the physics of low dimensional

7/24/2019 the Physics of Low dimensional

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THE PHYSICS OF

LOW-DIMENSIONAL

SEMICONDUCTORS


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TF-IE PI-IYSICS OF

LOW-DIMENSIONAL

SEMICONDUCTORS

AN INTRODUCTION

JOHN H. DAVIES

Glasgow University

CAMBRIDGE

UNIVERSITY PRESS


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CAMBRIDGE UNIVERSITY PRESS

Cambridge, New York, Melbourne, Madrid , Cape Tow n, Singapore, So Paulo

Cam bridge University Press

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Street, New York,

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0Cam bridge University Press 1998

This book is in copyright. Subject to statutory exception

and to the provisions of relevant collective licensing agreements,

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the written perm ission of Cambridge University Press.

First published

1998

Reprinted 1999, 2000, 2004, 2005

Printed in the United States of Am erica

A catalog record for this

publication

is

available from

the British Library.

Library

of

Congress

Cataloging in

Publication

Data

Davies, J. H.

(John H.)

The

physics

of

low-dimensional semiconductors

:

an introduction

/

John H. Dav ies.

p. cm.

Includes bibliographical

references

and index.

ISBN 0-521-48148-1 (hc) . ISBN

0-521-48491-X (pbk.)

1. Low-dimen sional semiconductors. I . T i tle .

QC611.8.L68039 1997

537.6'221 dc21

7-88

ISBN-13

978-0-521-48148-9

hardback

ISBN-10 0-521-48148-1 hardback

ISBN-13

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ISBN-10 0-521-48491-X

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or appropriate.


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To

Christine


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CONTENTS

Preface

age

xiii

Introduction

v

1 OUNDATIONS

1.1

Wave Mechanics

and the Schr6dinger

Equation

1.2

Free

Particles

1.3

Bo und Par t ic les: Quantum

Wel l

1.4

Charge and

Current Densities

1.5

Operators and

Measurement

3

1.6 Mathematical Properties

of Eigenstates

0

1.7

Counting

States

2

1.8

Filling

States: The Occupation Function

0

Further Reading

0

Exercises

1

2

LECTRONS AND

PHONONS

IN

CRYSTALS

5

2.1

Band Structure in One Dimension

5

2.2

Motion of

Electrons in Bands

0

2.3 Density of States

4

2.4

Band Structure in

T w o and Three

Dimensions

5

2.5

Crystal Structure of the

Common Semiconductors

7

2.6 Band Structure of the

Common Semiconductors

1

2.7 Optical Measurement

of Band

Gaps

9

2.8

Phonons 0

Further Reading

6

Exercises

6

vil


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viii

ONTENTS

3

ETEROSTRUCTURES

0

3.1 General Properties of Heterostructures 0

3.2

Growth

of Heterostructures 2

3.3

Band Engineering

5

3.4

Layered St ructures:

Quantum Wells and

Barriers

8

3.5 Doped Heterostructures

2

3.6

Strained Layers

6

3.7 Si l icon Germanium Heterostructures 00

3.8 W i r e s and Dots 02

3.9

Optical Confinement

05

3.10 Effective-Mass Approximation

07

3 .11

Effect ive-Mass Theory in Heterostructures 11

Further Reading

14

Exercises

14

4

UANTUM

WELLS AND LOW-DIMENSIONAL SYSTEMS

18

4.1

Infinitely

Deep Square

W e l l

18

4.2

Square

Wel l

of

Finite Depth

19

4 .3 Parabol ic W el l

25

4.4

Triangular Well 28

4 .5 Low-Dimensional Systems

30

4 .6

Occupation of Subbands 33

4.7

Two- and

Three-Dimensional Potential Wel l s

35

4.8

Further Confinement

B e y o n d T wo

Dimensions

40

4.9 Quantum Wells in Heterostructures

42

Further Reading

46

Exercises 46

5

UNNELLING TRANSPORT

50

5 .1 Potential Step 50

5 .2 T-Matrices 53

5.3

More on T-Matrices

58

5 .4

Current and Conductance

62

5 .5 Resonant Tunnelling

67


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CONTENTS

x

5 .6

Superlattices

and

Minibands

77

5 .7 Coherent

Transport

with Many Channels

83

5 .8

Tunnelling

in

Heterostructures

95

5.9 What Has Been Brushed Under

the Carpet?

99

Further Reading

00

Exercises 01

ELECTRIC

AND

MAGNETIC

FIELDS

06

6.1 The Schrtidinger Equation with Electric and

Magnetic Fields

06

6 .2

Uniform Electric

Field 08

6 .3 Conductivity and Resistivity Tensors

16

6 .4 Uniform Magnetic

Field

19

6 .5 Magnetic

Field in a

Nar ro w

Channel

33

6 .6

The Quantum H all Effect

38

Further

Reading

45

Exercises

46

7

PPROXIMATE METHODS

49

7.1

The

Matrix

Formulation of Qu antum

Mechanics 49

7 .2

Time-Independent

Perturbation Theory 52

7.3 k p

Theory

61

7 .4 W K B

Theory

63

7 .5

Variational Method

70

7 .6

Degenerate

Perturbation

Theory

73

7 .7

Band S tructure:

Tight Binding

75

7 .8 Band S tructure:

Nearly Free Electrons 80

Further

Reading

84

Exercises

84

8 CATTERING

RATES: THE GOLDEN RULE

90

8.1

Golden Rule for

Static Potentials

90

8 .2

Impurity

Scat tering

95

83

Golden Rule for Oscillating Potentials

01

8 .4

Phonon Scat tering

02


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CONTENTS

8.5

Optical Absorption

08

8.6 Interband Absorption 13

8.7

Absorption in a Quantum

Well

16

8.8 Diagrams and the

Self-Energy 21

Further Reading

24

Exercises

24

9

HE

TWO-DIMENSIONAL ELECTRON GAS

29

9.1

Band Diagram of M odulation-Doped Layers 29

9.2 Beyond the

Simplest Model

36

9.3

Electronic Structure of a 2DEG 42

9.4

Screening by

an

Electron Gas

49

9.5

Scattering by Remote Impurities

56

9.6

Other Scattering Mechanisms

62

Further Reading

65

Exercises

66

10

OPTICAL PROPERTIES OF

QUANTUM

WELLS

71

10.1

General Theory

71

10.2 Valence-Band Structure: The Kane M odel

77

10.3 Bands in a Quantum Well

84

10.4 Interband Transitions in a Quantum

Well

87

10.5 Intersubband

Transitions in a Quantum Well

93

10.6 Optical Gain and Lasers

95

10.7

Excitons

97

Further Reading

06

Exercises

06

A TABLE

OF PHYSICAL C ONSTANTS

09

A2

PROPERTIES

OF IMPORTANT SEMICONDUCTORS 10

A3

PROPERTIES

OF

GaAsAlAs

ALLOYS

AT

ROOM TEMPERATURE

12

A4

HERMITE S

EQUATION: HARMONIC OSCILLATOR

13


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CONTENTS

i

AS

AIRY

FUNCTIONS: TRIANGULAR WE LL

15

AB

KRAMERS KRONIG RELATIONS AND

RESPONSE FUNCTIONS

17

A6.1

Derivat ion

of the Kram ersKronig Rela t ions 17

A6.2

Model Response Functions

19

Bibliography

23

Index

27


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PREFACE

I joined the Department of Electronics and

Electrical

Engineering at

Glasgow Uni-

versity

som e ten years ago. M y research was performed in a group w orking

on

advanced sem iconducting devices for both electronic

and optical applications.

It

soon became

apparent that advances in physics

and

technology had left

a gap be-

hind them in the education

of

postgraduate students. These students

came

from a

wide range of backgrounds, both in physics and engineering; some had received

extensive instruction in quantum me chanics

and

solid state physics, w hereas others

had only

the

smattering of semiconductor physics needed to explain the operation

of

classical

transistors. Their projects were equally

diverse,

ranging from

quantum

dots and electro-optic mo dulators to Bloch oscillators and ultrafast

field-effect

tran-

sistors. Some excellent reviews were available, but

most started at

a level beyond

many

of the

students. The

same was true of the

proceedings

of

several summ er

schools. I therefore initiated a lecture course

with

John

Barker on

nanoelectronics

that instantly attracted

an

enthusiastic audience. The course

was given

for several

years

and evolved into this book.

It was difficult to keep

the length of the lecture course manageable, and a book

faces the same problem. The applications of heterostructures and

low-dimensional

semiconductors continue to grow steadily, in

both physics and engineering.

Should

one display the myriad ways in which the properties

of

heterostructures

can be

harnessed,

or

concentrate

on

their physical foundations? T here seeme d to be a

broad gap in the literature, between

a

textbook

on quantum

mechanics

and solid

state physics illustrated with semiconductors,

and an analysis of the devices that can

be made.

I have aimed towards the textbook,

a fortunate decision as there are now

some excellent books describing the applications. The experience of

teaching at a

coup le of sum mer schools also convinced me that a more

introductory treatment

would b e useful,

one

that concentrated

on the basic

physics.

This book addresses

that need.

Acknowledgements

Several colleagues contributed to

the course out of

which this

book developed.

John

Barker, Andrew

Long, and Clivia Sotomayor-Torres

shared the

lecturing at various

times

and

helped to shape

the syllabus. Several students and

postdoctoral research


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xlv

REFACE

assistants encouraged

m e

to continue the course and learn some topics that were

new to me. I would particularly like to thank A ndrew Jen nings, Michael and Frances

Laughton, Alistair Meney, and John Nixon. It is also

a

pleasure to thank An drew

Long and

my w ife

for

their helpful comments

on the

manuscript.

M any colleagues

have

kindly provided data that I have been allowed to replot

in

a

convenient w ay to illustrate

the text. I

am

very grateful

for their help, particularly

to those who generously supplied unpublished measurem ents and

calculations,

and

to Mike Burt ,

who also gave

advice on effective-mass theory.

It has taken a long time to com plete this

book.

I don 't imagine that I am the first

author who has sadly und erestimated

the effort required to turn

a pile

of lecture

notes

into

a

coherent man uscript. Most of the work has been done in evenings, between

reading bedtime stories to my daug hters and feeling

exhaustion setting in.

A s most

parents

with youn g children w ill appreciate, this interval is

short and

frequently non-

existent. I

am

very grateful to my family

for

their forbearance and encouragement.

I would also like to thank the publishers

for

their tolerance,

as they might well have

despaired of ever receiving

a finished manu script.

The final

proofreading was carried

out at the Center for Quantized Electronic Structures (QUEST) in the University of

California at

Santa Barbara. It is a

pleasure to ack nowledg e their hospitality

as

well

as the financial support of QUEST and the Leverhulme Trust during this period.

I wo uld like to finish with a quotation from the preface by E R eif to his book,

Fundamentals of statistical

and

thermal physics. It must reflect many authors' feel-

ings

as

their

books

approach

publication.

It has been said that 'an author nev er finishes a book, he merely

abandons

it'. I have

com e to appreciate vividly

the truth of this

statement

and

dread to see

the

day w hen, looking at

the manuscript

in print, I am sure to realize that so many things could have

been done

better

and explained

more clearly. If I

abandon the book nevertheless,

it is in the m odest hope that it may be useful to others despite its

shortcomings.

John Davies

Milngavie, September 1996


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INTRODUCTION

Low-dimensional systems have revolutionized sem iconductor physics. They rely

on

the technology of heterostructures, where the composition of a semiconductor can

be changed

on the

scale

of a nanometre. For

example,

a sandwich of

GaAs between

two layers

of A l , Ga

,As

acts like an

elementary quantum well. The

energy levels

are

widely separated if the well is narrow, and all electrons may be trapped

in the

lowest level. Motion

parallel to

the

layers is not affected, however, so

the

electrons

remain free in

those directions. The result is

a two-dimensional electron

gas, and

holes can be trapped in the same way.

Optical measurements provide

direct

evidence for the

low-dimensional behaviour

of electrons and holes

in a quantum well. The

density of

states

changes from a

smooth parabola

in three

dimensions to

a

staircase in a

two-dimensional system.

This

is seen clearly

in

optical

absorption, and the

step at

the

bottom

of the

density

of

states enhances

the

optical properties. This is put to practical

use in

quantum-well

lasers, whose threshold current is lower than that

of a three-dimen sional device.

Further

assistance

from technology is needed to harness low-dimensional systems

for transport.

Electrons and

holes

must

be introduced by doping, but the carriers

leave charged impurities behind, which limit their mean

free path. The solution to

this problem is

modulation doping, where carriers are

removed

in space from

the

impurities that

have provided them. This

has raised

the mean

free

path of

electrons

in a

two-dimensional electron gas to around 0.1 mm

at low tem perature. It is now

possible to fabricate

structures

inside which electrons are

coherent

and must

be

treated as

w aves rather than particles. Observations of

interference attest to

the

success

of this approach. Again, there

are

practical

applications such as field-effect

transistors in

direct-broadcast

satellite receivers.

A s these examples

show,

complicated technology underpins experiments

on low-

dimensional systems. In con trast ,

it turns out

that most

of the

physics can be under-

stood with relatively straightforward

concepts. The

aim

of

this

book

is to explain

the

physics that underlies

the

behaviour of

most low-dimensional systems

in

semicon-

ductors, considering both

transport and

optical properties.

The

me thods described,

such

as perturbation theory,

are standard but have

immediate application the

quantum-confined

Stark effect , for

example, is both a

straightforward illustration of

perturbation theory and the basis

of a

practical

electro-optic modulator.

The most

XV


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xvi

NTRODUCTION

advanced technique used is Fermi's golden rule, which marks a traditional dividing

line between 'elementary' and 'advanced' quantum mechanics.

The

disadvantage

of

this approach is that it is impossible to describe

more than a

tiny

fraction of the applications of the basic

theory. M any topics

of

current

research,

such as the chaotic behaviour of

electrons in microcavities or the

optical properties

of self-organized quantum dots, have to be om itted. Fortunately there are several

surveys of the applications of

low-dimensional semiconductors,

and

also more ad-

vanced theoretical descriptions. This book provides the foundations on

which they

are built.

Outline

Chapters 1 and 2 provide

the

foundations

of quantum mechanics and solid state

physics

on

which

the

rest

of the book

builds.

The

presentation is intended only

as a

refresher course, and there

are

some

suggestions for textbooks

if

much of

the

ma terial is unfamiliar.

A survey

of heterostructures is given in Chapter 3, and

Chapter

4

covers the basic theory of low-dimensional systems. This entails the

solution of simple

quantum-well problems, and an appreciation of how trapping

in

such

a well makes a three-dimensional electron behave as though it is only tw o-

dimen sional. Chapter 5

is devoted to tunnelling, with applications

such

as resonant-

tunnelling

diodes.

Electric and magnetic fields provide important probes of

many

systems

and are

discussed

in

Chapter

6. Perhaps

the m ost drama tic result is

the

quantum Hall

effect

in a

magnetic

field,

which is found to be of value as a standard

of resistance w hile continuing to tax our und erstanding. Chapter

7

contains a range

of approximate m ethods used to treat systems

in a

steady state. These have wide

application,

notably to band structure.

An other example is

th e W K B

method, w hich

can be used to find

the

energies of allowed states in a quantum well or to estimate

the rate of

tunnelling through

a

barrier. Fermi's golden

rule is derived

in

Chapter

8

and used to calculate

the

scattering

of electrons by im purities

and phonons. Optical

absorption is another

major application. The final

chapters are

devoted to

the

two

principal

low-dimensional systems.

The

two-dim ensional electron gas

in

Chapter

9

is used prim arily for its transport properties, whereas the optical properties

of a

quantum well

in

Chapter 10 find employment

in

devices such as semiconductor

lasers.

The book is pitched at

the

level

of beginning postgraduate and advanced under-

graduate students. All

the basic techniques in this

book

were

in

my undergraduate

physics

courses,

although few

of the applications

had been invented It is assumed

that

the

reader has had a

glimpse

of quantum mechanics and

solid state physics,

but

only at

the

level covered

in

many courses

quaintly nam ed 'mode rn physics'.

The

first two chapters will be rather hard going

if

this is not

the case . My

experience has

been that students with a degree in

physics find the basic theory familiar, but the

applications

improve their understanding imm ensely. The background of students

from electrical

engineering varies

widely,

but the lecture course evolved to address


18/457

INTRODUCTION

vii

this,

and the

level

of the book

should be suitable

for

them too.

The theoretical level

has deliberately been k ept low and should not provide any impediment.

Exercises

Each chapter has around twenty exercises. Their difficulty

varies

considerably; some

are trivial, whereas others require numerical

solution.

I did almost all

the

calculations

for

this

book with an obsolete spreadsheet (Trapeze, dating from 1988), so the

numerical aspects are n ot serious. How ever, a symbolic-manipulation program such

as

M aple, Mathematica,

or the l ike w ould m ake these tasks easier. Occasionally

some integrals

or special functions appear

and references for

these are given

in the

bibliography.

Units

I

have used SI units throughou t this

book,

with the exception of the

electron volt

(eV), which is far too conven ient to

abandon. The main problems

for

users

of

CGS

units

are in

equations concerning electric

and

magnetic

fields. Removing

a factor

of 47r

c

o from equations

for

electrostatics

in SI

units should give the

corresponding

result in CGS units.

The

magnetic

field or flux

density B

is measured

in tesla (T) in

SI

units,

and formulas should be divided by

the

velocity of light c wherever B or the

vector potential A appears to give their form in

CGS units. Finally, I

use nanometres

rather than Angstrom units for lengths; 1 nm 10

A .

Notation for

vectors

It is often necessary to distinguish betwe en two- and

three-dime nsional vectors in

low-dimensional systems, particularly

for position and

wave vectors. I have tried

to follow a consistent notation

throughout this

book, the penalty being that some

familiar

formulas look

slightly odd.

Most

low-dimensional

structures are

grown

in

layers

and the z-axis is taken

as

the direction of growth,

normal to

the

layers. Vectors in the xy-plane, parallel to

the layers, are

denoted with lower-case letters. Thus

the position in the plane

is

r =

(x,

y).

Upp er-case letters

are

used

for the

corresponding three-dimensional

vector,

so

for position

R = (r, z =

(x, y, z).

Similarly, wave vectors

are

written

as

K = (k, k

z ) = (kx , k b ., k z

).

The

only o ther quantity that needs to be d istingu ished

in this way is

Q, which is used

for the

wave vector

in

scattering.

This notation

requires some familiar results to be w ritten w ith upper-case let-

ters

for

consistency.

The

energy of free electrons in three

dimensions

is E 0

(K)

h2K2/2ino,

for exam ple. I hope that

the

clarity offered b y

consistent usage

offsets

this sm all disadvantage.

References

and

further reading

It is not appropriate

in a

textbook to give detailed references to

original

papers. In-

stead there is a bibliography of more

advanced

books and articles from review jour-

nals.

In one or

two

cases

I

have referred to

original

papers when I w as unable to find


19/457

xviii

NTRODUCTION

an appropriate review. There are also several summ er schools

on

low-dimensional

semiconductors that publish their proceedings; naturally I

have

provided a reference

to one that I edited.

There

are

several

books

that develop

the

material

in

this

book

further. A mon g

them, Bastard (1988) gives a lucid account of the electronic structure of heterostruc-

tures, including

a

thorough description of the Kane model , with their electronic

and

optical properties. W eisbuch and

Vinter's book (1991) is

an

enlargement of

an earlier review

in

Willardson and

Beer (19664 They describe

the applications

of

heterostructures

as

well

as

their physics, with a particularly good

section on

quantum-well lasers. Their list of references exceeds 600 entries, which gives some

idea of the activity in

this

field. Finally, K elly's book

(1995)

is

notable for the

breadth

of its coverage.

H e

describes the technology of fabrication and an

enormous range

of applications of

heterostructures

in

physics

and engineering.

Just

a

glance at

the

topics covered

in

this survey leaves

one with

no

doub t that this

field

will advance

vigorously into the

next century.


20/457

FOUNDATIONS

This book is about low-dimensional semiconductors, structures in

which electrons

behave as though they

are free

to move in only two

or

fewer free dimensions. Most

of

these

structures are

really

heterostructures,

meaning that they

comprise more

than one

kind

of material. Before we c an investigate the

properties

of a heterostruc-

ture,

we need to understand

the

behaviour of electrons in a uniform semiconduc tor.

This in turn rests on the foundations of quantum mechanics, statistical mechan-

ics, and the band theory of crystalline solids. The

first two chapters

of this

book

provide a review of these foundations. U nfortunately i t is impossible

to provide

a full tutorial within the space available, so the reader should consu lt

one of the

books suggested at the end of the appropriate chapter

if

much

of the

material is

unfamiliar.

This

first chapter covers

quantum

mechanics

and

statistical physics. Some topics,

such

as the

theory

of angular mom entum, are

not included althoug h they

are vital

to

a thorough course on quantum mechanics.

The

historical background, treated

at length

in

most textbooks

on quantum

m echanics, is also om itted. There is little

attempt to justify quantum

mechanics, although the rest

of the book could be said to

provide

support because we

are able to explain numerous experimental observations

using the basic theory developed in this chapter.

1.1

Wave Mechanics and

the

Schrbdinger

Equation

Consider the motion of a single particle, such

as an

electron, moving in one dimen-

sion

for

simplicity. Elementary classical mechanics is based on the concept of a

point

particle, whose position x and momentum p (or velocity

v =

plm) appear in

the

equations of motion. These quan tities

are

given directly by New ton's laws,

or

can be calculated from the Lagrangian or Hamiltonian functions in more

advanced

formulations.

W ave mechanics is

an elementary

formulation of quantum theory that,

as

its

name implies, is centred on a wave function

t u

i (x, t). Quantities such

as position

and

momentum

are

not given directly,

but must be deduced from W . Instead of


21/457

1 . FOUNDATIONS

New ton's laws we

have a

wave equation that governs the evolution

of

tli(x, t). In

one dimension this takes the form

h 2

a

2

p(x,

t)

(x)tli (x , t) = iht P (x, t),

1.1)

2m

ax-

t

which is

the time-dependent Schrddinger equation. W e shall see

a partial

justifica-

tion of its form a little later. This equation describes a particle m oving in a region

of

varying po tential energy r(x);

forces do

not

enter directly.

The

potential energy

could

arise from an

electric

field expressed as a scalar potential, but the inclusion

of a

magnetic field is more

complicated and will be deferred to Chapter 6.

A

useful simplification is obtained by looking

for separable

solutions

where

the

dependence on

x

and t is decoupled, ) = ilf(x)T (t).

I shall

use capital

for the

t ime-dependent w ave function

and the

lower-case letter *'

for the

time-

independent function. Substituting the product into the time-dependent Schrtidinger

equation (1.1) and dividing by

T

gives

1 dT (t)

2

d

2

*(x)

T

ih

dt 2m

dx2

V(x)*(x)1

.

1.2)

Now

the left-hand side is a function of

t

only, whereas the right-hand side is a

function of

x

only, so

the

separation has succeeded.

This

makes sense only

if

both

sides

are equal to

a constant, E,

say.

Then

the left-hand side gives

T (t)

c x

exp

Et

h

xp(icot),

1.3)

where E = hco. This

is

s imple harmonic

variation in time. W e have no

choice

about the

com plex exponential function: i t cannot be replaced by

a real sine or

cosine, nor is exp(+icot)

acceptable. The

form of the time-dependent Schrtidinger

equation requires

exp(icot)

and this convent ion

is follow ed throughout quantum

mec hanics. It contrasts with other areas

of physics, where exp(icot) may be used

for the

dependence

of oscillations on time, or engineering,

where exp(+

jcot)

is

usual. Un fortunately this choice of sign has a far-reaching

influence and crops up in

unexpected places,

such

as the

sign of the imaginary

part of the complex dielectric

function E

r (W )

(Section

8.5.1).

The spatial part of the

separated Schrticlinger equation becom es

V(x)i/(x) =

2m dx 2

This

is

the

time-independent Schr6dinger equation.

The equation takes the same

form

in

three

dimensions

with

a-/ax-

replaced by V2 a2/ax2+a2/ay2+a2/az2.

Thus the solutions of the

time-dependent Schr6dinger equation take the form

iEt

(x, t)

= (x)

exp

1.5)

h2 d

2 lif

(1.4)


22/457

1.2

FREE

PARTICLES

Later we shall justify the identification of the

separation

constant

E as the

energy

of the particle. Thus

the solutions of the

time-independent

Schriidinger

equation

describe states of the particle with

a definite energy, known

as

stationary states.

Again, this term will be justified

later, but first we shall look at some simple but

important solutions of the

Schriidinger

equation.

1.2FreeParticles

The simplest example is a particle

(an

electron,

say) in free space, so

V(x) = 0

everywhere.

The

time-independent

Schrtidinger

equation is

h

2 d2

0

.

E*(x).

1.6)

2m dx

2

This is the standard (Helmholtz) wave equation, and is s imple

enough that we

can guess the possible solutions. One choice is to

use

complex exponential waves,

(x) = exp(-Fikx) or

exp(ikx).

Alternatively we could use real trigonometric

functions and

write

1r

(x)

= sin

kx

or cos

kx. It turns

out that

the

choice

of real

or complex functions has important

consequences.

Substitution shows

that any of

these functions is

a solution

with

h

2

k

2

E=

2m

(1.7)

Classically

the kinetic energy can be written as E = p

2

/2m so we deduce that the

momentum

p = hk.

Com bining this with

the relation between energy

and

frequency

yields

the two

central relations of old

quantum theory:

E = hco = hv Einstein),

1.8)

h

p = hk =

3

-

de Brogl ie) .

1.9)

D ividing the energy by

h gives the dispersion relation

between frequency

and wave

num ber to be co

= (h12m)k

2

.

This

is nonlinear, which means that the

velocity of

particle waves is

a

function of

their frequency

and must be defined carefully.

The

two

standard definitions are as

fol lows:

(1.10)

phase velocity)

h

- - =

Th

k 2m 2m

dco hk

(group velocity)

g =

dk

h

m

where v e l is the classical velocity. The w ave packet sketched

in Figure 1.1

shows the

s ignif icance of these two v elocit ies. The

w avelets ins ide the packet m ove along at

the


23/457

1.

FOUNDATIONS

FIGURE 1.1. A

wave packet, showing

the envelope that moves at v g

while

the wavelets inside

move at v p

h.

phase

velocity v p

h

while

the

envelope moves at

the

group velocity

v g

.

If

this packet

represents

a

particle such as an electron, we are usually interested in the behaviour

of the w ave function as a

whole rather than its internal mo tion. The group velocity

is then

the

appropriate

one and

it is

a relief

that this agrees with

the

classical result.

Even if

we

use a wave pack et to represent

an electron, it is still spread out

over

space rather than localized at a point as in classical mechanics. This is inevitable in

a picture based

on waves, and

mean s that we cannot give

the location of a

particle

precisely. W e'll look at this further in Section

1.5.3.

Finally, we found that

E

2k2/2m5

w hich m eans that E >

0

if k is a

real number. The wave num ber becomes imaginary, k

K , if

E 0.

This

contrasts with classical mec hanics,

where the

state of

lowest energy has

the

particle sitting still, anywhere

on the

bottom

of the well, with

no

kinetic energy and

E = 0.

Such behaviour would violate

the

uncertainty principle

in quantum me chanics, described

in Section 1.5,

so even

the

lowest state has

posit ive

zero-point

energy.

The wave functions

in the square

well

have an important

symmetry property.

Those w ith n

odd are

even functions of x

about the cen tre of the well ,

whereas those

with n even are odd functions

of x

(i t would be neater if the numbering of

states

started with

0

rather than

1). This symm etry holds for

any potential well where

V(x)

is

an

even function

of

x.

Sym m etry properties like this

are important in

providing

selection rules

for

many processes, such

as optical

absorption. This

is a very simple

examp le; group theory is needed to describe

more com plicated sym metries, such

as

those of a crystal.


25/457

(a)

Al GaAs GaAs

A lG a A s

barrier

ell arrier

6

.

FOUNDATIONS

Now that we have

calculated

the

energy levels

in a quantum w ell, an obvious

question is how do we measure them experimentally. Optical methods provide

the

most direct techniques, so w e shall next take a look

at optical

absorption in a quantum

well.

1.3.1

OPTICAL ABSO RPTION IN

A

QUANTUM WELL

The quantum

well looks like

an artificial model, which is at

home in a textbook but

has little

application in the

real world. Although

an

infinitely deep

well cannot be

made,

it is simple nowadays to grow

structures that

are close to ideal finite wells.

The energy levels in a finite well will be calculated in Section 4.2, but in practice

the

infinitely

deep

well is often used

as an approximation

because its results

are

so

simple.

A heterostructure consisting of a

thin

sandwich of GaAs between thick layers

of

AlGaAs provides a simple quantum w ell,

shown

in Figure 1.3(a). (`AlGaAs'

really means an alloy such as A10.3Ga0.7As, but the

abbreviation is

universal.) To

justify this, we need to anticipate some concepts

that will be explained more fully

later

on.

First,

look at

the

behaviour of

electrons. Free

electrons have

energy

so(k)

=

h2k2/2m0.

Electrons

in a

semicond uctor l ive

in the

conduction

band,

which changes

their energy

in

two w ays. First, energy

must

be measured f rom

the bottom of the band

FIGURE 1.3.

Optical absorption in a quantum wel l formed by

a layer of GaAs surrounded by

AlGaAs.

(a)

Potential well

in conduction and valence band,

showing two b ound states

in

each;

the

energy gap

of GaAs is really much larger than this diagram implies.

(b)

Transitions between states

in the quantum well produce absorption

lines between the band gaps of the GaAs w ell

and AlGaAs

barrier.


26/457

1.3

BOUND PARTICLES: QUANTUM WELL

at

E,

rather than from

zero. Second,

electrons behave

as

though their

mass

is

mom

e,

where the effective mass m

e

0.067 in

GaAs. Thus s(k) = Ee +h2 k 2

12m o t t l

e

. The

sandwich acts like

a quantum well because

E,

is higher in AlGaAs

than

in

GaAs,

and the

difference A E provides

the

barrier that

confines the

electrons. Typically

A E

0.2-0.3

eV,

which is not large.

How ever, we shall approximate it

as

infinite

to find

the

energy levels in a well

of

width a. Ad apting equation (1.12)

shows

that

the

energy of the boun d states, labelled w ith n e

,

is

2m

o m e

a 2

W e could m easure these energy levels by shining light on the

sample and

de-

termining which frequencies were absorbed.

A photon

is absorbed by exciting

an

electron from

a

lower level to

a

higher

one, and the

energy

of the photon ma tches the

difference

in electronic energy levels. W e m ight therefore hope to see absorption at

a

frequency given by

hw =

Ee2 Eel,

for

example. Unfortunately this is

a difficult

experiment and a different technique

is usually used.

Semiconductors

have energy levels

in

other bands. The most

important of

these is

the

valence band which

l ies below

the conduction band. The top of this is at

E

v

and

the band curves downwards as a function

of

k, giving

s(k) =E, h 2 k 2 /2momh,

which contains another

effec tive mass m h (mh =

0.5 in

GaAs). The conduction and

valence bands are separated by an

energy called the

band gap

given by E

g = E,

E.

Ag ain there is a quantum well because

E, is at a different level in the

GaA s well

and AlGaAs barriers.

The energies

of the bound states

are

h

2 7

2

n 2

Ehnh

vGaAs

1.14)

2m0mha

2

Everything is 'upside down' in the valence band,

as

shown

in Figure 1.3(a).

The valence band

is com pletely

full, and the conduction band

com pletely empty,

in a pure semiconductor at

zero

tempe rature. Optical absorption must therefore

lift

an

electron from

the valence band

into

the conduction band. In a

bulk sample

of

GaAs this can occ ur provided that

hw > EgG a A s ,

the band gap of

GaAs. Similarly we

need

hco > E

AlGaAs

in AlGaAs. This

process leaves behind

an

empty state or hole

in the valence band,

so

a

subscript

h

is used to identify parameters

of the valence

band.

Now

look at the quantum well. Although the well is of

GaAs, absorption

cannot

start at

hw =

E

gG a A s because

the

states in the well are quantized. The lowest en-

ergy at which

absorption can occur is given by the

difference

in energy

Eel Eh]

between

the lowest state in the

well

in the conduction band and the

lowest state

in the

well

in the valence band.

A bsorption can occur at higher energies by us-

ing other states, and we shall see later that the strongest transitions occur between

corresponding states in the

two bands, so

set n e

=n

= n.

Therefore strong

h 2

7 2

n

2

Eene

cGaAs

(1.13)


27/457

h

2 z 2 n 2

1

1

1

=

E

G a A s

g m0a2m

Mh

(1.15)

1 .

FOUNDATIONS

absorption occurs at the frequencies given by

ntO

n

=

(

Een

_

Ehn

=E

GaAs

h2z2 n

2

2mom

e

a 2

(E

v

GaAs

h2

7 .

t

2

n 2

2momha

2 )

The

energies look like those

in a quantum well where

the effective mass

is men,

given by 1 /

Meh =

1/M

e 1/Mh.

This is called the optical effective mass; almost

every process has its own

effective

mass

If the

wells really were infinitely

deep,

there would be

an

infinite series

of

l ines

with frequencies given by equation

(1.15).

The barriers in the

semiconductor

are

finite,

and absorption occurs in the

AlGaAs barriers

for all frequencies where

n

o )

> EAlGaAs.

The resulting spectrum is shown in Figure

1.3(b), assuming that

there

are

two bound states

in

both

the conduction and valence

bands. No absorp-

tion is

possible for hco < E

G aA

s,

and there is

a

continuous band of absorption

f

or

h c o

FA1GaAs.

Between these two frequencies lie

two discrete lines produced

'g

by transitions between states in the quantum well,

with energies given by equa-

tion (1.15). The width of

the well can be inferred from the energy of

these lines

if

the effective m asses are known.

This

is

a routine check to see that layers

have

been

grow n co rrectly .

In

practice a slightly different experiment is usually performed, called photo-

luminescence

(PL).

Light with hco > E

g

AlG

a

A s is shone o n the sample, which excites

many electrons from

the valence

to

the conduction band

everywhere. Some of

these electrons become trapped

in the quantum well, and the same thing happens

to the

holes in the valence band.

It is then

possible for an

electron to fall from

the conduction band

into

a hole in the valence band and release the difference

in

energy

as

light. This luminescence

is

the reverse

process to

absorption and

can

occur at

the

same energies. Only

the

lowest levels

are

usually seen, so the PL

spectrum should show a

line at

ha)]

.

A n

example of a photoluminescence spectrum

is shown in Figure

1.4. The

sample has four wells of different widths, each of

which contributes

a peak to

the PL. A

detailed analysis of this spectrum is left

as

an exercise.

Unfortunately the true picture is slightly

more

complicated.

One problem is that

the valence band is not as simple as we have assumed .

A

better model is to

assume

that there are two varieties of

holes, heavy and

light. Thus two

sets of spectral

lines should be seen, although

the

heavy holes

are

much more

prominent.

A

further

complication is that electrons

and

holes bind together to form

excitons,

analogous

to hydrogen atoms,

and this

modifies the

energies slightly.

This problem will be

addressed in Section 10.7.


28/457

12

10

8 -

6

4 -

2 -

c

o

n

s

1

Ait

(a)

1.4

CHARGE AND

CURRENT DENSITIES

9 nm 6 nm 4 nm 2 nm

650

00 50

00

wavelength / n m

FIGURE 1.4.

Photoluminescence as a function of wavelength for a sample with four quantum

wells of different widths, whose conduction and valence bands are shown on the right. The barriers

between the w ells

are mu ch thicker than drawn.

[Data

kindly supplied by Prof. E. L. Hu, Un iversity

of California at Santa B arbara.]

1.4 Charge

and

Current Densities

The Schrbdinger equation yields a wave function x l i ( x , t ) , and we should now con-

sider how to deduc e quantities

of interest from it.

First we would like to know the

location of the

particle. This follows from

the

squared modulus

of the

wave function:

1W (x ,

01

2

OC

probability density

of

finding the

particle at x. 1.16)

This

does n ot strictly im ply that the particle itself is spread out:

it should be inter-

preted

as a

statement about our know ledge

of the

particle. H owever, this

distinction

will not be important for the topics covered in this book,

and we can

put the relation

in a more physically transparent

form by using

the charge density. If the

particle

has

charge

q,

this becom es

qlw (x

,

t ) 1

2

a

charge

density of particle,

1.17)

or

q 1W ( x , t)1 2 dx c x charge in a region d x around x.

1.18)

If the

electron is bound w ithin some volum e,

we know that

the total charge

enclosed

in that volume must be

q The

propo rtionality can then be turned into an

equality:

q 1

1 P

(x,

0 1

2 =

p

x) =

charge

density

of particle. 1.19)


29/457

1 .

FOUNDATIONS

A n integral over

the total volume must

recover the total charge:

f

p(x)dx

qlkli(x,t)1

2

dx = q.

1.20)

Removing the charge q

from this equation gives

f

T(x, t)1

2

dx = 1.

1.21)

This is the standard condition for normalizing the wave function,

and

means that

t ) 1

2 is

the probability density of finding the particle.

Not all wave functions can be normalized in this way. T he free electron is an

obvious example because

the integral over all space would

diverge. In this

case

one

can talk only

about

relative

probabilities.

In

practice, we can get around this

difficulty by starting with

the

electron

in a large but

finite

box, for which there

are no

problems, and

letting

the volume of the box go to infinity at

the end of the

calculation. We shall do this in Section 1.7

to calculate

the density of

states.

Normalization gives physical

dimensions to the wave function . Take the

infinitely

deep well as an

example . The

wave functions O

n

(x) =

A

n

sin(turx/a),

and the

condition for normalization is

1 = f

an7441

sin

2

dX =

0

a

2

Thus

the normalized wave function,

if A ,

is taken to be real, is

2

u r x

On(x)=

in

a

(1.22)

(1.23)

Normalization has given

the

wave function

dimensions of (length) -1 / 2 in one

di-

mension. This

is often useful

as a check.

A plane wave such

as

Ok(x)

=

A e

i

"

n an

infinite

volume

can be normalized

in a

slightly different

way.

Here

the

densi ty I

( / ) k ( x ) 1 2

=

I A

2

,

which can be

set

to

a

given density

of

particles.

Now that we have a charge density, there should be a

current density

J (or

just current in one dimension)

associated with

i t .

These must obey the continuity

equation

a J

(1.24)

a x t

to ensure

the conservation of charge and

part icles. In

three

dimensions

8

Jlax

becomes

div

J .

To const ruct

a

current den sity, start w ith

the

time-dependent

Schrbdinger

equation

h 2 8

a

W

(x,

t)

+ v(x ,

t) tp( x, t) = ih tp(x, t).

2m ax 2

t

(1.25)


30/457

1 .4 CHARGE AND CURRENT DENSITIES

1

M ultiply both sides

on the

left by

the

complex conjugate of the wave function,

x l i * :

h2 92

2m

k W ' x2

+ kIJ*V = IP 1.26)

at

Obtain

a second equation by going back to

the

Schr6dinger equation, taking its

complex conjugate,

and multiplying from the left by

his gives

h

2

2

LP xlirkr =

1.27)

2m ax

2

t

Now subtract

(1.27)

from

(1.26).

The

terms w ith

the potential cancel provided that

V(x, t)

is real. The

two terms

on the

right-hand side add and are clearly the

derivative

of a product,

so the

difference becom es

h2

( *

O

2

v

IT 1

2

2m x

2

x

2

t

To simplify the left-hand

side, use the rule for the

derivative

of a product:

(1.28)

( .a

ax(

j

ax qj )

kr\ /tp

O x

()T*W2 2 T

ta

(1.29)

W hen this is applied to (1.28),

the

products of single derivatives cancel and

it reduces

to

h2

tp*

w

v*)

4JF .

2m ax

x

t

(1.30)

Finally, moving the factor of ih

to

the left and

multiplying throughout by

q to turn

the probab ility densities into charge densities gives

O

r h

q

(w*

* V 1 =

q

1 2

) =

ax

L2im

x

x 1

t

t

(1.31)

Com paring this w ith

the

continuity equation

(1.24) shows

that

the current density

is given by

J(x,t) = hq

(kr

tp

p .

1.32)

2im

x

x

In three dimensions

the

derivative atP/ax becomes the gradient

V

1 1 .

The dependence

on

t ime vanishes from both p

and J

for a stationary state be-

cause exp(icot) cancels between

k I J and k l i * . This partly explains

the

origin of the

term 'stationary', althoug h the states may still carry a

current (constant

in time)

so it

is slightly m isleading. H owever,

a

stationary state where

lk

(x) is purely real carries

no c urrent. This

applies to

a

particle in a box and to bound states

in

general.

A

su-

perposition

of

bound states is needed to generate a current. This feature emphasizes

that the

wave function in quantum

mechanics is

in general

a 'genuine' com plex

quantity.

This contrasts with

the

complex notation widely used for oscillations in


31/457

12

.

FOUNDATIONS

systems ranging from electric circuits to balls

and

springs. Here the

response is real

and the

comp lex form is used only for

convenience.

As an exam ple, consider tli(x,

t) =

A

exp[i(kx cot)],

which describes

a plane

wave m oving in the

+x

-direction. Its charge density

p = qIAl

2

,

uniformly over all

space, and the

current

J = q(hk1m)

IAl

2 . Now hklm = plm = v so

J =

pu,

which is the expected result (like

'J = nev').

The

Schrtidinger

equation is linear, so further wave func tions can be c onstructed

by superposing basic solutions. For example,

kIJ(x,

t) = [ A + exp(ikx) +

xp(ikx)] exp(icot)

1.33)

describes a superpo sition of

waves

travelling in opposite directions. The

quantum-

mec hanical ex pression for the

current gives

the

expected result

hqk

=(IA+I IA-1 2

).

1.34)

There is an interesting result for tw o counter-propagating

decaying

waves,

k V (x , t) = [B+ exp(Kx) + B_ exp(K x)] exp(ic.ot). 1.35)

Neither com ponent would

carry a

current by itself because it is real,

but the

super-

position gives

J =

hqic

(B B *

B* B_)

2hqK

Im(BB*).

(1.36)

The

wave

must contain components decaying in

both

directions, with a phase

dif-

ference between them , for a current to flow. This effect is shown

in Figure 1.5 for

a

wave hitting a barrier. We shall see in

Chapter 5 that an oscillating wave turns

into a decaying one

inside a

high barrier. If the

barrier is infinitely

long, it contains

a single

decaying wave

and

there is no net

current. A

finite barrier,

on the

other

hand, transmits

a

(small) current

and must

contain two counter-propagating decay-

ing waves.

The

returning wave (exp

Kx) from the far end of the

barrier carries

the

information

that the

barrier is finite and that a current flows.

(a)

FIGURE 1.5.

Current carried by counter-propagating decaying waves.

(a) An

infinitely thick

barrier contains a single

decaying exponential that carries

no current. (b)

A

finite barrier contains

both growing and

decaying

exponentials

and passes

current.

(The wave function is complex,

so

the

f igure

is only

a rough guide.)


32/457

1.5

OPERATORS

AND

MEASUREMENT

3

1 .5

Operators

and Measurement

It is now tim e to return to the theory of quantum mechanics in a

little

more

depth,

and

to see how physical quantities can be deduced from

the

wave function.

1.5.1

OPERATORS

It is

a

postulate

of quantum mechanics that observable quantities can be represented

by op erators that act on the

wav e function (although it is a

further postulate that

the

wave function itself is

not

observable). Operators will be denoted w ith

a

hat or

circumflex.

The position, mom entum, and total energy can be represented by the

following operators

on

kli(x,

t):

x

x,

1.37)

a

p ih, 1.38)

ax

E

ih

.

a t

(1.39)

A n important feature is that

the

momentum

/ 3

appears as a

spatial

derivative.

More

com plicated operators can be constructed from these compon ents.

For ex-

ample, the

Ham iltonian function

H

= p2 /2m V(x) gives the total

energy of a

classical particle

in the type of

system that we

have

studied, where energy is con-

served. This becomes a Hamiltonian operator

fi in quantum mechanics

and is given

by

2

a 2

= H (2 , )3) =

-

v(x).

2m ax

2

(1.40)

Equating the effect

of

this operator w ith that

of the energy operator gives 1

7

:141 =

EkIJ ,

or

[

2

h

a

W

(x, t). 1.41)

x)

x, t) =

2m ax2

t

W e

are

back to

the

time-depend ent Schr6dinger equation (1.1).

The

t ime-independen t Schriidinger equation can now be w ritten concisely

as

i f * (x) = (x),

where E is a number, not an operator.

This resembles

a matrix

eigenvalue e quation: there is an operator

acting on the

wave function

on one side,

and a constant

mu ltiplying it

on the

other.

The

ideas of eigenvectors and eigenvalues

work

in much the same way

for

differential operators

as for matrices, and

similar

terminology is used. Here 1t is called

an eigenfunction or eigenstate, and

E is

the

corresponding eigenvalue.

This

will be develop ed further

in Section 1.6.


33/457

14

.

FOUNDATIONS

The current density can be rewritten

in

terms of the momentum operator, giving

J(x,t) = 1 -

141)

(

1 -1

41)

2

This show s that

the

current is related to

the velocity

p/ in. A more

elaborate expres-

sion is needed

in a

magnetic field, which complicates the relation between velocity

and

momentum.

This

will be considered in

Chapter

6 .

N o w look

at the

effect of the momentum operator on a wav e funct ion.

A plane

wave

iii(x) =

A exp(ikx) gives

fr = in (Ae i k x) hkAe i k x = (hk4f.

1.43)

dx

Again this has reduced to

an

eigenvalue

equation. We interpret this

as

meaning that

the momentum has

a definite value

p =

fl k,

a result which we inferred earlier by

analogy with classical mechanics.

A

further postulate

of quantum

mechanics states that

the only

possible values

of a

physical observable are the eigenvalues of its corresponding operator. If the

wave function is

an

eigenfunction

of

this operator,

as in the case of the plane wave

and momentum, the observable has

a

definite

value. In general this is not

the case.

Consider the

effect o f the

momentumoperator

on a

particle

in a box:

d

ihnz

A ,

zx

(1),(x)

ih

dx

A

n

si

nzx

n

os

1.44)

a

These wave functions

are

not eigenfunctions

of f9,

and

therefore

do

not have a

definite

value of

momentum. Measurements of

momentum would yield a range of

values which we could characterize

in

te rms of an average value (zero

here) and

a

spread. Taking another derivative

shows

that O n (x)

is

an

eigenfunction of /3

2 .

It

therefore has

a

definite value of kinetic energy, whose operator

t = 13

2

12m.

Similar issues arise

when we measure the position of a

particle, which we shall

consider nex t .

1.5.2

EXPECTATION VALUES

Suppose

that we

are

given ( x , t) for some particle. Two simple quantities that

we might wish to know

are the average position of the

particle and how well it

is localized

about

that position. Note

that we cannot say that

the particle is at a

particular

point ,

unl ike

in classical mechanics, because we

are

using

a

picture based

on

waves .

W e know tha t

the probability density for finding a

particle is

P(x,t) cx

W

(x,

t)1

2 ,

and

can

use

this in the standard formula for

finding a m e a n value. This gives

( 1.42)

(x(t)) = f x P(x,t)dx,

1.45)


34/457

1.5 OPERATORS AND MEASUREMENT

5

where

angle

brackets

( )

are

used to denote exp ectation

values. For

normalized wave

functions this becomes

(

x(t))

=

f x141 (x , 01

2

dx = f W*(x

To answer

the question of how well

the particle is localized,

the standard

deviation

A x defined by

(Ax )2 =

(

x

2)

x

)2 ,

where (x

2

) is the expectation

value of x

2

, given in the

(x

2

) = f x 2 P(x) dx = f

T*(x, t)

Take the

lowest state of a particle in a box as an

example.

2

r X

dx

f

(X) = s i n

2

, t) x 41(x , t) dx .

a common

same way by

x 2 ) dx .

Then

a

2'

1 \

(1.46)

measure is

(1.47)

(1.48)

(1.49)

(1.50)

(1.5 1 )

a

which is obvious from sym me try,

and

2

TX

2

x 2

)

=

f x

2

s in

2 dx

a

a

Thus

3

1

7 2 )

18 a

1

A x

--

12 7 2

The

particle is most likely to be foun d in the middle

of the

well, but

with considerable

spread around this (which increases for higher states).

The same questions can be asked about

the

momentum

of the

particle

and

can be

answered

in the same way using

the momentum operator.

The general expression

for the expectation value

(q)

of some physically observable quantity

q

is

(q) = f W*(x, t) W(x, t) dx ,

1.52)

where is

the

corresponding operator.

For example, the average value of the mo-

mentum is given by

in

w x ,

x

8x

This can be ex tended to quantities such

as

( p

2

)

and

A p

as

was done

for

x.

The results of these expressions are physical quantities, and must

therefore be

mathematically real numbers. This

requires that physical quantities be rep resented

by Hermitian operators. Such operators

have real

eigenvalues, which guarantees

( p) = f x , t)

x , t)dx =

(1.53)


35/457

16

.

FOUNDATIONS

that measurements

on the

wave function will yield real values. Their properties are

reviewed briefly

in Section 1.6. Non-H ermitian operators are important in other

applications, notably

as

creation

and annihilation operators

in field

theory, but will

not be used

in

this

book.

Expectation

values of stationary states

are constant in time,

because their depen-

dence

on t ime cancels between

k I J and W". For example, (x) is constant for

any

stationary state, so the

particle appears to be 'stationary'. A superposition of states

is required for the

particle to 'move' in the sense that (x) varies with t ime.

Going

back to the one-dimensional well again, we can construct

a

moving w ave function

from the first two states,

t

= 0) = 2414)1(x) + A202(x)

A s

this wave function evolves

in time, the average position

becomes

a

2a,41,42

(82 e i)t i

.

(x(t))

=

os

2

7 1 . 2

(1.54)

(1.55)

The

particle oscillates back

and

forth in the

well at angular frequency

(82 1)

h

given by

the

difference

in energy

of the

two levels.

The

analysis is left

as an exercise.

1.5.3

MOTION OF A

WAVE PACKET

Elementary classical mechanics rests on the concept of point

particles, whose po-

sition and mo men tum can be specified precisely,

but

this is not tenab le in

wave

mechanics.

The

natural

analogue

is a

wave packet like that in Figure 1.1, a wave

that is restricted to

a finite region by

an

envelope. This

also provides another illumi-

nating example of expectation

values. Start with

a plain carrier wave

exp(ip

o

x/h),

and m odulate it with

a

Gaussian envelope at

t

= 0:

1

x

)

(x x0)

2

1

(x ,

t

= 0) =

xp

xp

(27d

2

)

1 /

4

d 2

(1.56)

The probability density of this is

a

normalized Gaussian function with mean xo and

standard deviation

d:

[

(x xo)

2

IT(x, t =

0

) 1

2

=

(27rd2)1/2

ex

2d

2

(1.57)

It is clear from this that (x) = x o

and Ax =

d at t

= 0.

W e can make

the wave

packet

as

localized as

we d esire by choosing

an

appropriate

value of d.

The carrier

has defini te m omentum po but

we

have

had to m ix many waves

together to get the wave p acket, so there is now a range of momenta in the

wave

function. There are

two ways of

extracting

(p)

and Ap. One

is to

use the definitions


36/457

1.5

OPERATORS

AND MEASUREMENT 7

of the

expectation

values

like equation

(1.53)

given earlier. The

other way is to

write the

wave function

as a

function

of mo mentum rather than

position.

Since

we kno w that a plane wave exp(ipx/h) has definite m omentum p, the distribution

of

mo menta w ithin W is given by resolving i t into

plane

waves

-

just a Fo urier

transform. Thus

the wave function

in

mom entum space (I)

(p, t)

is related to that

in

real space by

(+_px\ dp

t) = f

I ) (p, t) exp

h 12.7T

'

dx

(I)(p,

t) = f

)

exp

h )

N

/27rh

(1.58)

(1.59)

The factors of

,,,[2

-

th

ensure that

s z l )

has the

same normalization

as

W .

Taking

the Fourier transform

of

equation (1.56)

for the

Gaussian wave packet

gives

1

-i (p - po)xo]

exp

[

( Po) 2 1

(1)(p,

t

= 0) =

[27 (h 12d)2]1/4

exp

(h/2d)

2

(1.60)

whose p robability density is

1

(

2

P PO)

4(p, t

= 0)1 =

xp [

1.61)

[27 (h/2d)

2 ]I/2

(h/2d)

2

j

This is another norm alized Gaussian with m ean (p) = p

from the carrier and

standard

deviation Ap =

h I2d .

A n important

result comes from the

product of the standard deviations in

space

and

momentum:

h

Ax Ap =

d

d

= 1.62)

Thus

the

better we localize

the

particle to fix its position in

real space,

the more

waves we need

and the wider

the

spread in mom entum becomes. This

is

the

fa-

mous Heisenberg uncertainty principle: we cannot m easure both

the position and

momentum of a particle to arbitrary precision. It contrasts with

the

classical picture

where both x

and

p could be know n precisely. Gaussian wave p ackets happen to

give the minimum uncertainty, an d in

general

the result is

Ax Ap > 1.63)

As one example, a plane wave

exp(ipo x

h) definitely has mom entum po

, so

Ap = 0,

but it is spread even ly over all space giving

Ax =

Do.

The uncertainty principle also forces the lowest state

in a quantum

well to have

non zero kinetic energy, unlike classical mechanics where it wo uld be

still. The

momentum

p

wo uld be know n exactly (zero) if the particle were at rest,

giving


37/457

1 B

. FOUNDATIONS

Ap = 0,

while

A x

is finite because w e know that

the

particle is somewhere in the

well . Thus

Ax Ap = 0, which is not allowed. The only w ay around this is for the

particle to

have a

zero-point energy in the lowest state so that both Ax and A p are

non-zero.

This

can be used to estimate the zero-point energy. C onsider, for

example,

an

infinitely

deep

potential well of width

a. W e know that

the

particle is in the well,

so roughly A x a/4.

This

me ans that Ap h/2Ax =

2h/a.

Taking the kinetic

energy as

(Ap)

2 /2m

gives

an estimate

of (h

2 /2m)(2/a) 2 for the

energy of the

ground state. The exact

result has

7 r

instead of

2.

This

explains

the

dependence

of

the

energy levels on the

width a:

making

the well narrower reduces the spread

of

the

particle in

real space

and

therefore increases its

range of momenta and hence

the

energy. This

principle can be extended to estimate

the zero-point

energy

in any

well by including

the

mean potential energy.

Returning to

the

Gaussian wav e packet, we foun d that it has

minimum uncertainty

(in the sense

of the

product

Ax Ap)

at t = 0, but this changes as it evolves in

t ime.

W e know that a plane wave

exp(i px Ih) evolves

in

time like exp(i tot) with

hco =

p

2 /2m.

This

applies to each Fourier component

of the

wave pack et, so the

wave function

in Fourier space for t >

0

is

1

szti(p, t) =

[27 (h12d)21114

exp

[-i(P-

h

13)x exp

(h/2d)2

P P())2

1 exp (

2h

iP2t).

m

(1.64)

W e

must

transform this back to real space to find

(x ,

t). A little rearrangement

gives

(

p

o (x po t 12m)1

x, t)

xp [

[27 (h12,d)

2

]

1 1 4

x

f

exp

[

i po)(x

x o pot/m)1

h

x exp

(p

p

0

) 2

ht

p

L 4(h/2d) 2

m d 2

LI I23

Th

(1.65)

The prefactor gives

a ca rrier

wave with momentum po, moving at the phase velocity

v

p

h = p0/2m. Inside

the

integral,

the first exponential

shows

that

the

wave packet

is now centred

on x o

+

p

o t I

m and

therefore m oves at

the

group velocity v p =

po/m.

The second

exponen tial, which controls the width of the wave packet, is also

modified. Evaluation

of the

integral

shows

that

ht

Ax(t)

1.66)

2,n2d)


38/457

1.5

OPERATORS AND MEASUREMENT

The pulse spreads

out in space

as

it propagates. The mom entum remains unchanged

if there are no forces acting o n the

particle, so

the product

A x

zip grows

and

our

information abou t the

particle deteriorates in time. This

is the

typical effect

of

dispersion as

seen,

for

example,

in communications.

Dispersion arises because a wave packet necessarily co ntains

a range of mom enta,

each of which propagates at a different velocity causing the wave p acket to spread.

Eventually this overwhelms the initial

width.

The range of velocities is (Ap)Im

so at

large times we expect Ax (Ap)t I

m = ht 12md,

in agreement with equa-

tion

(1.66).

A short pulse

contains

a wider

range of m omen ta than a longer pulse

and will eventually become longer.

1.5.4

FURTHER PROPER TIES

OF

OPERATORS

The uncertainty relation can be traced back to properties

of the operators involved.

W e are trying to measure both the position and momentum of the particle described

by the wave packet. A problem arises because of the

order of these operations.

Suppose we first measure

the

mom entum, then the position. The operators acting

on the wave function are

iptp = x ( ih

k p inx

aT

ax

x

The op posite

order gives

= ( i

w =

in (c a

(1.67)

(1.68)

The

last line follows from

the

derivative

of the produc t. Clearly the results

are

different

and the order

of the

operations is significant. Subtracting

the

two gives

i

1.69)

Since this equation holds

for any T,

we can w rite it

for the

operators alone

as

[2,

h.

1.70)

The no tation ri , p 1

is called a

commutator.

Two operators are

said to

commute

if

[A, B] =

0 since the order

of their operation is unimportant. It is

possible to

measure two physical quantities simultaneously to arbitrary accuracy only

if

their

operators commute. Clearly this does n ot apply to

x

and

p,

and their accuracy is

limited by

the

uncertainty p rinciple.

Similar relations apply to other coordinates

and their corresponding m om enta

such

as [j), fry] =

in.

On the other hand

[ 5 ' ,

fix

] = 0,

so these quantit ies can be

measured simultaneously. Some further examples are given

in the problems.


39/457

20

. FOUNDATIONS

A quantity w hose operator commutes with the

Ham iltonian is called a constant of

the motion because its value

does not change

with

time. For example, [P, = 0

for a free

particle so i ts m omentum remains constant. These constants usually

arise

from some symm etry

of the

system, translational

invariance in

this

case.

W e

have

seen that

the

order of

operators such

as

and p

is important and

that

they cannot be reordered like numbers. The

sam e is true

of matrices, and

we shall

see later that operators can be represen ted by

matrices instead

of the

differential

operators used here. Further, the choice of operators depends on the way in which

the

wave fun ction is represented. W e derived the wave function of a wave packet in

mom entum space before and could use the corresponding operators

a

= ih.

1.71)

ap

These obey the

same

commutation relation

] =

ih

as the

earlier forms

in x

and are

therefore

an equally valid choice.

1.6Mathematica Properties

of

Eigenstates

This is

a

brief

section on formal properties of

eigenstates,

w hich w ill be needed

later

in the construction of perturbation

theory. Further details can be found

in a

book on mathematical methods

for

physics such as

Mathews and

Walker (1970).

W e

have already seen that the

wave functions

in the

infinitely deep square

well

can be norm alized. Assume that w e are dealing with

a finite system, so we c an

ignore the

problems posed by plane waves

and the

like.

Let the eigenstates (wave

functions) of the

Ham iltonian be

(/), (x)

with corresponding

eigenvalues (energies)

E n

,

and norm alize each state such that

f

14),(x)1

2

dx =

1 .

1.72)

The range of

integration covers

the

region w ithin which the particle can m ove,

0


40/457


41/457

1 . FOUNDATIONS

W e know that each eigenstate

evolves

in

time like O n (x) exp(i E n tlh), and therefore

the

given state evolves

as

n

kI J (x, t) =

n

ck, (x)

exp

Et)

h

1.81)

n=1

This is how we followed

the

evolution of the wave packet

in Section

1.5.3.

1.7

Counting

States

A complete

description of a

system requires the energies

and wave functions

of

all

its states. Clearly this is

an impossible

task

for

anything

but the

simplest systems,

and

most

of the information would

in any

case be unwanted.

For many applications

the

density

of

states

N (E) is adequate.

The definition is that

N (E) BE is the number

of states of the system w hose energies

lie in the range E

to

E 3E.

Clearly this

tel ls u s nothing about the wave functions at all, just

the distribution of energies. W e

shall first calculate

the

density of states of a one-dimensional system before looking

at more

general results.

1.7.1

O NE DI M E NSI O N

A n imm ediate problem, as we saw

in the

previous section, is that

the wave functions

exp(ikx ) cannot be normalized in the

usual way

if the particles travel through all

space. The simplest way around this problem is to put the particles in a

finite box

of

length

L, and se t L

oc

at the end of the calculation. Hav ing put the particles

in a box, we need to choose boundary condit ions. Two are

comm only used.

(i )

Fixed

or box

boundary conditions, in

which the

wave function vanishes at

the

boundary:

f (0) = f(L) =

0.

1.82)

(ii) Periodic or Bornvon Karman

boundary

conditions, in which we

imagine

repeating

the system p eriodically with the same w ave function

in

each sys-

tem. The

wave function at

x =

L must match

sm oothly to that at

x = 0,

which requires

(0) =

(L),

a l f r

(1.83)

a x

x=0

x

x=L

Fixed boundary

conditions are

obviously

the same as the

particle

in a box studied

earlier.

The

energy levels are given by

E0(k) =

h

2

k

2 /2/12 and the allowed values of

k

are

J r

in

L

m = 1,

2, 3, . . . .

1.84)


42/457

1.7

COUNTING STATES

3

The wave functions are standing waves w ith this choice. They carry no

current, and

the

allowed values of k are all positive.

Periodic boundary

conditions require

a different choice

of

k.

W e can

use trav-

elling exponential waves rather than

sine waves, and

they

must

obey

exp(ikL) =

exp(i k0) = 1 = exp(27ni). This

also satisfies

the condition on the gradient, and

the

norm alized states

are O

n

(X)

= L - 1 1 2

exp(i

k

n

x).

The allowed

values of

k are

27n

=

= 0, 1,2,

(1.85)

These

are

twice as far apart

as

with fixed bound ary

conditions, but both signs

of k

are

permitted and there are two degenerate states at each energy level (excep t for

k =

0),

with opposite signs of k and velocity.

This

raises the

following

crucial question: does the density

of

states, which we

are

trying to

calculate, depend on

which boundary conditions

we choose to apply to our

artificial

box?

Fortunately it can be shown that

the

result is insensitive to bo undary

cond itions as

L co. It is usually

more appropriate to treat

free electrons

as

travelling

rather than

standing

waves, so periodic boundary conditions are

generally

used.

To turn these allowed

values of k and

E

into a density of

states,

plot the allowe d

values of k along a line

as in Figure

1.6. This

is a simple

one-dimensional

the physics of low dimensional

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