same sense as the u °rÞ, in that we do not expect to be able to ask the same

questions about probability densities that we might ask about an electron in

the state ji. It is instead just a useful device that allows us to insert Eq. (2.1.9)

into Eq. (2.1.8) and write

j 0 ih 00 j ¼ j 0 ih0j0ih 00 j

¼ °j 0 ih0jÞ°j0ih 00 jÞ:

We now de¬ne the combination j0ih 00 j as the operator c 00 that annihilates

any electron it ¬nds in the state j 00 i, and the conjugate combination

j 0 ih0j as the creation operator cy 0 for the state j 0 i. Then

j 0 ih 00 j cy 0 c 00 ;

and

X

h 0 jVj 00 icy 0 c 00 :

V°rÞ

0 ; 00

31

2.2 Occupation numbers

Similarly for the kinetic energy operator

1 X 0 2 00 y

p2

h jp j ic 0 c 00 :

2m 2m 0 ; 00

One might at this stage ask what use it serves to write the Hamiltonian for

an electron in this form. As long as there is only one electron present then this

way of representing operators is just a needless complication. It is when we

have a large number of identical particles present that this language comes

into its own.

2.2 Occupation numbers

Let us consider a collection of N identical free particles that do not interact.

The Hamiltonian for this system is just the sum of the Hamiltonians for the

individual particles,

X 1X2

H¼ Hi ¼ p:

2m i i

i

We could form eigenfunctions of H simply by multiplying together the eigen-

functions of the individual Hi . If

Hi ui °ri Þ ¼ E i ui °ri Þ

and

Y

N

0

È¼ ui °ri Þ ¼ u1 °r1 Þu2 °r2 Þ Á Á Á uN °rN Þ

i¼1

then

X

N

0

Ei È 0:

HÈ ¼

i¼1

The wavefunction È 0 however is not adequate as a physical solution of the

Schrodinger equation because of its lack of symmetry. We know that if we

¨

interchange any two of the coordinates, then the wavefunction È must either

remain unchanged or else be changed by this operation to ÀÈ. That is, È

must either be symmetric in the coordinates of the particles, which we then

call bosons, or else be antisymmetric, in which case we refer to the particles as

32 Second quantization and the electron gas

fermions. Electrons in particular are known to be fermions, and so È must be

expressed as an antisymmetrized combination of the È 0 . We can achieve this

by forming a determinant of the various ui °rj Þ and writing

u1 °r1 Þ u1 °r2 Þ u1 °rN Þ

ÁÁÁ

1 u2 °r1 Þ ÁÁÁ

È ¼ p¬¬¬¬¬¬ . :

.

N! .

.

.

.

u °r Þ Á Á Á uN °rN Þ

ÁÁÁ

N1

This form of the wavefunction (known as a Slater determinant) is very

cumbersome to handle, even in this simple case where the particles do not

interact. When interactions are present È will no longer be an eigenfunction

of H, but since the various È form a complete set any N-electron wavefunc-

tion É can be expanded in terms of N-electron Slater determinants. We could

then write

X

É¼ C°1 Á Á Á N ÞÈ°1 Á Á Á N Þ

1 ÁÁÁ N

where the i label the various states ui that occur in the determinant. If we

knew the complex constants C°1 . . . N Þ then the wavefunction would be

completely speci¬ed; it would, however, be grossly inconvenient in making

calculations, each È alone being a sum of N! terms.

The shorthand we use for describing the È is known as the occupation

number representation, and cuts out all the redundant information contained

in È. We know ¬rst of all that there are N particles, and that all the coordi-

nates ri come into the wavefunction on an equal footing. We also know that

È is antisymmetrized, and so we do not need to write this explicitly every

time we consider È. All we need specify are the states u that are occupied.

We de¬ne an occupation number, n , that is equal to 1 if the state appears in

the determinant describing È, and equal to 0 if it does not. Thus we specify

the two-particle state

u °r Þ u °r Þ

11 2

È; °r1 ; r2 Þ ¼ p¬¬¬

2 u °r1 Þ u °r2 Þ

by writing

n ¼ n ¼ 1; all other n ¼ 0:

33

2.2 Occupation numbers

We adopt the notation

j1; 1; 0; 0; . . .i

to signify the wavefunction in which

n ¼ 1; n ¼ 1; n

¼ 0; n ¼ 0; etc:

Clearly we can specify a È for any number of particles by means of this

notation. We might further abbreviate by writing È as jfn gi, where fn g is

understood to be the set of occupation numbers n ; n , etc.

Now we know that for any two distinct È describing states containing the

same number of particles

°

È* È 0 dr1 dr2 Á Á Á drN ¼ 0: °2:2:1Þ

We must also consider the È to be orthogonal when they specify di¬erent

numbers of particles, since this integral is a product of the integrals over the

various ri , and if È is a wavefunction for only N À 1 particles then it has zero

component in the space de¬ned by rN . (Note that we do not show this result

by taking an integral of the form (2.2.1) when the È have di¬erent numbers

of particles present, say N and N À 1. Rather we must convert the È for

N À 1 particles to a function in N-particle space by multiplying by some

function of rN . This function is of course zero, since there is no Nth particle!)

We denote this orthogonality by writing

hfn gjfn 0 gi ¼ 0

for the cases where the sets fn g and fn 0 g are not identical. For normalized

wavefunctions

hfn gjfn gi ¼ 1:

The many-particle wavefunctions that we describe with the notation jfn gi

are a generalization of the single-particle wavefunctions ji of Section 2.1.

The jfn gi form a complete set not just in one-particle space, as did the ji,

but also in the space known as Fock space, which can contain any number of

particles. Corresponding to Eq. (2.1.6), which only referred to one-particle

space, will be the relation

X

jfn gihfn gj 1:

fn g

34 Second quantization and the electron gas

The sum here is over all possibilities for the set of numbers fn g. If V is an

operator containing the coordinates of any number of particles we can write

the identity

X

V jfn gihfn gjVjfn 0 gihfn 0 gj: °2:2:2Þ

fn g;fn 0 g

If we further abbreviate hfn gjVjfn 0 gi by V 0 this becomes

X

V V 0 jfn gihfn 0 gj:

fn g;fn 0 g

Our next task is to interpret the many-particle operators jfn gihfn 0 gj in terms

of the annihilation and creation operators that were introduced at the end of

Section 2.1.

2.3 Second quantization for fermions

The notation jfn gihfn 0 gj that we adopted in the last section was an abbre-

viation for

0 0

jn1 ; n2 ; . . .ihn1 ; n2 ; . . . j: °2:3:1Þ

In the case where our identical particles are fermions the individual occupa-

tion numbers, ni , can only take on the values 0 and 1. Because each wave-

function jn1 ; n2 ; . . .i symbolizes a determinant of the wavefunctions ui for

which ni ¼ 1, we must be sure that we always take the various ui in the

same order. If we took them in a di¬erent order we would be doing the

equivalent of interchanging some columns of the determinant de¬ning È,

and we might end up with ÀÈ instead.

The simplest operator of the form (2.3.1) is one in which just one of the ni0

is di¬erent from ni . That is, we consider

jn1 ; n2 ; . . . ; np ¼ 0; . . .ihn1 ; n2 ; . . . ; np ¼ 1; . . . j: °2:3:2Þ

This clearly has something in common with the annihilation operator of

Section 2.1, since it acts only on a wavefunction that has the pth one-particle

state occupied, and gives a wavefunction with the pth state empty. If we

want an operator that acts on any wavefunction for which np ¼ 1, we should

have to sum expression (2.3.2) over all the possibilities for the other ni .

35

2.3 Second quantization for fermions

This would give

X

jn1 ; . . . ; np ¼ 0; . . .ihn1 ; . . . ; np ¼ 1; . . . j:

fni g°i6¼pÞ

Finally, we need to keep track of the sign of the new wavefunction that this

operator gives, as mentioned above. This can be achieved by de¬ning a

number

X

pÀ1

Np ¼ nj

j¼1

and multiplying the operator by °À1ÞNp . We de¬ne the result of this as the

annihilation operator for the fermion system.

X

cp ¼ °À1ÞNp j . . . ; ni ; . . . ; np ¼ 0; . . .ih. . . ; ni ; . . . ; np ¼ 1; . . . j: °2:3:3Þ

fni g°i6¼pÞ

Now consider the e¬ect of cp upon a wavefunction in which the pth state is

empty. Since

h. . . np ¼ 1; . . . j . . . np ¼ 0; . . .i ¼ 0

we have

cp j . . . np ¼ 0; . . .i ¼ 0:

If the pth state is occupied, however, there will be one term in the summation

that will not be orthogonal to the wavefunction, and

cp jn1 ; n2 ; . . . ; np ¼ 1; . . .i ¼ °À1ÞNp jn1 ; n2 ; . . . ; np ¼ 0; . . .i:

To operate twice with cp would be to try and destroy two particles from the

same state, and so

c2 ¼ 0;

p

as may also be seen from the de¬nition of cp .

The creation operator is the operator conjugate to cp . It is de¬ned by

X

y

cp ¼ °À1ÞNp j . . . np ¼ 1; . . .ih. . . np ¼ 0; . . . j;

fni g°i6¼pÞ

36 Second quantization and the electron gas

and has the e¬ect of introducing a particle into the formerly empty pth state.

cy j . . . np ¼ 0; . . .i ¼ °À1ÞNP j . . . np ¼ 1; . . .i

p

cy j . . . np ¼ 1; . . .i ¼ 0:

p

Any more complicated operator of the form jfn gihfn 0 gj can be expressed

in terms of the various cp and cy , for we can always write

p

jfn gihfn 0 gj ¼ jfn gihn 00 gjfn 00 gihfn 000 gj Á Á Áihfn 0 gj;

choosing jfn 00 gi to di¬er from jfn gi in only one occupation number, and so

on. Thus we could write

X

j . . . np ¼ 0; nq ¼ 0; . . .ih. . . np ¼ 1; nq ¼ 1; . . . j

fni g°i6¼p;qÞ

X

¼ j . . . np ¼ 0; nq ¼ 0ih. . . np ¼ 0; nq ¼ 1; . . . j

fni g°i6¼p;qÞ

j . . . np ¼ 0; nq ¼ 1; . . .ih. . . np ¼ 1; nq ¼ 1; . . . j

¼ °À1ÞNq cq °À1ÞNp cp : °2:3:4Þ

Alternatively we might have inserted a term

h. . . np ¼ 1; nq ¼ 0; . . . j . . . np ¼ 1; nq ¼ 0 . . .i

into the operator. Then we would have found it to be equal to

°À1ÞNp cp °À1ÞNq cq : °2:3:5Þ

Now cp destroys the particle in the pth state, and so the value of Nq depends

on whether we evaluate it before or after operating with cp if we assume

q > p. We then ¬nd that

°À1ÞNq cp ¼ cp °À1ÞNq þ1 : °2:3:6Þ

Np , on the other hand, does not depend on nq . We thus ¬nd, combining Eqs.

(2.3.4), (2.3.5), and (2.3.6),

cp cq þ cq cp ¼ 0 °p 6¼ qÞ: °2:3:7Þ

37

2.3 Second quantization for fermions

We say that cp and cq anticommute, and abbreviate Eq. (2.3.7) by writing

fcp ; cq g ¼ 0 °p 6¼ qÞ: °2:3:8Þ

By similar arguments it can also be shown that

fcy ; cy g ¼ fcy ; cq g ¼ 0 °p 6¼ qÞ: °2:3:9Þ

pq p

The combination cy cp , in which cp ¬rst operates followed by cy , is particu-

p p

larly important. It is easy to see that it has eigenvalue zero when it operates

on a state for which np is zero, and has eigenvalue unity when it operates on a

state that has np ¼ 1. We can consequently identify the operator cy cp as the

p

number operator,

c y c p ¼ np :

p

Now if we ¬rst operate with cy upon a state with np ¼ 0, we obtain the state

p

with np ¼ 1. Thus when the combination cp cy operates upon a wavefunction

p

in which the pth state is empty it gives unity. Similarly we know that

cp cy j . . . np ¼ 1; . . .i ¼ 0

p

since cy cannot create another particle in an already occupied state. We are

p

thus led to the conclusion that

cp cy ¼ 1 À np ;

p

from which

cy cp þ cp cy ¼ 1:

p p

Thus in the notation of Eq. (2.3.8) we have

fcy ; cp g ¼ 1:

p

In summary the commutation relations for cp are

fcy ; cy g ¼ fcp ; cq g ¼ 0

pq

fcp ; cy g ¼ pq : °2:3:10Þ

q

38 Second quantization and the electron gas

We are now in a position to use Eq. (2.2.2) to write any operator in terms

of annihilation and creation operators. Let us ¬rst consider a single-particle

operator, such as an ordinary potential, V°rÞ. This will enter the many-body

Hamiltonian in the form

X

N

V°ri Þ

i¼1

since it acts equally on all the particles. We then ¬nd that the only matrix

elements, given by

°

0 0

hfn gjVjfn 0 gi ¼ È*°1 . . . N ÞV°ri ÞÈ°1 . . . N Þ dr1 Á Á Á drN ; °2:3:11Þ

that do not vanish are those in which not more than one of the i0 is di¬erent

from i . Then the integral reduces to

°

u*°rÞV°rÞui 0 °rÞ dr:

i

In the occupation number representation we should write this as

Vii 0 hn1 ; . . . ; ni ¼ 1; ni0 ¼ 0; . . . jVjn1 ; . . . ; ni ¼ 0; ni0 ¼ 1; . . .i:

Then from Eq. (2.2.2)

X

V Vii 0 j . . . ni ¼ 1; ni0 ¼ 0; . . .ih. . . ni ¼ 0; ni0 ¼ 1; . . . j

i ;i0

fnj g° j6¼i ;i0 Þ

X y

¼ Vii 0 ci ci0 :

i ;i0

We could do the same with the kinetic energy part, T, of the Hamiltonian,

and write the Hamiltonian as a sum of T and V. It is usually most convenient

to choose a set of functions u such that T only has diagonal matrix elements.

The plane wave representation of Section 2.1 satis¬es this criterion, so that if

we choose

u °rÞ ¼ À1=2 eik Á r °2:3:12Þ

39

2.3 Second quantization for fermions

as the states out of which to build our determinants È, we shall have

°

À02 2 ik 0 Á r 02 k2

Tkk 0 ¼ À1 eÀik Á r re dr ¼ 0;

2m kk

2m

and

°

0

À1

ei°k ÀkÞ Á r V°rÞ dr:

Vkk 0 ¼

The Hamiltonian then becomes

X X

y

Vkk 0 cy ck 0

H¼ E k ck ck þ k

k;k 0

k

where we have written E k for Tkk . Note that the annihilation and creation

operators always appear in pairs; a potential cannot remove a particle from a

state without putting it back in some other state.

As long as there are no interactions between the particles “ that is, as long

as the Hamiltonian can be split up into a sum of parts each of which refers to

only one particle “ there is little pro¬t in rephrasing the problem in this

notation, which is, for irrelevant reasons, known as second quantization.

When, however, we introduce interactions between particles this formalism

provides the only workable approach. Consider, for example, an interaction

between particles that is expressible as a simple potential. That is, we add to

the single-particle Hamiltonian terms of the form

X

V ¼2 V°ri À rj Þ; i 6¼ j

1

i; j

1

(the factor of prevents us from counting interactions twice). Then

2

hfn gjVjfn 0 gi

X° 0 0

¼ 2 È*°1 Á Á Á N ÞV°ri À rj ÞÈ°1 Á Á Á N Þ dr1 Á Á Á drN :

1

i; j

As in the case of Eq. (2.3.11), this integral may be simpli¬ed. The determi-

nants are sums of products of the functions u , and so the integral is a sum of

terms of the form

°

u*°r1 Þu*°r2 Þ Á Á Á u*°rN ÞV°ri À rj Þua 0 °r1 Þ Á Á Á uz 0 °rN Þ dr1 Á Á Á drN :

a b z

40 Second quantization and the electron gas

This can be separated into a product of integrals over one coordinate,

° ° °

u*°r1 Þua 0 °r1 Þ dr1 u*°r2 Þub 0 °r2 Þ dr2 Á Á Á ; °2:3:13Þ

a b

with the exception of the integration over dri and drj , which cannot be

separated, and gives a term of the form

°

V

¼ u* °ri Þu* °rj ÞV°ri À rj Þu

°rj Þu °ri Þ dri drj : °2:3:14Þ

Because the u°rÞ are orthogonal, the integrals (2.3.13) vanish unless

a ¼ a 0 ; b ¼ b 0 , etc. We are thus left with the fact that V can only alter the

occupation of the states ; ;

and . Thus V may be expressed as

X

V

cy cy c

c :

V °2:3:15Þ

1

2

;;

;

It is not obvious that we have the correct numerical factor and the correct

order of the operators c

and c in this expression. Because these operators

anticommute, incorrect ordering would describe the interaction °ÀVÞ. We

can check the validity of the expression by considering the simplest case,

where V ¼ 1. Then the contribution of V to the Hamiltonian is

X

1 ¼ 1 N°N À 1Þ; °2:3:16Þ

1

2 2

i6¼j

with N the total number of particles present. From (2.3.14) we have

V

¼

and so from Eq. (2.3.15)

X

cy cy c c

V ¼1

2

;

X

cy °c cy À Þc

¼1

2

;

¼ 1 °N 2 À NÞ

2

in agreement with Eq. (2.3.16).

41

2.3 Second quantization for fermions

The matrix elements of V take on a particularly simple form when the u°rÞ

are the plane waves de¬ned in (2.3.12). Then

°

000

ÀkÞ Á r1 i°k 00 Àk 0 Þ Á r2

Vk;k 0 ;k 00 ;k 000 ¼ À2 ei°k V°r1 À r2 Þ dr1 dr2

e

°

000

Àkþk 00 Àk 0 Þ Á °r1 þr2 Þ 000

ÀkÀk 00 þk 0 Þ Á °r1 Àr2 Þ

¼ À2 e2i°k

1 1

e2i°k

‚ V°r1 À r2 Þ dr1 dr2 :

If we change to relative coordinates by writing r ¼ r1 À r2 and R ¼ 1 °r1 þ r2 Þ

2

we have

° °

i°k 000 Àkþk 00 Àk 0 Þ Á R 000

ÀkÀk 00 þk 0 Þ Á r

À2 1

Vk;k 0 ;k 00 ;k 000 ¼ dR e2i°k

e V°rÞ dr:

The integral over dR vanishes unless k 000 À k þ k 00 À k 0 ¼ 0, in which case it

gives , so that

°

0 00

À1

ei°k Àk ÞÁr

Vk;k 0 ;k 00 ;k 000 ¼ kþk 0 ;k 000 þk 00 V°rÞ dr:

The -function is no more than an expression of the conservation of momen-

tum in a scattering process, since for a plane wave p ¼ 0k. The integral over

dr is the Fourier transform of the interparticle potential; we shall call it

Vk 0 Àk 00 . Then

X

Vk 0 Àk 00 cy cy 0 ck 00 ckþk 0 Àk 00 :

V 1 kk

2

k;k 0 ;k 00

We can make it clearer that momentum is being transferred by de¬ning

k 0 À k 00 ¼ q and renaming the other variables. This gives

X

Vq cy cy 0 þq ck 0 ck :

V 1

kÀq k

2

k;k 0 ;q

It is often useful to interpret this product of operators pictorially. Particles

in the states k and k 0 are destroyed, while particles in the states k À q and

k 0 þ q are created. This can be seen as a scattering of one particle by the

other “ a process in which an amount of momentum equal to 0q is transferred

(Fig. 2.3.1).

42 Second quantization and the electron gas

Figure 2.3.1. This diagram shows electrons in states k and k 0 being scattered into

y y

states k À q and k 0 þ q, and represents the product of operators ckÀq ck 0 þq ck 0 ck .

2.4 The electron gas and the Hartree“Fock approximation

Our principal motivation in studying the theory of systems of interacting

fermions is the hope that we might in this way better understand the

behavior of the conduction electrons in a metal. Accordingly the ¬rst

system to which we apply this formalism is that of unit volume of a gas

of N spinless electrons, interacting by means of the Coulomb electrostatic

repulsion. Then

e2

V°ri À rj Þ ¼ ;

jri À rj j

and

°

e2

À1 iq Á r

Vq ¼ e dr

jrj

° °

2e2 1

d eiqr cos r sin

¼ dr

0 0

°

4e2 1

¼ sin qr dr:

q 0

This integral does not converge, owing to the long range of the Coulomb

potential, and so one uses the trick of supposing that the potential does not

vary merely as rÀ1 , but as rÀ1 eÀr and then takes the limit as tends to zero.

43

2.4 The electron gas and the Hartree“Fock approximation

One ¬nds

°1

4e2

eÀy=q sin y dy

Vq ¼ lim

q2 !0 0

4e2

¼ ;

q2

provided q 6¼ 0. The divergence of Vq for q ¼ 0 is a di¬culty that we might

have anticipated had we looked more closely at the physics of the situation

we are considering. Because the electrons are described by traveling waves the

box may be considered as a conductor carrying a charge Ne, and it is an

elementary result of electrostatics that charge always resides at the outer

surface of a conductor.

It is now clear that our model of a metal as a mere gas of electrons was too

simpli¬ed to be useful. We must take into account the presence of the posi-

tively charged ions that maintain the overall electrical neutrality of the metal.

This, however, makes the problem a very di¬cult one indeed. Even for a

single electron it is not trivial to solve the Schrodinger equation for a periodic

¨

lattice potential, as we shall see in Chapter 4, and so it is necessary to keep as

simple a model as possible. This is achieved by replacing the lattice of posi-

tively charged ions by a ¬xed uniform distribution of positive charge, and

investigating the interaction of the electrons in the presence of this back-

ground charge. This simpli¬ed model of a metal is sometimes known as

jellium. The positive charge background adds to the Hamiltonian an extra

one-particle potential term V þ which, as we saw in the previous section, can

be written

X

þ

V þcy ck :

V q kÀq

k;q

þ

As this charge density is uniform, the Fourier transform Vq of the potential

due to it vanishes unless q ¼ 0. Thus

X

Vþ ¼ þ þ

V0 nk ¼ NV0

k

and no scattering is caused by this term, which is just a constant. Now if we

look back to our transcription of the electron interaction potential in second-

quantized form, we see that the troublesome coe¬cient V0 also occurs in

44 Second quantization and the electron gas

terms that cause no scattering. The part of H containing V0 was

X X

yy

V0 cy °ck cy 0 À kk 0 Þck 0

0c 0c ¼

1 1

V0 c k c k k k 2 k k

2

k;k 0 k;k 0

X

¼ V0 °nk nk 0 À nk kk 0 Þ

1

2

0

k;k

¼ 1 N°N À 1ÞV0 ;

2

which is again a constant. It is then reasonable to suppose that we can choose

the density of the positive charge background in such a way that these terms

cancel. We write

þ þ

N°N À 1ÞV0 þ NV0 þ W0 ¼ 0;

1

2

þ

where the energy W0 of interaction of the positive charge with itself has been

included in the sum of divergent terms that must cancel. The details of this

cancellation are left as an exercise, and it su¬ces for us to know that the

divergence can be removed. We shall assume that this has been done in what

follows, and always omit the term for q ¼ 0. The Hamiltonian then becomes

X X 2e2 y y

E k cy ck

H¼ þ c c 0 c 0c ; °2:4:1Þ

q2 kÀq k þq k k

k

k;k 0 ;q

k

where once again we have written E k for 02 k2 =2m. We are interested in

¬nding the eigenfunctions of this Hamiltonian and the corresponding eigen-

values. Such a task proves to be immensely di¬cult, as many of the techni-

ques that are used for single-particle problems fail in this instance. We shall

not go too deeply into the complicated procedures that can be developed to

get around these di¬culties, but instead shall just examine some simple

approximation methods.

A zero-order solution can be arrived at by neglecting the interaction term

altogether. Then

X X

y

H0 ¼ E k c k ck ¼ E k nk :

k k

We can easily construct eigenfunctions of the operator nk . First let us start

with the state j0i in which there are no particles at all. This is just the vacuum

state de¬ned in Section 2.1. For all k

nk j0i ¼ 0:

45

2.4 The electron gas and the Hartree“Fock approximation

We can next create one particle in the state k 0 by operating upon the vacuum

state with cy 0 ,

k

jk 0 i ¼ cy 0 j0i:

k

Then

nk jk 0 i ¼ cy ck cy 0 j0i

k k

¼ cy °kk 0 À cy 0 ck Þj0i

k k

from the commutation relations. But we know that ck operating upon the

vacuum state gives zero, since there is no particle there to destroy, and so

nk jk 0 i ¼ kk 0 jk 0 i:

We can go on to build up a wavefunction jÈi containing N particles by

repeatedly operating upon the vacuum state with di¬erent cy 0 , so that

k

Y

N

cy i

jÈi ¼ j0i:

k

i¼1

We can similarly show that

X

N

nk jÈi ¼ kki jÈi:

i¼1

Then

X

H0 jÈi ¼ E k nk jÈi

k

X

N

¼ E ki jÈi:

i¼1

P

Thus jÈi is an eigenfunction of the Hamiltonian with eigenvalue i E ki . The

solution of this kind having the lowest energy is clearly that in which the

wavenumbers ki represent the N single-particle states for which the indivi-

dual energies E ki are the lowest. Since

02 k2

i

E ki ¼ ;

2m

46 Second quantization and the electron gas

this is no more than our picture of a Fermi surface in momentum space. If

there are N states for which jkj < kF , then all such states are ¬lled to give the

ground state of a gas of noninteracting fermions. This picture is sometimes

known as a Sommerfeld gas (Fig. 2.4.1).

Now that we have found the eigenfunctions of H0 we can calculate an

approximation to the energy of the interacting system. The exact energy

we know to be hÉjHjÉi, where É is the exact wavefunction. If we assume

É and È to be not too dissimilar we can calculate an approximate energy,

E HF , by forming hÈjHjÈi. Using Eq. (2.4.1) we ¬nd

E HF ¼ hÈjH0 þ VjÈi

X 2e2 y y

¼ E 0 þ hÈj c c 0 c 0 c jÈi;

q2 kÀq k þq k k

k;k 0 ;q

where we call the energy of the noninteracting system E 0 .

Now the e¬ect of the potential is to take particles out of the states k and k 0

and put them into the states k À q and k 0 þ q. If these states are di¬erent

from the original ones, then the wavefunction formed in this way will be

orthogonal to È, and the matrix element will be zero. This means that the

only terms that do not vanish will be those that do not change the occupation

numbers of jÈi. Then either

k0 ¼ k0 þ q and k ¼ k À q

Figure 2.4.1. The Fermi surface of the Sommerfeld gas separates k-states for which

nk ¼ 1 from those for which nk ¼ 0.

47

2.4 The electron gas and the Hartree“Fock approximation

or else

k0 ¼ k À q and k ¼ k 0 þ q:

The ¬rst possibility implies that q ¼ 0, and we have agreed to omit this term,

which does not correspond to any scattering at all (Fig. 2.4.2(a)). This leaves

us with the second possibility, Fig. 2.4.2(b), known as exchange scattering, in

which the particle that was in the state k is scattered into the state k 0 , and vice

versa. The correction this gives to E 0 is known as the exchange energy. We

¬nd

X 2e2 yy

E HF ¼ E0 þ hÈj ck 0 ck ck 0 ck jÈi

jk À k 0 j2

k;k 0

X 2e2

¼ E0 þ hÈj °Ànk 0 nk ÞjÈi: °2:4:2Þ

jk À k 0 j2

0

k;k

This we can write as

X 2e2

E HF ¼ E 0 À

jk À k 0 j2

k;k 0 occupied

Figure 2.4.2. In the Hartree“Fock approximation only direct scattering (a) and

exchange scattering (b) between identical states can occur.

48 Second quantization and the electron gas

where now the summation is over states k and k 0 that are both occupied. This

method of treating the electron gas is a special case of an approach known as

the Hartree“Fock approximation, and is the simplest way in which we can

take the interactions into account. Even this approach can become very

complicated, however, if in addition to the interactions there is some addi-

tional one-particle potential applied to the system.

The interesting point to note is that in this approximation the energy is

reduced below that of the Sommerfeld gas, E 0 . It is not paradoxical that the

repulsive interaction should decrease the energy of the system, for we must

not forget that we have also e¬ectively added a uniform background of

positive charge when we eliminated the interaction Vq¼0 . The reduction in

energy comes from the fact that the particles are kept apart by the antisym-

metrization of the wavefunction, and so are acted upon more by the positive

charge background than by their neighbors.

So far we have considered only a system of spinless particles. In fact we

know that the electron has a spin angular momentum of 1 0, which means

2

that a single electron can occupy a state k in two ways, either with spin up or

with spin down. We denote this by naming the states k" and k#. This

modi¬es the energy we ¬nd for the Sommerfeld gas, since now the Fermi

wavenumber kF is determined by the condition that there be only 1 N values

2

of k for which jkj < kF . The inclusion of the spin of the electron in our model

must necessarily introduce a number of other complications, for the spin will

be accompanied by a magnetic moment, and the electrons will interact by

virtue of their magnetic ¬elds. However, we ignore these e¬ects, retaining

only the Coulomb interaction, and ask how the Hartree“Fock energy is

modi¬ed by the inclusion of spin.

The ¬rst point we note is that since k" and k# denote separate states all

anticommutators for states of opposite spin vanish. Thus for example

fck" ; cy g ¼ 0:

k#

Next we see that since the Coulomb interaction does not contain the spin

coordinates it cannot cause an exchange of particles with opposite spin. The

integration over rj , for example, in Eq. (2.3.14) demands that the spin of state

u be the same as that of u

. The Hamiltonian thus becomes

X X

E k cy cks Vq cy cy 0 þq;s 0 ck 0 ;s 0 ck;s

H¼ þ 1

ks kÀq;s k

2

k;k 0 ;q;s;s 0

k;s

and only terms of the form nk 0 " nk" and nk 0 # nk# will come into the expression

49

2.4 The electron gas and the Hartree“Fock approximation

for the exchange energy. We again ¬nd

X 2e2

E HF ¼ E 0 À

jk À k 0 j2

but now the summation proceeds only over states k and k 0 that are occupied

and that have the same spin. This reduces the amount by which the Hartree“

Fock energy E HF is less than E 0 . One may express this by saying that now the

antisymmetrization only keeps apart the electrons having parallel spins. We

can guess that in the exact solution to the problem all the electrons will try to

keep apart from one another regardless of their spins; they will reduce the

potential energy of the system by doing so. We say that their motions will be

correlated, and that the di¬erence between the exact energy and E HF is the

correlation energy.

One might now be tempted to ask how the energy of a single electron is

altered by the interaction. A little re¬‚ection shows this question to be mean-

ingless, since the energy of interaction of two particles cannot be associated

with either one, and it makes no sense to share out the interaction energy

between the particles in some arbitrary way. One can only talk about the

total energy of the system. However, as long as we are within the Hartree“

Fock approximation we can ask how the total energy changes when we

change the approximate wavefunction È. In particular we might ask how

E HF changes when we remove an electron. If we consider hÈjnp jÈi as the

number hnp i (as opposed to the operator np ¼ cy cp ), then to take away an

p

electron in the pth state is to reduce hnp i by 1. The energy change is conse-

quently @E=@hnp i. From Eq. (2.4.2) this gives for the example where we

neglected spin

( )

X X

@E HF @ 2e2

¼ E hn i À hnk nk 0 i

02

@hnp i @hnp i k k k 0 jk À k j

k;k

X 4e2

¼ Ep À hnk i: °2:4:3Þ

jk À pj2

k

We might then put this electron back into another state p 0 . The work

required to take the electron from p to p 0 would then be

@E HF @E HF

ÁW ¼ À :

@hnp 0 i @hnp i

If the states were separated by a very small momentum di¬erence, so that

50 Second quantization and the electron gas

p 0 ¼ p þ d we should have

@ @E HF

ÁW ¼ d Á : °2:4:4Þ

@p @hnp i

Because of the spherical symmetry in momentum space of the interactions

and of the function hnp i, we know that the derivative with respect to p (by

which we mean the gradient in p-space) must be in the direction of p, so that

only the component, p , of d that is parallel to p enters Eq. (2.4.4). In the limit

that p ! 0 we have

ÁW @ @E HF

dW

! ¼ ;

p @p @hnp i

dp

which from (2.4.3) is

@ X 4e2

dW @E p

¼ À hn i: °2:4:5Þ

@p @p k jk À pj2 k

dp

This is of interest because it tells us the energy of the lowest-lying group of

excitations of the system within the Hartree“Fock approximation. When we

took an electron below the Fermi surface of the noninteracting system and

put it in a higher energy state, we said that we had created a particle“hole

pair, which was an elementary excitation. In the limiting case that we had

taken an electron from a vanishingly small distance =2 below the Fermi

surface, and put it in a state =2 above the surface, the excitation would

have had energy ‚ °@E=@pÞp¼pF , corresponding to the ¬rst term in (2.4.5).

In the Hartree“Fock approximation the elementary excitation has the energy

‚ °dW=dpÞp¼pF , which includes the second term in (2.4.5). However, inspec-

tion of (2.4.5) shows that as p ! pF ; dW=dp becomes in¬nite. This has con-

sequences that are at variance with the experimentally determined properties

of the electron gas as found in metals, and is a ¬rst indication that the

Hartree“Fock approximation may be inadequate where Coulomb forces

are involved.

2.5 Perturbation theory

The only system whose wavefunctions we have studied has been the gas of

noninteracting fermions described by the Hamiltonian

X

H0 ¼ E k nk :

k

51

2.5 Perturbation theory

In the Hartree“Fock approximation we merely took the wavefunctions jÈi

of the noninteracting system, and worked out the expectation value in the

state jÈi of a Hamiltonian containing interactions. While we could look

upon this as a variational approach “ we guess that the wavefunction

might be like jÈi and we work out the energy it would give “ we could

also consider it as the ¬rst term in a perturbation expansion. Let us now

quickly look at the methods of perturbation theory, and see how they apply

to many-body systems.

We start by stating a solution of a simple problem,

H0 ji ¼ E 0 ji; °2:5:1Þ

and consider the solutions of

°H0 þ VÞj i ¼ Ej i: °2:5:2Þ

From the perturbed Eq. (2.5.2) we have that

hj°H0 þ VÞj i ¼ hjEj i

and so if we normalize j i with the condition

hj i ¼ 1;

then because hjH0 ¼ hjE 0 we have

E À E 0 ¼ hjVj i: °2:5:3Þ

To proceed further we need to de¬ne what we mean by a function of an

operator. From (2.5.1), for instance, we can ¬nd by operating with H0 on

both sides that

H0 H0 ji ¼ H2 ji ¼ E 2 ji;

0 0

and in general that

°H0 Þn ji ¼ °E 0 Þn ji:

Thus if we interpret f °H0 Þ as a power series expansion in H0 we should have

f °H0 Þji ¼ f °E 0 Þji: °2:5:4Þ

52 Second quantization and the electron gas

We generalize this de¬nition to include functions like H1=2 that have no

power series expansions. In particular, the operator function °z À H0 ÞÀ1 is

the operator whose eigenfunctions are ji and whose corresponding eigen-

values are °z À E 0 ÞÀ1 , provided z 6¼ E 0 .

There are various ways in which we can write the solution of (2.5.2) using

the expression for the perturbed energy (2.5.3). We could write

j i ¼ ji þ °E 0 À H0 ÞÀ1 °1 À j ihjÞVj i; °2:5:5Þ

which is the starting point of Rayleigh“Schro¨dinger perturbation theory.

Another possibility is

j i ¼ ji þ °E À H0 ÞÀ1 °1 À jihjÞVj i; °2:5:6Þ

which is the starting point for Brillouin“Wigner perturbation theory. These

equations can be veri¬ed by operating upon them with °E 0 À H0 Þ and

°E À H0 Þ, respectively. Because the right-hand sides still contain the unknown

j i, one iterates these by using the equation itself to substitute for j i. Thus if

we write

°E À H0 ÞÀ1 °1 À jihjÞ ¼ G0

then the Brillouin“Wigner formula becomes

j i ¼ ji þ G0 Vji þ G0 VG0 Vji þ Á Á Á :

The Rayleigh“Schrodinger expansion, for example, allows us to write the

¨

energy, to second order in V, as

E ¼ E 0 þ hjVji þ hjV°E 0 À H0 ÞÀ1 °1 À jihjÞVji:

This sort of expression is evaluated by remembering that

X

jp 0 ihp 0 j 1

p0

when the jp 0 i form a complete set. Then for the energy of the state j pi

corresponding to the unperturbed state jp i of energy E p one ¬nds

X

hp jVjp 0 i°E p À E p 0 ÞÀ1 hp 0 jVjp i:

E ¼ E p þ hp jVjp i þ °2:5:7Þ

p 0 °p 0 6¼pÞ

53

2.5 Perturbation theory

Again, this expression is valid only to second order in V. The term p 0 ¼ p is

excluded from the summation by the projection operator, 1 À jp ihp j. This

removes one term for which the energy denominator would vanish.

We can interpret the ¬nal term in expression (2.5.7) in the following way.

First the interaction V causes the system to make a transition from the state

jp i to the state jp 0 i. This process does not conserve energy, and so the

system can only remain in the intermediate state a time of the order of

0=°E p À E p 0 Þ, which is all that the Uncertainty Principle allows. It must

then make a transition back to the original state, again by means of the

perturbation V. The higher-order expansions become very complicated, espe-

cially in the Rayleigh“Schrodinger formula, where j i occurs twice. In fact, it

¨

turns out that the Brillouin“Wigner expression is less well suited to many-

body systems than the Rayleigh“Schrodinger one from the point of view of

¨

convergence. Although there are a number of elegant methods that mitigate

the awkwardness of keeping track of the terms in the Rayleigh“Schrodinger

¨

expansion the approach remains a di¬cult one, and we shall for the most

part leave this approach to the more specialized texts.

In the case of the electron gas in particular, one runs into di¬culty even in

the second-order expansion for the energy, as this turns out to diverge. We

noticed the danger signals ¬‚ying in the ¬rst-order term, which was the

Hartree“Fock approximation. There we found in Eq. (2.4.5) that although

the energy W required to add another electron to the system was ¬nite, its

derivative, dW=dp, was in¬nite. This was due to the fact that for the

Coulomb interaction Vq / qÀ2 , and re¬‚ects the long-range nature of the

Coulomb force. If we try to take this to second order in perturbation theory

by using (2.5.7) we ¬nd that the total energy itself diverges logarithmically.

To see this we note from Eq. (2.5.7) that the contribution to the energy that is

of second order in V takes on the form

X

°2Þ

hÈ jVjÈ i°E À E ÞÀ1 hÈ jVjÈ i;

E ¼

°6¼Þ

where È and È are the initial and intermediate wavefunctions describing

the N independent electrons. Now

X

Vq cy cy 0 þq ck 0 ck

V¼ 1

kÀq k

2

k;k 0 ;q

and so È di¬ers from È in having electrons removed from states k and k 0

and put back in states k 0 þ q and k À q. Now the second time that V appears

54 Second quantization and the electron gas

in E °2Þ it must transform È back into È . If we now write

X

Vq 0 cy 0 cy 0 þq 0 cp 0 cp

V¼1 pÀq p

2

p;p 0 ;q 0

then there are only two possible ways in which this can happen. These are

illustrated in Fig. 2.5.1. We must either have p ¼ k 0 þ q and p 0 ¼ k À q as in

Fig. 2.5.1(a) or else p ¼ k À q and p 0 ¼ k 0 þ q as in Fig. 2.5.1(b). In the

former case the operators cp 0 and cp annihilate the electrons that were scat-

tered into states k À q and k 0 þ q, respectively, and electrons are created in

states k À q þ q 0 and k 0 þ q À q 0 . In order for the net result of all this to be

the original state È we must either have q ¼ q 0 or else k 0 þ q À q 0 ¼ k. We

call the ¬rst possibility a ˜˜direct™™ term and the second an ˜˜exchange™™ term.

We shall investigate only the contribution to E °2Þ of the direct term, as this

turns out to be the more important one. If we follow up the alternative

Figure 2.5.1. The two possible second-order scattering processes.

55

2.5 Perturbation theory

possibility for the choice of p and p 0 , we ¬nd that this also leads to a direct

term and an exchange scattering term of the same magnitude as before. The

total contribution of direct terms to E °2Þ is thus

X 1 2

E °2Þ ¼ 2 hÈ jcy 0 cy ckÀq ck 0 þq

Vq kk

direct

2

k;k 0 ;q

1

cy cy 0 þq ck 0 ck jÈ i:

‚

E k þ E k 0 À E kÀq À E k 0 þq kÀq k

We may use the fact that E k ¼ 02 k2 =2m to simplify the energy denominator

and then commute the c-operators into pairs that form number operators.

We then have

mX 2 °1 À nkÀq Þ°1 À nk 0 þq Þnk 0 nk

E °2Þ ¼ Vq hÈ j jÈ i; °2:5:8Þ

Àq Á °q þ k 0 À kÞ

direct

202 k;k 0 ;q

and we now take È to be the noninteracting ground state. If we include the

spin of the electron we should have to multiply this result by a factor of 4,

since both states k and k 0 can have spin either up or down.

The di¬culty with expression (2.5.8) lies in the terms for which q is small.

The factor of °1 À nkÀq Þnk then restricts the summation over k to a thin layer

of states of thickness q on one side of the Fermi surface, as indicated in Fig.

2.5.2. The summation over k 0 is similarly restricted to a layer on the opposite

side. The two summations thus contribute a factor of order q2 . The volume

element for the summation over q will be 4q2 dq, and Vq is proportional to

Figure 2.5.2. The product °1 À nkÀq Þnk vanishes everywhere outside the shaded

volume.

56 Second quantization and the electron gas

Ð

qÀ2 . The net result is that E °2Þ contains a factor 0 qÀ1 dq, which diverges

direct

logarithmically.

We might also have guessed that perturbation theory could not be applied

in any straightforward way from our semiclassical approach to the electron

gas. There we found that a form of collective motion “ the plasma oscillation

“ played an important role in the dynamics of the system. It is clear that we

could not arrive at a description of collective motion by just taking a couple

of terms of a perturbation expansion starting with a scheme of independent

particles. As it turns out, rather sophisticated methods have been devised

whereby one can sum an in¬nite number of terms selected from the perturba-

tion expansion, and arrive at a picture of collective behavior. We, however,

shall ¬rst take a simpler approach. We know that plasma oscillations repre-

sent density ¬‚uctuations, and so we shall deliberately search for a solution of

the Schrodinger equation that describes these.

¨

2.6 The density operator

So far we have used the occupation number representation to de¬ne opera-

tors that create particles in various states, u . When these are plane wave

states the probability of ¬nding the particle is constant at all points in space.

We now ask whether it is possible to de¬ne operators that will create or

destroy particles at one particular point in space.

For the plane wave states contained in a cubical box of volume , we

found in Section 2.1 that the allowed values of k were given by

2mx 2my 2mz

k¼ ; ; ;

L L L

where the m were integers. We now ¬rst show the important relation

X

eik Á r ¼ °rÞ °2:6:1Þ

k

where °rÞ is the three-dimensional Dirac delta-function, which is zero for

Ð

r 6¼ 0 and for which f °rÞ °rÞ dr ¼ f °0Þ when r ¼ 0 is within . To see this

we ¬rst substitute for k to ¬nd

1 1

X X 2imx x X 2imy y X

1

2imz z

eik Á r ¼ :

exp exp exp

L L L

¼À1 m ¼À1 m ¼À1

mx

k y z

57

2.6 The density operator

Now

°2M þ 1Þx

sin

X

M

2imx L

¼

exp

x

L

m¼ÀM sin

L

so that

°X °

M

L sin½°2M þ 1Þ

2imx L

f °xÞ dx ¼

exp f d:

sin

L

m¼ÀM

As M ! 1 the term sin½°2M þ 1Þ begins to oscillate so rapidly that only

the region near ¼ 0 gives any contribution to the integral. Then we can

replace sin by and f °L=Þ by f °0Þ so that the integral becomes

°1

sin½°2M þ 1Þ

L

f °0Þ d°MÞ ¼ Lf °0Þ:

M

M¼À1

Thus

° X

eik Á r dr ¼ L3 f °0Þ;

f °rÞ

k

which proves (2.6.1).

y

°rÞ by

This relation suggests that we de¬ne an operator

X

y

°rÞ ¼ À1=2 eÀik Á r cy : °2:6:2Þ

k

k

This operator, not to be confused with a one-particle wavefunction, is known

as a fermion ¬eld operator. Its conjugate is

X

°rÞ ¼ À1=2 eik Á r ck : °2:6:3Þ

k

58 Second quantization and the electron gas

Then

X 0

Á r 0Þ

y

°r 0 Þg ¼ À1 ei°k Á rÀk fck ; cy 0 g

f °rÞ; k

k;k 0

X 0

À1

eik Á °rÀr Þ

¼

k

¼ °r À r 0 Þ: °2:6:4Þ

The fact that the ¬eld operators have these anticommutation relations shows

that they are the operators we are looking for. If °rÞ does indeed annihilate a

particle at r we should expect that

y

°r 0 Þ °rÞj0i ¼ 0

h0j

and

y

°r 0 Þj0i ¼ °r À r 0 Þ;

h0j °rÞ

for °rÞ will always give zero when operating on the vacuum state j0i unless

we ¬rst operate with y °rÞ. This is compatible with (2.6.4).

The operators that we expressed in terms of the cy and ck may equally well

k

y

be expressed in terms of the °rÞ and °rÞ. For instance,

°

y

V°rÞ °rÞV°rÞ °rÞ dr;

and

°

V°r; r 0 Þ y y

°r 0 ÞV°r; r 0 Þ °r 0 Þ °rÞ dr dr 0 ;

°rÞ

as may be veri¬ed by substitution from the de¬nitions (2.6.2) and (2.6.3) and

use of (2.6.1).

In particular, we can use the ¬eld operators to represent the density of

particles, °rÞ, at the point r. The density is de¬ned as the sum over particle

coordinates, ri ,

X

°rÞ ¼ °r À ri Þ;

i

59

2.6 The density operator

and in terms of the ¬eld operators this becomes

°

y

°r 0 Þ °r À r 0 Þ °r 0 Þ dr 0

°rÞ

y

¼ °rÞ °rÞ: °2:6:5Þ

The Fourier transform of the particle density is

°

À1

eÀiq Á r °rÞ dr

q ¼

° X 0

À2 Àiq Á r

ei°k ÀkÞ Á r cy ck 0 dr

¼ e k

k;k 0

X

¼ À1 cy ck 0 k 0 Àk;q

k

k;k 0

X

À1

cy ckþq :

¼ k

k

Because °rÞ ¼ y °rÞ it follows that y ¼ Àq .

q

As an example of the usefulness of the density operator, we show how the

Hamiltonian for the electron gas can be expressed in terms of the number and

density operators. From Eq. (2.4.1) we have