13 978-0-521-56583-7 paperback
ISBN-IO 0-521-56583-9 paperback
Transferred to digital printing 2005

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Marmara Onive~itesi
KOWphane ve DokLinl~I(J:i;a;von
DairEf Ba:,;ka nll 91 •.

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Dedication

Unlike so many other supportive families, mine did not proof-read, or
type this manuscript, or anything else. In fact they played absolutely no
part in the preparation of this book, and distracted me from it at every
opportunity. They are not the slightest bit interested in physics and know
nothing of relativity and quantum theory. Their lack of knowledge in
these areas does not worry them at all. Furthermore they undermine one
of the central tenets of the theory of relativity, by providing me with a
unique frame of reference. Nonetheless, I would like to dedicate this book

to them, Jo, Jessica, Susanna and Elizabeth.

• '!:'""

vii
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Contents

Preface

xv

1.1
1.2
1.3
1.4
1.5
1.6
1.7

The Theory of Special Relativity
The Lorentz Transformations
Relativistic Velocities
Mass, Momentum and Energy
Four-Vectors
Relativity and Electromagnetism
The Compton Effect
Problems


1
2
7
8
14
15
18
21

2
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10
2.11
2.12
2.13
2.14

Aspects of Angular Momentum
Various Angular Momenta
Angular Momentum and Rotations
Operators and Eigenvectors for Spin 1/2

Operators for Higher Spins
Orbital Magnetic Moments
Spin Without Relativity
Thomas Precession
The Pauli Equation in a Central Potential
Dirac Notation
Clebsch-Gordan and Racah Coefficients
Relativistic Quantum Numbers and Spin-Angular Functions
Energy Levels of the One-Electron Atom
Plane Wave Expansions
Problems

23
24
29
30
33
35
38
41
46
50
51
58
60
62
63

3
3.1


Particles of Spin Zero
The Klein-Gordon Equation

64
65

1

IX

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x

Contents

3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9

Relativistic Wavefunctions, Probabilities and Currents
The Fine Structure Constant
The Two-Component Klein-Gordon Equation

Free Klein-Gordon Particles/Antiparticles
The Klein Paradox
The Radial Klein-Gordon Equation
The Spinless Electron Atom
Problems

4

The Dirae Equation
The Origin of the Dirac Equation
The Dirac Matrices
Lorentz Invariance of the Dirac Equation
The Non-Relativistic Limit of the Dirac Equation
An Alternative Formulation of the Dirac Equation
Probabilities and Currents
Gordon Decomposition
Forces and Fields
Gauge Invariance and the Dirac Equation
Problems

99
100
103
109
111
118
121
123
125
127

128

5.4
5.5
5.6
5.7
5.8

Free Particles! Antiparticles
Wavefunctions, Densities and Currents
Free-Particle Solutions
Free-Particle Spin
Rotations and Spinors
A Generalized Spin Operator
Negative Energy States, Antiparticles
Classical Negative Energy Particles?
The Klein Paradox Revisited
Lorentz Transformation of the Free-Particle Wavefunction
Problems

130
130
131
133
139
142
145
150
152
154

156

6
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9

Symmetries and Operators
Non-Relativistic Spin Projection Operators
Relativistic Energy and Spin Projection Operators
Charge Conjugation
Time-Reversal Invariance
Parity
~P/!r
Angular Momentum Again
Non-Relativistic Limits Again
Second Quantization

157
158
160
162
165
167

168
171
175
176

4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
5
5.1
5.2
5.3

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67
72

73
77

85
88

91
98


Contents

xi

6.10
6.11
6.12

Field Operators
Second Quantization in Relativistic Quantum Mechanics
Problems

186
189
198

7

Separating Particles from Antiparticles

7.1
7.2
7.3
7.4
7.5
7.6


The Foldy-Wouthuysen Transformation for a Free Particle
Foldy-Wouthuysen Transformation of Operators

199
200
203
206
214
216
226

Zitterbewegung

Foldy-Wouthuysen Transformation of the Wavefunction
The F-W Transformation in an Electromagnetic Field
Problems

8

One-Electron Atoms

8.1
8.2
8.3
8.4
8.5
8.6
8.7


The Radial Dirac Equation
Free-Electron Solutions
One-Electron Atoms, Eigenvectors and Eigenvalues
Behaviour of the Radial Functions
The Zeeman Effect
Magnetic Dichroism
Problems

9

Potential Problems

9.1
9.2

9.6
9.7

A Particle in a One-Dimensional Well
The Dirac Oscillator
The Non-Relativistic Limit
Solution of the Dirac Oscillator
Expectation Values and the Uncertainty Principle
Bloch's Theorem
The Relativistic Kronig-Penney Model
A One-Dimensional Time-Independent Dirac Equation
A Potential Step
A One-Dimensional Solid
An Electron in Crossed Electric and Magnetic Fields
An Electron in a Constant Magnetic Field

An Electron in a Field for which IEI = clBI
Non-Linear Dirac Equations, the Dirac Soliton
Problems

10

More Than One Electron

10.1
10.2
10.3
10.4

The Breit Interaction
Two Electrons
Many-Electron Wavefunctions
The Many-Electron Hamiltonian

9.3
9.4

9.5

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227
228
231
233
243

248
254
261
263
264
269
269
271
276
280
283
283
286
289
295
298
309
311
315
317
318
322
329
333


~,

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\

I
xii
10.5

I

Contents

10.15
10.16
10.17
10.18

Dirac-Hartree-Fock Integrals
Single-Particle Integrals
Two-Particle Integrals
The Direct Coulomb Integral
The Exchange Integral
The Dirac-Hartree-Fock Equations
The One-Electron Atom
The Many-Electron Atom
Koopmans' Theorem
Implementation of the Dirac-Hartree-Fock Method
Introduction to Density Functional Theory
Non-Relativistic Density Functional Theory
The Variational Principle and the Kohn-Sham Equation
Density Functional Theory and Magnetism
Density Functional Theory in a Weak Magnetic Field

Density Functional Theory in a Strong Magnetic Field
The Exchange-Correlation Energy
Relativistic Density Functional Theory (RDFT)
RDFT with an External Scalar Potential
RDFT with an External Vector Potential
The Dirac-Kohn-Sham Equation
An Approximate Relativistic Density Functional Theory
Further Development of RDFT
Relativistic Exchange-Correlation Functionals
Implementation of RDFT

334
336
337
339
341
345
346
348
351
352
363
365
370
374
374
376
380
386
387

389
391
395
397
398
400

11

Scattering Theory

11.1
11.2
11.3
11.4
11.5
11.6
11.7
11.8
11.9
11.10
11.11
11.12
11.13
11.14
11.15
11.16

Green's Functions
Time-Dependent Green's Functions

The T -Operator
The Relativistic Free-Particle Green's Function
The Scattered Particle Wavefunction
The Scattering Experiment
Single-Site Scattering in Zero Field
Radial Dirac Equation in a Magnetic Field
Single-Site Scattering in a Magnetic Field
The Single-Site Scattering Green's Function
Transforming Between Representations
The Scattering Path Operator
The Non-Relativistic Free Particle Green's Function
Multiple Scattering Theory
The Multiple Scattering Green's Function
The Average T -Matrix Approximation

407
408
411
412
413
416
418
420
426
428
433
435
437
440
443

447
450

10.6

10.7
10.8
10.9
10.10
10.11
10.12

10.13
10.14

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,1

Contents

11.17 The Calculation of Observables

xiii

The Band Structure
The Fermi Surface

The Density of States
The Charge Density
Magnetic Moments
Energetic Quantities
11.18 Magnetic Anisotropy
The Non-Relativistic Limit, the RKKY Interaction
The Origin of Anisotropy

452
452
456
457
461
462
465
467
472
475

12
12.1
12.2
12.3
12.4
12.5
12.6
12.7
12.8
12.9
12.10

12.11

Electrons and Photons
Photon Polarization and Angular Momentum
Quantizing the Electromagnetic Field
Time-Dependent Perturbation Theory
Photon Absorption and Emission in Condensed Matter
Magneto-Optical Effects
Photon Scattering Theory
Thomson Scattering
Rayleigh Scattering
Compton Scattering
Magnetic Scattering of X-Rays
Resonant Scattering of X-Rays

480
481
483
485
494
497
505
509
511
514
522
533

13


Superconductivity
Do Electrons Find Each Other Attractive?
Superconductivity, the Hamiltonian
The Dirac-Bogolubov-de Gennes Equation
Solution of the Dirac-Bogolubov-de Gennes Equations
Observable Properties of Superconductors
Electrodynamics of Superconductors

536
537
539
542
544
547
551

13.1
13.2
13.3
13.4
13.5
13.6

The Uncertainty Principle

556

Appendix B The Confluent Hypergeometric Function
Relations to Other Functions
B.1


559
560

Appendix C

Spherical Harmonics

562

Appendix D

Unit Systems

567

Appendix E

Fundamental Constants

569

Appendix A

References

570

Index


585

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1


,1

Preface

I always thought I would write a book and this is it. In the end, though,
I hardly wrote it at all, it evolved from my research notes, from essays I
wrote for postgraduates starting work with me, and from lecture handouts
I distribute to students taking the relativistic quantum mechanics option in
the Physics department at Keele University. Therefore the early chapters
of this book discuss pure relativistic quantum mechanics and the later
chapters discuss applications of relevance in condensed matter physics.
This book, then, is written with an audience ranging from advanced
students to professional researchers in mind. I wrote it because anyone
aiming to do research in relativistic quantum theory applied to condensed
matter has to pull together information from a wide range of sources
using different conventions, notation and units, which can lead to a lot of
confusion (I speak from experience). Most relativistic quantum mechanics
books, it seems to me, are directed towards quantum field theory and
particle physics, not condensed matter physics, and many start off at
too advanced a level for present day physics graduates from a British
university. Therefore, I have tried to start at a sufficiently elementary
level, and have used the SI system of units throughout.
When I started preparing this book I thought I might be able to write

everything I knew in around fifty pages. It soon became apparent that
that was not the case. Indeed it now appears to me that the principal
decisions to be taken in writing a book are about what to omit. I have
written this much quantum mechanics and not used the word Lagrangian.
This saddens me, but surely must make me unique in the history of
relativistic quantum theory. I have not discussed the very interesting
quantum mechanics describing the neutrino and its helicity, another topic
that invariably appears in other relativistic quantum mechanics texts.
However, as we are leaning towards condensed matter physics in this
book, there are sections on topics such as magneto-optical effects and
xv
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I


XVI

Preface

magnetic anisotropy which don't appear in other books despite being
intrinsically relativistic and quantum mechanical in nature. In the end,
what is included and what is omitted is just a question of taste, and it is
up to the reader to decide whether such decisions were good or bad.
This book is very mathematical, containing something like two thousand
equations. I make no apology for that. I think the way the mathematics
works is the great beauty of the subject. Throughout the book I try to
make the mathematics clear, but I do not try to avoid it. Paraphrasing
Niels Bohr I believe that "If you can't do the maths, you don't understand
it." If you don't like maths, you are reading the wrong book.

There are a lot of people I would like to thank for their help with, and
influence on, my understanding of quantum mechanics, particularly the
relativistic version of the theory. They are Dr E. Arola, Dr P.J. Durham,
Professor H. Ebert, Professor W.M. Fairbairn, Professor J.M.F. Gunn, Prof
B.L. Gyarify, Dr R.B. Jones, Dr P.M. Lee, Dr lB. Staunton, Professor
J.G. Valatin, and Dr W. Yeung.
Several of the examples and problems in this book stem from projects
done by undergraduate students during their time at Keele, and from
the work of my Ph.D students. Thanks are also due to them, C. Blewitt,
H.J. Gotsis, O. Gratton, A.C. Jenkins, P.M. Mobit, and E. Pugh, and
to the funding agencies who supported them (Keele University physics
department, the EPSRC, and the Nuffield foundation).
There are several other people I would like to thank for their general
influence, encouragement and friendship. They are Dr T. Ellis, Professor
M.l Oillan, Dr M.E. Hagen, Dr P.W. Haycock, Mr 1 Hodgeson and Mr
B.O. Locke-Scobie. I would also like to thank R. Neal and L. Nightingale
of Cambridge University Press for their encouragement of, and patience
with, me. Finally, my parents do not have a scientific background, nonetheless they have always supported me in my education and have taken a
keen interest in the writing of this book. Thanks are also due to them,
R.J. and v.A. Strange.
I hope you enjoy this book, although I am not sure 'enjoy' is the right
word to describe the feeling one has when reading a quantum mechanics
textbook. Perhaps it would be better to say that I hope you find this book
informative and instructive. What I would really like would be for you to
be inspired to look deeper into the subject, as I was by my undergraduate
lectures many years ago. Many people think quantum mechanics is not
relevant to everyday life, but it has certainly influenced my life for the
better! I hope it will do the same for you.

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. ,';,":

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1
The Theory of Special Relativity

Relativistic quantum mechanics is the unification into a consistent theory
of Einstein's theory of relativity and the quantum mechanics of physicists
such as Bohr, Schrodinger, and Heisenberg. Evidently, to appreciate relativistic quantum theory it is necessary to have a good understanding of
these component theories. Apart from this chapter we assume the reader
has this understanding. However, here we are going to recall some of
the important points of the classical theory of special relativity. There is
good reason for doing this. As you will discover all too soon, relativistic
quantum mechanics is a very mathematical subject and my experience has
been that the complexity of the mathematics often obscures the physics
being described. To facilitate the interpretation of the mathematics here,
appropriate limits are taken wherever possible, to obtain expressions with
which the reader should be familiar. Clearly, when this is done it is useful
to have the limiting expressions handy. Presenting them in this chapter
means they can be referred to easily.
Taking the above argument to its logical conclusion means we should
include a chapter on non-relativistic quantum mechanics as well. However,
that is too vast a subject to include in a single chapter. Furthermore, there
already exists a plethora of good books on the subject. Therefore, where
it is appropriate, the reader will be referred to one of these (Baym 1967,
Dicke and Wittke 1974, Gasiorowicz 1974, Landau and Lifschitz 1977,
Merzbacher 1970, and McMurry 1993).

This chapter is included for revision purposes and for reference later
on, therefore some topics are included without much justification and
without proof. The reader should either accept these statements or refer
to books on the classical theory of special relativity. In the first section of
this chapter we state the fundamental assumptions of the special theory
of relativity. Then we discuss the Lorentz transformations of time and
space. Next we come to discuss velocities, momentum and energy. Then
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2

1 TIle Theory of Special Relativity

we go on to think about relativity and the electromagnetic field. Finally,
we look at the Compton effect where relativity and quantum theory are
brought together for the first time in most physics courses.
1.1 The Lorentz Transformations


Newton's laws are known to be invariant under a Galilean transformation
from one reference frame to another. However, Maxwell's equations are
not invariant under such a transformation. This led Michelson and Morley
(1887) to attempt their famous experiment which tried to exploit the noninvariance of Maxwell's equations to determine the absolute velocity of
the earth. Here, I do not propose to go through the Michelson-Morley
experiment (Shankland et al. 1955). However, its failure to detect the
movement of the earth through the ether is the experimental foundation
of the theory of relativity and led to a revolution in our view of time
and space. Within the theory of relativity both Newton's laws and the
Maxwell equations remain the same when we transform from one frame
to another. This theory can be encapsulated in two well-known postulates,
the first of which can be written down simply as
( 1) All inertial frames are equivalent.

By this we mean that in an isolated system (e.g. a spaceship with no
windows moving at a constant velocity v (with respect to distant stars or
something)) there is no experiment that can be done that will determine v.
According to Feynman (1962) this principle has been verified experimentally (although a bunch of scientists standing around in a spaceship not
knowing how to measure their own velocity is not a sufficient verification) ..
Here, we are implicitly assuming that space is isotropic and uniform. The
second postulate is
(2) TIlere exists a maximum speed, c. If a particle is measured to have
speed c in one inertial frame, a measurement in any other inertial frame will
also give the value c (provided the measurement is done correctly). That is,
the speed of light is independent of the speed of the source and the observer.

The whole vast consequences of the theory of relativity follow directly
from these two statements (French 1968, Kittel et al. 1973). It is necessary
to find transformation laws from one frame of reference to another that
are consistent with these postulates (Einstein 1905). Consider a Cartesian

frame S in which there is a source of light at the origin. At time t = 0
a spherical wavefront of light is emitted. The distance of the wavefront

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I


1.1 The Lorentz Transformations
z

zz

z

z'

oo'}-----xx'

3
z'

xx'

yy'
(b)

(a)

(e)


Fig. 1.1. (a) At t = t' = 0 the two frames are coincident and observers 0 and 0'
are at the origin. At this time a spherical wavefront is emitted. (b) At a time t > 0
as viewed by an observer stationary in the unprimed frame. Observer 0 is at the
centre of the wavefront. (c) At a time t' > 0 as viewed by an observer stationary
in the primed frame. Observer 0' is at the centre of the wavefront. Note that in
(b) and (c) it is not possible for the observer not at the centre of the wavefront
to be outside the wavefront.

from the origin at any subsequent time t is given by
x2 +

l

+z2 = c2~

(1.1)

Now consider a second frame S' moving in the x-direction with velocity v
relative to S. Let us set up S' such that its origin coincides with the origin
of S at t' = t = 0 when the wavefront is emitted. Now the equation giving
the distance of the wavefront from the origin of S' at a subsequent time
t' as measured in S' is
(1.2)
So, at all times t, t' > 0, observers at the origin of both frames would
believe themselves to be at the centre of the wavefront. However, each
observer would see the other as being displaced from the centre. This is
illustrated in figure 1.1. It can easily be seen that a Galilean transformation
relating the coordinates in equations (1.1) and (1.2) does not give consistent
results. A set of coordinate transformations that are consistent with (1.1)

and (1.2) is

x-vt
x' - --;:===
v2jc 2'

)1-

y' =y,

z' = z,

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(1.3a)


4

1 The Theory of Special Relativity

and the inverse transformations are
x=

x' + vt'

V1- v /e
2

y = y',


z

=

z',

(1.3b)

2'

These equations are known as the Lorentz transformations. Under these
transformations the interval 8 defined by
82

= (et'f -

x'2 - y,2 _ z'2

= (et)2 -

x2 -

y2 - z2

(1.4)

is a constant in all frames.
It is conventional to adopt the notation
1

'Y - -,====

- V1-v 2 /e 2 '

f3

= v/e

(1.5)

Equations (1.3) lead to some startling conclusions. Firstly consider measurements of length. If we measure the length of a rod by looking at the
position of its ends relative to a ruler, then if in the S frame the rod is
at rest we can measure the ends at Xl and X2 and infer that its length
is L = X2 - Xl. Now consider the situation in the primed frame. The
observer will measure the ends as being at points x~ and Xz and hence
L' = Xz - x~. We want to know the relation between these two lengths.
The rod is moving at velocity -v in the x-direction relative to the observer
in S'. To find the length this observer must have measured the position
of the ends simultaneously (at t'o) in his frame. So, considering the first of
equations (1.3b) we have
(1.6)
Subtracting these equations leads directly to
L'

= Xz - x~

=

V(l- v2/e2)(x2 - xd = V(l- v2/e2)L


(1.7)

This is the famous Lorentz-Fitzgerald contraction and is illustrated in
figure 1.2. It shows that observers in different inertial frames of reference
will measure lengths differently. The length of any object takes on its
maximum value in its rest frame. Let us emphasize that nothing physical
has happened to the rod. Measuring the length of the rod from one reference frame is a different experiment to measuring the length from another
reference frame, and the different experiments give different answers. The
process of measuring correctly gives a different result in different inertial
frames of reference.
The above description of Lorentz-Fitzgerald contraction depended crucially on the fact that the observer in S' performed his measurements of

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5

1.1 The Lorentz Transformations
z

z'

L' =L/y

f========-.. . .

L

-1)


}-------x'

x

L

y

L' =L/'Y

(a)

(b)

Fig. 1.2. Here are two rods that are identical in their rest frames with length
L. In (a) we are in the rest frame of the lower rod. The upper rod is moving in
the positive x-direction with velocity v and is Lorentz-Fitzgerald contracted so
that its length is measured as Lfy. In (b) we are in the rest frame of the upper
rod and the lower rod is moving in the negative x-direction with velocity -v. In
this frame of reference it is the lower rod that appears to be Lorentz-Fitzgerald
contracted.

the position of the end points simultaneously. It is important to note that
simultaneous in S' does not mean simultaneous in S. So the fact that the
light from the ends of the rod arrived at the observer in S' at the same
time does not mean it left the ends of the rod at the same time. This is
trivial to verify from the time transformations in equations (1.3).
Next we consider intervals of time. Imagine a clock and an observer in
frame S at rest with respect to the clock. The observer can measure a time
interval easily enough as the time between two readings on the clock

(1.8)
Now we can use the Lorentz time transformations to find the times t'2 and
t'1 as measured by an observer in S' again moving with velocity v in the
x-direction relative to the observer in S:
,
t1 - (XIV/C 2)
(1.9)
t1 =
,
v2 /c 2

)1-

We can subtract one of these from the other to discover how to transform
time intervals from one frame to another:
"
t2 - t1
(1.10)
t2 - t1 =
= y(t2 - tr)
2
2
v /c

)1-

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.6


1 The Theory of Special Relativity

where we have set X2-XI = O. This is obviously true as the clock is defined
as staying at the same coordinate in S. What we have found here is the
time dilation formula. The time interval measured in S' is longer than the
time interval measured in S. Another way of stating the same thing is to
say that moving clocks appear to run more slowly than stationary clocks.
This, of course, is completely counter-intuitive and takes some getting used
to. However, it has been well established experimentally, particularly from
measurements of the lifetime of elementary particles. It is also responsible
for one of the most famous of all problems in physics, the twin paradox.
Next, let me describe a thought experiment that one can do, which demystifies time dilation to some extent, and shows explicitly that it arises
from the constancy of the speed of light. Consider a train in its rest frame
S as shown in the top diagram in figure 1.3 (with a rather idealized train).
Light is emitted from a transmitter/receiver on the floor of the train in a
vertical direction at time zero. It is reflected from a mirror on the ceiling
and the time of its arrival back at the receiver is noted. The ceiling is at
a height L above the floor, so the time taken for the light to make the
return journey is
2L
t=(1.11)
c

Now suppose there is an observer in frame S', i.e. sitting by the track
as the train goes past while the experiment is being done, and there is
a series of synchronized clocks in this frame. This is shJwn in the lower
part of figure 1.3. The observer in S' can also time the light pulse. Using
Pythagoras's theorem it is easy to see from the figure that when the light
travels a distance L in S, it travels a distance (L 2 + (!vi')2)1/2 in S', and

it goes the same distance for the reflected path. So the total distance
travelled as viewed by the observer in S' is
d = 2(L2 + (!vt')2)1/2

But the velocity of light is the same in all frames. So
d2 c2t'2 = 4L2 + v 2t'2

(1.12)

(1.13)

Rearranging this
,
t

=

2L
2L
Y = yt
(c 2 _ v2)1/2 =

c

(1.14)

Thus if the clock in the train tells us the light's journe.y time was t, the
clocks by the side of the track tell us it was yt > t. Se, to the observer
at the side of the track, the clock in S will appear to be running slowly.
Equation (1.14) is exactly the same as equation (1.10) wbich was obtained

directly from the Lorentz transformations.

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1.2 Relativistic Velocities

7

s

b

L

s
L

~
Ut'

Q

Q

I

G

~u


/2

Q

Q

Q

Q

Q

Q

Q

Fig. 1.3. Thought experiment illustrating time dilation, as discussed in the text.
The upper figure shows the experiment in the rest frame of the train and the
lower figure shows it in the frame of an observer by the side of the track.

Equations (1.3) are easy to derive from the postulates, and easy to apply.
However, their meaning is not so clear. In fact they can be interpreted in
several ways. Depending on the circumstances, I tend to think of them in
two ways. Firstly, a rather woolly and obvious statement. At low velocities
non-relativistic mechanics is OK because the time taken for light to get
from the object to the detector (your eye) is infinitesimal compared with
the time taken for the object to move, so the velocity oflight does not affect
your perception. However, when the object is moving at an appreciable
fraction of the speed of light, the time taken for the light to reach your

eye does have an appreciable effect on your perception. Secondly, a rather
grander statement. Let us consider space and time as different components
of the same thing, as is implied by equations (1.1) and (1.3). Any observer
(Observer 1) can split space-time into space and time unambiguously,
and will know what he or she means by space and time separately. Any
observer (Observer 2) moving with a non-zero velocity with respect to
Observer 1 will be able to do the same. However, Observer 2 will not split
up time and space in the same way as Observer 1. Observers in different
inertial frames separate time and space in different ways!
1.2 Relativistic Velocities
Once we have the Lorentz transformations for position and time, it is an
easy matter to construct the velocity transformation equations. As before,

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I


8

1 The Theory of Special Relativity

we have a frame S in which we measure the velocity of a particle to have
three components ux ,. uy and Uz . Now let the frame S' be moving with
velocity v relative to S in the x-direction.
We can write the Lorentz transformations (1.3) in differential form:
A

I


oX -

dx - vdt
v2/c 2'

dy

- )1-

dz

'

= dz,

dt'

=

I

= dy,
2

(1.15)

dt - (dxv/c )
v2 /c 2

)1-


Now velocity in the x-direction in the S frame is given by dx/ dt and
in the S' frame by dx ' / dt' , and similarly for the other components. So
we simply divide each of the space transformations in (1.15) by the time
transformation, and divide top and bottom of the resulting fraction by dt
to obtain
I

U
X

Ux-V

=----:-;::-

1- vux /c 2

(1.16a)

(1.16b)

Equations (1.16) enable us to find the velocity of an object in any other
Lorentz frame given its velocity in one such frame (see figure 1.4). These
equations are certainly consistent with the postulate that c is the ultimate
speed. If, for example, we substitute U x = c in (1.16a) it is trivial to see
that u~ = c as well for any value of v, the relative velocity of the frames.
If, instead of choosing our photon velocity parallel to the relative motion
of the frames, we choose it in an arbitrary direction, the magnitude of the
velocity as measured in the S' frame also works out as c. However, the
angle the photon makes to the axes of S as measured in S is, in general,

different to the angle it makes to the axes of S' as measured in S'.
1.3 Mass, Momentum and Energy

Perhaps the most famous equation in the whole of physics, and certainly
one of the most important and fundamental in the theory of relativity, is
the equivalence of mass and energy described by
(1.17)
where m is the mass of a particle as measured in its rest frame. It is also
known that, for photons with zero rest mass, the energy E and frequency

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9

1.3 Mass, Momentum and Energy
z

z

z·

ll-U

- - _ . 11,,:'

-u
---------~

l-11xU/C


llx' =llx -

U

(Galilean)

r---r-----------x·
x

r------x
y

= - x - - 2 (Lorentz)

y

(a)

z

z

z·

\J ' ,
u;c'=-u

lly =lly{1-u/c,on
)


lly

-u

X

y

x

y
(b)

x'

y'

Fig. 1.4. Velocity transformation between different frames. (a) The upper left
frame contains a particle moving at velocity Vx ' In the upper right figure the
dashed line indicates the velocity in the primed frame· found from a Galilean
transformation and the full line indicates the velocity from a Lorentz transformation using (1.16a). (b) In the lower left figure we have a particle moving parallel
to the y-axis of an unprimed frame. Its velocity as viewed from a frame moving
in the x-direction relative to the unprimed frame has components in both the x'
and y' directions.
v are related by

he
(1.18)
A

where p and A are the photon momentum and wavelength respectively.
Putting these two equations together, the total energy of any free particle
is given by
(1.19)
E =hv

= - =pe

Obviously these statements do not constitute a derivation of (1.19). For
a full discussion of the origin of this equation the reader should refer
to standard texts on relativity. Equation (1.19) is usually developed for a

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10

1 The Theory of Special Relativity

single particle. However, if we have a collection of particles and observe
them in a frame S, we can measure their energy E and the magnitude
of the vector sum of their momenta p and we can form the quantity
E2 - p2c2• The same thing can be done again in a frame S', and we find
(1.20)

This is not just a statement about the rest mass of the particles because
there is no necessity for them all to be at rest in the same frame. The
quantity on both sides of equation (1.20) is described as being relativistically invariant, i.e. it doesn't change under a Lorentz transformation.
Compare equation (1.20) with the fundamental definition of an interval
given by (1.4) which is also a Lorentz invariant. From equation (1.19) it

can be shown that both the energy and mass as measured in a frame
moving with velocity v are given in terms of their rest frame values by
E(v)

= ./

mc 2
2

V 1- v jc

2

= ymc

2

( 1.21)

and
m(v)

111

--==== =ym

J1

(1.22)


v2 jc 2

where v = Ivl. Equation (1.22) is the inertial property of a body moving
with velocity v such that the momentum is given by
p = m(v)v = ymv

(1.23)

E = m(v)c 2

(1.24)

and

It is not immediately clear that equations (1.21-1.23) are consistent with
(1.19). The easiest way to prove this is to substitute (1.22) and (1.23) into
(1.19) and we find

(1.25)

Taking the square root of this gives (1.21) directly.
Note that now velocity and momentum are no longer proportional to
each other as they are in classical mechanics. The velocity is bounded by
-c < v < c, but the momentum can take on any value -00 < p < 00. In
figure 1.5 we show the relativistic momentum and the classical momentum

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1.3 Mass, Momentum and Energy


11

3.0 ...---..,.--...,.----,--,..----.----,-----.,..----..,.--...,.----.,

2.5

2.0

.z
o

1.5

R.

1.0

0.5

0.2

0.4

0.8

0.6

1.0


1.2

1.4

1.6

1.8

2.0

vie
Fig. 1.5. Graph showing the classical mv momentum (full line) and the relativistic
momentum (dotted line) given by equation (1.18). The momentum is divided by
mc, where In is the rest mass, so the plotted numbers are independent of the rest
mass of the particle. Note the agreement between the two curves at low velocities.

as a function of velocity. Clearly they agree at low velocities but diverge
strongly when the velocity becomes an appreciable fraction of c. The
velocity of a particle in terms of its energy and momentum is
pc 2

(1.26)

v=E:

It is easy to see that this has the correct non-relativistic limit. As c -+
we have

pc2
V = Vp 2c 2 + m 2c4


-

pc2
mc2J1 + p2jm2c4 ~ m(1

00

P
+ p2j2m2c2)

p

(1.27)

~-=V

m

Newton's laws are still valid within a relativistic framework. By direct
substitution of equation (1.23) we can define a relativistic force

F = dp = d
dt
dt

mv
1 - v2 j c2

J


and by integrating this over· a path we can define work as
K = mc2(y -1)

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(1.28)

(1.29)


12

"'I

1 The Theory oJ Special Relativity

I

!

Now to ensure that we are on the right track it is useful to take the nonrelativistic limit of some of these equations to make sure they correspond
to the quantity we think they do. Firstly, let us look at the total energy,
as given by equation (1.21):
E(v) =

me 2

V1 -v /e
2


=

me2(1- v2/e 2)-1/2

2

1

(1.30)
3v4 /8e 4 ) = me2 + -mv 2 - 3mv 4 /8e 2
2
This is just the rest mass energy plus the kinetic energy in the nonrelativistic limit. Rest mass energy does not appear in non-relativistic
physics and only corresponds to a redefinition of the zero of energy. So,
we have correctly found the non-relativistic limit of the total energy. In a
similar way we find the non-relativistic limit of K as
~ me2(1

+ v2/2e 2 -

K = !mv 2
2

(1.31a)

This, of course, is the kinetic energy as we expect. Let us consider the first
correction to this due to relativistic effects. It is
(1.31b)

This expression will be useful in interpreting the non-relativistic limit

of our relativistic wave equations in later chapters. In figure 1.6 we can
see the classical (full line) and relativistic (lower dotted line) expressions
for kinetic energy plotted as a function of velocity. This illustrates rather
clearly the agreement between the two up to velocities of order 0.5e, and
the continually increasing divergence between them as v - ? e. We also
include on this figure the total energy of equation (1.21) (upper dotted·
line) directly. Clearly, from equation (1.19), this must take on the value
me2 when the particle is at rest.
Energy and momentum become united in special relativity in the same
way as space and time. The energy-momentum Lorentz transformations
in one dimension are
I
Px -vE/e2
I
I
(1.32)
Py = Py,
pz = Pz,
Px=
1- v2 /e 2

V

'

The conservation of momentum and energy unite into one law, which is
the conservation of the four-component energy-momentum vector. At this
stage we should recall that mass and energy also are no longer separate
concepts. A photon moving in the x-direction has
E =hv,


Px

= h/).. = hv/e

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(1.33)