Relativistic quantum mechanics; with applications in condensed matter and atomic physics

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 advance

par-of quantum theory to condensed matter physics then follow Relevant theory for the one-electron atom is explored The theory is then de- veloped to describe the quantum mechanics of many electron systems, including Hartree-Fock and density functional methods Scattering the- ory, band structures, magneto-optical effects and superconductivity are amongst other significant topics discussed Many exercises and an exten- sive 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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*M0038435*

Relativistic Quantum Mechanics

RELATIVISTIC

QUANTUM MECHANICS

WITH APPLICATIONS IN CONDENSED MATTER

AND ATOMIC PHYSICS

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 of America 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, Relativistic quantum mechanics I Paul Strange

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

• '!:'""

Contents

2.11 Relativistic Quantum Numbers and Spin-Angular Functions 58

3.2 Relativistic Wavefunctions, Probabilities and Currents 67

4.4 The Non-Relativistic Limit of the Dirac Equation 111 4.5 An Alternative Formulation of the Dirac Equation 118

5.7 Lorentz Transformation of the Free-Particle Wavefunction 154

6.2 Relativistic Energy and Spin Projection Operators 160

Contents xi

6.11 Second Quantization in Relativistic Quantum Mechanics 189

7.1 The Foldy-Wouthuysen Transformation for a Free Particle 200

8.3 One-Electron Atoms, Eigenvectors and Eigenvalues 233

Expectation Values and the Uncertainty Principle 276

A One-Dimensional Time-Independent Dirac Equation 283

9.5 An Electron in Crossed Electric and Magnetic Fields 295

An Electron in a Constant Magnetic Field 298

An Electron in a Field for which IEI = clBI 309 9.6 Non-Linear Dirac Equations, the Dirac Soliton 311

10.11 The Variational Principle and the Kohn-Sham Equation 370

Density Functional Theory in a Weak Magnetic Field 374

Density Functional Theory in a Strong Magnetic Field 376

10.15 An Approximate Relativistic Density Functional Theory 395

11.13 The Non-Relativistic Free Particle Green's Function 440

,1 1

The Non-Relativistic Limit, the RKKY Interaction 472

12.1 Photon Polarization and Angular Momentum 481

12.2 Quantizing the Electromagnetic Field 483

12.3 Time-Dependent Perturbation Theory 485

12.4 Photon Absorption and Emission in Condensed Matter 494

13.1 Do Electrons Find Each Other Attractive? 537

13.2 Superconductivity, the Hamiltonian 539

13.3 The Dirac-Bogolubov-de Gennes Equation 542

13.4 Solution of the Dirac-Bogolubov-de Gennes Equations 544

13.5 Observable Properties of Superconductors 547

13.6 Electrodynamics of Superconductors 551

Appendix B The Confluent Hypergeometric Function 559

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

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, nonethe-less 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

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

rel-ativistic 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

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

non-invariance 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

experimen-tally (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

~i

I

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

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

)1-v2jc2'

4 1 The Theory of Special Relativity

and the inverse transformations are

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-v2/e2 ' f3 = v/e (1.5) Equations (1.3) lead to some startling conclusions Firstly consider mea-surements 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

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 refer-ence 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 cially on the fact that the observer in S' performed his measurements of

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:

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

de-mystifies 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 =

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

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

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,

I

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 u x , uy and U z 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:

oX - dy = dy,

- )1-v2/c2'

dz' = dz, dt' = dt - (dxv/c 2 )

)1-v 2 /c 2

(1.15)

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

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

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

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

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 - p 2 c 2• 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 relativis-tically 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

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

12 1 The Theory oJ Special Relativity

Now to ensure that we are on the right track it is useful to take the

non-relativistic 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,

This is just the rest mass energy plus the kinetic energy in the

non-relativistic limit Rest mass energy does not appear in non-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

2 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

me 2 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

Px= V 1- v 2 /e 2 '

I

Py = Py, pz I = Pz, (1.32)

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

"'I I

!

5

+ +

mc 2 where m is the rest mass, thus making the plotted numbers independent of the rest mass of the particle

and substituting these into the last equation of (1.32) gives

source For a relativistic effect this has a surprising number of mundane applications It is the physics underlying the change in pitch in the sound

of a train passing you as you stand on a railway station platform It is also used to measure the speed of motor vehicles by the police In physics perhaps its most important uses are astrophysical where the famous red shift enables us to estimate stellar and galactic relative velocities

Finally in this section we mention briefly the problem of defining a relativistic centre of mass We will not do any mathematics associated

14 1 The Theory of Special Relativity

with this, but just point out the problem Consider two particles (A and

B) of equal mass (in their rest frames), moving uniformly towards each other We can certainly imagine sitting in a frame of reference in which they move with equal and opposite velocities, and the point at which they collide will be the centre of mass That will surely be the centre of mass frame Now consider the same situation from the rest frame of particle

A In that frame A has mass m, but particle B has mass ym, from (1.22), and so the centre of mass is closer to B than to A An observer in the rest frame of B will see A as having the greater mass and so will think that the centre of mass is nearer to A Obviously these two points of view cannot be reconciled and we are left with the conclusion that the centre

of mass depends upon the inertial frame in which observations are made

1.4 Four-Vectors

Next we will discuss four-vectors (see Greiner 1990 for example) They are

a familiar concept in relativity, and do provide a very neat way of writing down equations However, they are used sparingly in this book There are two reasons for this They are only a mathematical nicety, and for people inexperienced in using them they can obscure the physical meaning

of the equations they represent Furthermore, they are not used much in the condensed matter physics literature On the other hand, 4-vectors are very much used in many applications of relativity and an understanding

of them is undoubtedly useful Therefore they are introduced here and used occasionally in this book To really understand 4-vectors and the mathematical advantage gained from them, the reader should go through the work in this book that uses 4-vectors and then redo it without 4-vector notation

A 4-vector is defined as a set of four quantities that can be written

in four component form such as r = (x, y, Z, et) Other familiar examples

of 4-vectors are the energy-momentum 4-vector pf1 = (px,py,pz,E/e), the 4-current Jf1 = (lx, Jy, J z, ep) with p the charge density, and the vector potential Af1 = (Ax,Ay,Az,<p/e) where <P is the scalar potential Using 4-vectors is easier if we simplify the notation We write the space-time 4-vector as

(1.35)

A general component of the space-time 4-vector can be written xf1 with

J1 = 1,2,3,4 Any vector with four components that transforms under Lorentz transformations like xf1 is written with an upper greek suffix, e.g

a f1 Now, we may lower the suffix according to the rule

2

and similarly for any other 4-vector The xP are called the contravariant components of x, and the X/I are called the covariant components of x For two arbitrary 4-vectors a and b we may define the scalar products

apb P = a4b4 + alb 1 + a2b2 + a3 b3

This illustrates another important convention If on one side of an equation the same greek letter appears as a lower and upper suffix it is automatically summed over This is known as the summation convention Let us define the fundamental tensor gPI' such that

g44 = 1, gll = g22 = g33 = -1, gPI' = 0 fl, =1= v (1.38) then we can use this to relate the covariant and contravariant elements of

a 4-vector

(1.39) where the summation convention is used These definitions of 4-vectors and scalar product mean we can often write equations in a very neat and succinct way For example, the definition of the space-time interval, equation (1.4), can be written

(1.40) and, using the definition of the momentum 4-vector above, we can write equation (1.20) in 4-vector form as

(1.41)

In this book, 4-vectors will only be used a few times, and then only when they really do simplify the mathematics, without unduly obscuring the physics of what they describe

1.5 Relativity and Electromagnetism

The whole of classical electromagnetism is described by the Maxwell equations together with the conservation of charge (Reitz and Milford

A particle with charge e and velocity v in an electromagnetic field feels

the Lorentz force

F = eE+ev x B The current density J is related to the electric field by Ohm's law

J = erE

(1.47)

(1.48) The electric and magnetic fields E and B are often written in terms of the scalar and vector potentials

B=VxA

E = _ aA - V<I>

at

(1.49) (1.50)

It is clear, though, that these definitions of the fields in terms of potentials

A and <I> are not unique In electromagnetism, observables depend· upon the forces which are written in terms of the electric and magnetic fields

in equation (1.47) Any alternative expressions for the potentials that

leave the fields unchanged are equally acceptable In fact we can define equivalent potentials toA and <1> using any function e such that

A' =A-Ve, <1>' = <1> + ae

Substituting these back into (1.49) and (1.50) gives us back (1.49) and (1.50) because V x ve = 0, which follows directly from the properties of the V operator, and the terms including e we find when we substitute into (1.50) cancel immediately This freedom to choose appropriate forms for A and <1> is known as gauge freedom It is often possible to make calculations significantly easier by choosing to work in a suitable gauge

We will make use of this in chapter 9

Maxwell's equations can provide us with our first non-trivial use of the 4-vector formalism Let us define the electromagnetic field tensor Written out explicitly this is

- axil - ax' ay' az' act (1.53)

Next we operate with a/axil on FIlV, and two of Maxwell's equations can

18 1 The The01Y of Special Relativity

A natural question to ask is how the electromagnetic fields behave under a

Lorentz transformation We answer that here Firstly we write the

space-time Lorentz transformations for relative motion in the x-direction in

which just defines the Lorentz tensor L~ We can use the Lorentz tensor

to write the transformations of the electromagnetic field as

In component form this amounts to

Clearly the electromagnetic fields transform in a way somewhat

remi-niscent of the space-time transformations In actual fact, though, these

transformations are quite different to (1.3) In particular, note that it is

the fields in the direction of the relative motion that are unaffected by

the transformation and the fields perpendicular to the motion that are

transformed Equation (1.58a) is a neat way to write the Lorentz

trans-formations, and is more general than (1.58b) because it does not require

us to choose a particular axis for the relative motion of the reference

frames However, it is also rather abstract, and when we implement the

transformations the form of equations (1.58b) is simpler to use

1.6 The Compton Effect

The Compton effect is an illustration of the application of relativity

theory to the properties of fundamental physics It is a very elegant and

convincing illustration of the particle nature of electromagnetic radiation

Furthermore, it is useful to include it here in its elementary form as we

shall be considering a more sophisticated and complete (but less physically

transparent) description of it in chapter 12

,

' ~ I

The physics of the Compton effect is shown in figure 1.7 An magnetic wave is incident upon a stationary free electron The photon has frequency Vj (in the X-ray region of the spectrum) After the collision the photon has been scattered through an angle e and has a reduced frequency Vf The electron recoils making an angle 4J with the direction

electro-of the initial photon Let us write down the conservation electro-of energy and momentum for this collision The initial energy of the electron is mc 2 and the final energy ymc 2, so

Now we want to eliminate 4J between (1.60a) and (1.60b) The best way

to do this is to rearrange them so that the term in 4J is on its own on one side of the equations The equations can be squared and added Using cos2 4J + sin2 4J = 1 then leads to

(1.61) The conservation of energy (1.59) can be manipUlated to give a rather cumbersome expression for /32

y2 which can be substituted into (1.61) to

20 1 The Theory of Special Relativity

Fig 1.8 Schematic results of Compton scattering experiments at different angles

(a) The line of the incident radiation In (b), (c) and (d) the left hand peak shows

the incident photon wavelength and is indicated by the vertical dashed lines The

right hand peaks are the Compton scattered radiation The angles of scatter are

45° in (b), 90° in (c) and 135° in (d) The shift in wavelength of the scattered

radiation obeys equation (1.63)

wavelength Ae Numerically Ae = 2.43 x 10-12 m

This scattering was first observed by Compton (1923a and b) The

scattered radiation consists of two components Firstly there is radiation

scattered with the same frequency as the incident X-ray This occurs

because the radiation sets the electron oscillating at its own frequency

This radiation is emitted in all directions There is also radiation scattered

at a lower energy which corresponds to that which is Compton scattered

according to the formula above (Wichman 1971) This is illustrated in

I

1.7 Problems 21 figure 1.8 which is a schematic picture of Compton scattering results at

several different scattering angles Note that the scattering will be largest

for () = 180° as (1.63) requires and is zero for () = 0° Equation (1.63)

contains both hand c and so depends on both quantum mechanics

and relativity for its derivation The fact that the angular distribution

of the scattered radiation was found to obey (1.63) lent strong support

to the photon picture of light Furthermore, it was necessary to use the

relativistic equations of motion (1.59) and (1.60) to arrive at (1.63), so this

experiment also helps to validate relativistic mechanics on a microscopic

scale Unfortunately this is not the end of the story of Compton scattering

and we will return to it in chapter 12

1.7 Problems

(1) In a certain inertial frame two events occur a distance 5 kilometres

apart and separated by 5 J.lS An observer, who is travelling at velocity

v parallel to the line joining the two points where the events occurred,

notes that the events are simultaneous Find v

(2) Observer A sees two events occurring at the same place and separated

in time by 10-6 s A second observer B sees them to be separated by

a time interval 3 x 10-6 s What is the separation in space of the two

events according to B? What is the speed of B relative to A?

(3) In a scattering experiment an electron with speed 0.85c and a proton

of speed 0.7c are moving towards each other (a) What is the velocity

of the electron as seen by an observer in the rest frame of the proton?

(b) What is the velocity of the proton as seen by an observer in the

rest frame of the electron?

(4) In frame A two particles of equal mass move at equal velocity towards

each other They collide and stick together Clearly momentum is

conserved in this frame (a) Perform a Lorentz velocity transformation

to the rest frame of one of the particles (frame B) and show that

momentum is not conserved in such a frame (b) Use the Lorentz

energy-momentum transformations to show that conservation of

mo-mentum in frame B depends upon conservation of energy in frame A

(c) How would an observer in frame A explain that energy is conserved

in the collision?

(5) (a) Show that the potentials felt by a charged particle in electric and

magnetic fields E = (0, E, 0) and B = (0,0, B) respectively are

1

A = 2:(-yB,xB,O) <I> = -yE

22 1 The Theory of Special Relativity

(b) Show that the potentials in (a) describe the same electromagnetic fields as the potentials

A' = (-yB,O,O) <I> = -yE

(6) Starting from the electrostatic field due to a point particle at rest, use the Lorentz transformations for the electromagnetic field to find the electric and magnetic fields due to a point particle moving with velocity v

(7) Use equations (1.54) and (1.56) and the definition of the netic field tensors to write down Maxwell's equations in their more familiar form of equation (1.42)

electromag-2

Aspects of Angular Momentum

Spin is well known to be an intrinsically relativistic property of particles Nonetheless its effects are seen in many physical situations which are not obviously relativistic, perhaps the most obvious examples being magnets and the quantum mechanically permitted electronic configurations of the elements in the periodic table The view of spin adopted in these and other problems is that the electron has a quantized spin (8 = 1/2) and that is a fundamental tenet of the theory, rather than something that has

to be explained In this chapter we are going to discuss the behaviour of

spin-1/2 particles (electrons, protons and neutrons for example) without much direct reference to relativistic quantum theory and the origins of spin This chapter should be instructive in its own right and as a guide

to understanding spin when we come to discuss it in a fully relativistic context at various stages throughout this book Unless otherwise specified,

we will refer to electrons, but it should be borne in mind that the theory

is equally applicable to any spin-1/2 particle

Students of quantum theory cannot delve very deeply into the ject without coming across the quantization of angular momentum The orbital angular momentum usually surfaces in the theory leading up to the quantum description of the hydrogen atom Spin is often introduced through the Stern-Gerlach experiment and the Goudsmit-Uhlenbeck hy-pothesis, and the anomalous Zeeman effect (Eisberg and Resnick 1985) One of the things that makes non-relativistic quantum mechanics simpler than its relativistic counterpart is that, whereas only total angular momen-tum is conserved in relativistic quantum mechanics, in the non-relativistic theory both orbital angular momentum and spin angular momentum are conserved separately The simplest way to appreciate this is to observe that spin does not appear in the Schrodinger equation and therefore there is no mechanism within non-relativistic quantum theory to change

sub-it Non-relativistic quantum theory does conserve angular momentum,

2

Aspects of Angular Momentum

Spin is well known to be an intrinsically relativistic property of particles Nonetheless its effects are seen in many physical situations which are not obviously relativistic, perhaps the most obvious examples being magnets and the quantum mechanically permitted electronic configurations of the elements in the periodic table The view of spin adopted in these and other problems is that the electron has a quantized spin (8 = 1/2) and that is a fundamental tenet of the theory, rather than something that has

to be explained In this chapter we are going to discuss the behaviour of spin-1/2 particles (electrons, protons and neutrons for example) without much direct reference to relativistic quantum theory and the origins of spin This chapter should be instructive in its own right and as a guide

to understanding spin when we come to discuss it in a fully relativistic context at various stages throughout this book Unless otherwise specified,

we will refer to electrons, but it should be borne in mind that the theory

is equally applicable to any spin-I/2 particle

Students of quantum theory cannot delve very deeply into the ject without coming across the quantization of angular momentum The orbital angular momentum usually surfaces in the theory leading up to the quantum description of the hydrogen atom Spin is often introduced through the Stern-Gerlach experiment and the Goudsmit-Uhlenbeck hy-pothesis, and the anomalous Zeeman effect (Eisberg and Resnick 1985) One of the things that makes non-relativistic quantum mechanics simpler than its relativistic counterpart is that, whereas only total angular momen-tum is conserved in relativistic quantum mechanics, in the non-relativistic theory both orbital angular momentum and spin angular momentum are conserved separately The simplest way to appreciate this is to observe that spin does not appear· in the Schrodinger equation and therefore there is no mechanism within non-relativistic quantum theory to change

sub-it Non-relativistic quantum theory does conserve angular momentum,

but as orbital angular momentum is all that appears in the Schrodinger equation, that is all that it is able to conserve Spin is simply not a non-relativistic phenomenon It is possible, however, to 'fix up' non-relativistic

quantum theory by including spin ad hoc Non-relativistic quantum theory including spin is a theory based on the Pauli equation, to which we shall

be devoting some of this chapter

Firstly we will revise the properties of general angular momentum operators To gain further insight into the nature of quantum mechanical angular momentum we go on to show that the operator necessary to generate rotations can be written in terms of the angular momentum operators Next we look at the operators and eigenvectors that describe spin-1/2 particles, and through this we introduce the Pauli matrices As

an aside, in section 2.4 we examine a topic not often covered in quantum theory textbooks, the matrix representation of higher spin operators Then

we go on to look at the interaction of the orbital angular momentum of

an electron in an atom with an external magnetic field, leading to Larmor precession Next we write down the Pauli equation using a plausibility argument to justify it, and see that this leads directly to the concept of spin-orbit coupling We see how the relativistic velocity transformations lead us to postulate Thomas precession of the electron rest frame relative

to the nucleus, and how this affects the spin-orbit coupling term in the Pauli Hamiltonian Next we look at the solutions of the Pauli Hamiltonian

in a central field The eigenfunctions can be separated into spin-angular functions and radial parts, and we show this explicitly This leads to the coupling of angular momentum using the Clebsch-Gordan coefficients, and we examine some of their properties It is convenient to introduce some new quantum numbers when considering the solution of the Pauli equation in a central field, and we explain this and show their relation with the more familiar angular momentum quantum numbers Also in this section we look at the properties of the spin-angular functions Finally

we discuss how to expand a plane wave in terms of the spin-angular functions, as this will be useful when we come to discuss scattering theory This chapter provides a grounding in the quantum theory of angular momentum An understanding of the material covered here should enable the reader to cope easily with angular momentum operators where they appear later in the book, in particular when we come to discuss atoms

2.1 Various Angular Momenta

In this chapter we will assume that the reader is familiar with angular momentum up to the level required to solve the hydrogen atom in non-relativistic quantum theory The application of quantum mechanics to the

2.1 Various Angular Momenta 25 description of atoms cannot proceed much beyond the hydrogen atom without a discussion of spin

The name spin is very suggestive, but we do not necessarily mean by

it that an electron spins about its own axis We know, however, that the spin operator obeys the commutation relations

A A A A A A A A A "'2 A

[Sx, Sy] = inSz, [Sy, Szl = inSx, [Sz, Sx] = inSy, [8, Sa = 0 (2.1a)

which are written in analogy to the orbital angular momentum operator commutation relations

In (2.1) subscript i = x, y, z The orbital angular momentum operator is usually defined in terms of the momentum and position operators as

L" = " r x p " = -;-r n" X V

I

(2.2) with the usual definition of the momentum operator The definition of the spin and total angular momentum in terms of position and momentum is usually not discussed In non-relativistic quantum theory we think of the electron (for example) as a point particle, so it is not possible to define spin in an analogous way to equation (2.2) as r = 0 and what would

be meant by the linear momentum at the surface of a spinning point particle is anybody's guess Hence we can define spin only as a degree

of freedom that obeys the same commutation relations as orbital angular momentum The orbital angular momentum operators have the following commutation relations with the position and momentum operators:

(2.3a) (2.3b)

and all cyclic permutations of (2.3) are also valid Now that we have the orbital and spin angular moment a, the total angular momentum operator

is

(2.4a)

and

[.t"/y] = inlz, [ly,lzl = inlx, [lz,lx] = mly, [J2 ,ld = 0 (2.4b)

The operators in (2.4a) and (2.4b) have eigenvalues which we denote

J2tp71 = j(j + l)n2tp? (2.5a)

(2.5b)

26 2 Aspects of Angular Momentum

where lP'ji are eigenfunctions of both J2 and lz, and we have arbitrarily

chosen the z-direction as the quantization direction Let us define the

raising and lowering operators

(2.6) The reason for this notation will soon become obvious and the reader

may well anticipate it Let us state that

where we have made use of the commutation rules (2Ab) This proves

that if lP7 j is an eigenfunction of lz and J2 then with the definitions

(2.6) and (2.7) l±lP;lj is also an eigenfunction of these operators with the

same value of j but with mj raised or lowered by one unit of n We have

not proved that (2.7) is valid in any absolute way because we used it in

(2.8) Equation (2.8) only shows that equation (2.7) is consistent with the

more familiar properties of angular momentum operators Let us assume

all eigenfunctions of J2 are normalized and try to find the value of the

constant a The functions lP7 j form an orthonormal set and we note that

(l+)t = J.+ form a pair of hermitian conjugate operators The strategy is

to-write the normalization of lP7 j ±1 in terms of the normalization of lP7 j

This will then tell us the value of the constant a

J( lPj mj+l)*( mj+1)d - lPj r - a 2 J(JA +lPj /Ilj)*JA +lPj /Iljd r

where we have set the final integral equal to one So, the right

normal-ization means that we can replace a in equation (2.7) using (2.9) and we

get

(2.10a)