Docsity
Docsity

Prepare for your exams
Prepare for your exams

Study with the several resources on Docsity


Earn points to download
Earn points to download

Earn points by helping other students or get them with a premium plan


Guidelines and tips
Guidelines and tips

Quantum Mechanic Perturbation Theory, Study notes of Quantum Mechanics

Quantum mechanic perturbation theory in explain time independent perturbation theory, degenerate perturbation theory, second quantized form of operators and canonical transfromation.

Typology: Study notes

2021/2022

Uploaded on 03/31/2022

anandamayi
anandamayi 🇺🇸

4.2

(9)

250 documents

1 / 67

Toggle sidebar

This page cannot be seen from the preview

Don't miss anything!

bg1
G
Quantum Mechanical Perturbation Theory
Quantum mechanical perturbation theory is a widely used method in solid-
state physics. Without the details of derivation, we shall list a number of basic
formulas of time-independent (stationary) and time-dependent perturbation
theory below. For simplicity, we shall use the Dirac notation for wavefunctions
and matrix elements.
G.1 Time-Independent Perturbation Theory
Assume that the complete solution (eigenfunctions and eigenvalues) of the
Schrödinger equation
H0ψ(0)
i"=E(0)
iψ(0)
i"(G.1.1)
is known for a system described by a simple Hamiltonian H0. If the system
is subject to a time-independent (stationary) perturbation described by the
Hamiltonian H1 which can be an external perturbation or the interaction
between the components of the system –, the eigenvalues and eigenfunctions
change. The method for determining the new ones depends on whether the
unperturbed energy level in question is degenerate or not.
G.1.1 Nondegenerate Perturbation Theory
We now introduce a fictitious coupling constant λ, whose value will be treated
as a parameter in the calculations and set equal to unity in the final result,
and write the full Hamiltonian H=H0+H1as
H=H0+λH1.(G.1.2)
The parameter λis purely a bookkeeping device to keep track of the relative
order of magnitude of the various terms, since the energy eigenvalues and
eigenfunctions will be sought in the form of an expansion in powers of λ:
pf3
pf4
pf5
pf8
pf9
pfa
pfd
pfe
pff
pf12
pf13
pf14
pf15
pf16
pf17
pf18
pf19
pf1a
pf1b
pf1c
pf1d
pf1e
pf1f
pf20
pf21
pf22
pf23
pf24
pf25
pf26
pf27
pf28
pf29
pf2a
pf2b
pf2c
pf2d
pf2e
pf2f
pf30
pf31
pf32
pf33
pf34
pf35
pf36
pf37
pf38
pf39
pf3a
pf3b
pf3c
pf3d
pf3e
pf3f
pf40
pf41
pf42
pf43

Partial preview of the text

Download Quantum Mechanic Perturbation Theory and more Study notes Quantum Mechanics in PDF only on Docsity!

G

Quantum Mechanical Perturbation Theory

Quantum mechanical perturbation theory is a widely used method in solid-

state physics. Without the details of derivation, we shall list a number of basic

formulas of time-independent (stationary) and time-dependent perturbation

theory below. For simplicity, we shall use the Dirac notation for wavefunctions

and matrix elements.

G.1 Time-Independent Perturbation Theory

Assume that the complete solution (eigenfunctions and eigenvalues) of the

Schrödinger equation

H 0

∣ψ (0) i

= E

(0) i

∣ψ (0) i

(G.1.1)

is known for a system described by a simple Hamiltonian H 0. If the system

is subject to a time-independent (stationary) perturbation described by the

Hamiltonian H 1 – which can be an external perturbation or the interaction

between the components of the system –, the eigenvalues and eigenfunctions

change. The method for determining the new ones depends on whether the

unperturbed energy level in question is degenerate or not.

G.1.1 Nondegenerate Perturbation Theory

We now introduce a fictitious coupling constant λ, whose value will be treated

as a parameter in the calculations and set equal to unity in the final result,

and write the full Hamiltonian H = H 0 + H 1 as

H = H 0 + λH 1. (G.1.2)

The parameter λ is purely a bookkeeping device to keep track of the relative

order of magnitude of the various terms, since the energy eigenvalues and

eigenfunctions will be sought in the form of an expansion in powers of λ:

580 G Quantum Mechanical Perturbation Theory

∣ψ i

∣ψ (0) i

∞ ∑

n=

λ

n

∣ψ (n) i

Ei = E

(0) i

∞ ∑

n=

λ

n E

(n) i

(G.1.3)

The series is convergent if the perturbation is weak, that is, in addition to the

formally introduced parameter λ, the interaction Hamiltonian itself contains

a small parameter, the physical coupling constant.

By substituting this expansion into the Schrödinger equation and collect-

ing the same powers of λ from both sides, we obtain

H 0

∣ψ (0) i

= E

(0) i

∣ψ (0) i

H 0

ψ

(1) i

+ H 1

ψ

(0) i

= E

(0) i

ψ

(1) i

+ E

(1) i

ψ

(0) i

H 0

ψ

(2) i

+ H 1

ψ

(1) i

= E

(0) i

ψ

(2) i

+ E

(1) i

ψ

(1) i

+ E

(2) i

ψ

(0) i

(G.1.4)

and similar equations for higher-order corrections. The corrections to the en-

ergy and wavefunction of any order are related to the lower-order ones by the

recursion formula

(H 0 − E

(0) i )

ψ

(n) i

+ (H 1 − E

(1) i )

ψ

(n−1) i

− E

(2) i

∣ψ (n−2) i

−... − E

(n) i

∣ψ (0) i

(G.1.5)

Multiplying the second equation in (G.1.4) (which comes from the terms

that are linear in λ) by

ψ

(0) i

∣ (^) from the left, the first-order correction to the

energy is

E

(1) i

ψ

(0) i

∣H

1

∣ψ (0) i

. (G.1.6)

To determine the correction to the wavefunction, the same equation is multi-

plied by

ψ

(0) j

(j = i):

E

(0) j

ψ

(0) j

∣ψ (1) i

ψ

(0) j

∣H

1

∣ψ (0) i

〉 = E

(0) i

ψ

(0) j

∣ψ (1) i

. (G.1.7)

Since the eigenfunctions of H 0 make up a complete set, the functions

ψ

(n) i

can be expanded in terms of them:

∣ψ (n) i

j

C

(n) ij

∣ψ (0) j

. (G.1.8)

The coefficients C

(n) ii are not determined by the previous equations: their val-

ues depend on the normalization of the perturbed wavefunction. Substituting

the previous formula into (G.1.7), we have

E

(0) j

C

(n) ij

ψ

(0) j

∣H

1

∣ψ (0) i

〉 = E

(0) i

C

(n) ij

, (G.1.9)

582 G Quantum Mechanical Perturbation Theory

which projects onto the subspace that is orthogonal to the state

∣ψ (0) i

. The

nth-order energy correction can then be written as

E

(n) i =^

ψ

(0) i

H 1

ψ

(n−1) i

, (G.1.16)

where the matrix element is to be taken with the wavefunction

ψ

(n) i

E

(0) i

− H 0

Pi

[

H 1 − E

(1) i

ψ

(n−1) i

−E

(2) i

∣ψ (n−2) i

−... − E

(n−1) i

∣ψ (1) i

]

(G.1.17)

which is in the subspace mentioned above.

Formally simpler expressions can be obtained when the Brillouin–Wigner

perturbation theory is used. The perturbed wavefunction in the Schrödinger

equation ( H 0 + H 1

∣ψ i

= Ei

∣ψ i

(G.1.18)

is then chosen in the form

∣ψ i

= C 0

∣ψ (0) i

∣Δψ i

, (G.1.19)

where

Δψi

is orthogonal to

ψ

(0) i

, and C 0 takes care of the appropriate

normalization. After some algebra, the eigenvalue equation reads

H 0 − Ei

∣Δψ i

+ H 1

∣ψ i

= C 0

Ei − E

(0) i

∣ψ (0) i

. (G.1.20)

By applying the projection operator Pi, and exploiting the relations

Pi

∣ψ (0) i

= 0 , Pi

∣ψ i

∣Δψ i

(G.1.21)

as well as the commutation of Pi and H 0 ,

Ei − H 0

∣Δψ i

= PiH 1

∣ψ i

(G.1.22)

is obtained. Its formal solution is

ψi

= C 0

ψ

(0) i

Pi

Ei − H 0

H 1

ψi

. (G.1.23)

Iteration then yields

∣ψ i

= C 0

∞ ∑

n=

Pi

Ei − H 0

H 1

)n ∣ ∣ψ (0) i

, (G.1.24)

and

ΔEi =

∞ ∑

n=

ψ

(0) i

∣H

1

Pi

Ei − H 0

H 1

)n ∣ ∣ψ (0) i

(G.1.25)

G.1 Time-Independent Perturbation Theory 583

for the energy correction. In this method the energy denominator contains the

perturbed energy Ei rather than the unperturbed one E

(0) i

. To first order in

the interaction,

Ei = E

(0) i

ψ

(0) i

∣H

1

∣ψ (0) i

, (G.1.26)

while to second order,

Ei = E

(0) i +^

ψ

(0) i

H 1

ψ

(0) i

j =i

ψ

(0) i

H 1

ψ

(0) j

ψ

(0) j

H 1

ψ

(0) i

Ei − E

(0) j

(G.1.27)

It is easy to show that by rearranging the energy denominator and expanding

it as

Ei − H 0

E

(0) i − H^0 + ΔEi

E

(0) i − H^0

∞ ∑

n=

−ΔEi

E

(0) i − H^0

)n

, (G.1.28)

the results of the Rayleigh–Schrödinger perturbation theory are recovered.

The formulas of time-dependent perturbation theory can also be used to

determine the ground-state energy and wavefunction of the perturbed sys-

tem, provided the interaction is assumed to be turned on adiabatically. The

appropriate formulas are given in Section G.2.

G.1.2 Degenerate Perturbation Theory

In the previous subsection we studied the shift of nondegenerate energy levels

due to the perturbation. For degenerate levels a slightly different method has

to be used because the formal application of the previous formulas would yield

vanishing energy denominators.

Assuming that the ith energy level of the unperturbed system is p-fold

degenerate – that is, the same energy E

(0) i belongs to each of the states

∣ψ (0) i 1

ψ

(0) i 2

ψ

(0) ip

–, any linear combination of these degenerate eigenstates

is also an eigenstate of H 0 with the same energy. We shall use such linear

combinations to determine the perturbed states. We write the wavefunctions

of the states of the perturbed system that arise from the degenerate states as

ψ

k

cik

ψ

(0) ik

n=i

cn

ψ

(0) n

, (G.1.29)

where the ci k are of order unity, whereas the other coefficients cn that specify

the mixing with the unperturbed eigenstates whose energy is different from

E

(0) i are small, proportional to the perturbation. By substituting this form

into the Schrödinger equation, and multiplying both sides by

ψ

(0) ij

from the

left,

G.2 Time-Dependent Perturbation Theory 585

cij (t) = c

(0) ij (t) +

∞ ∑

r=

λ

r c

(r) ij (t) , (G.2.4)

where, naturally, the zeroth-order term is a constant:

c

(0) ij (t) = δij. (G.2.5)

Substituting this series expansion into the Schrödinger equation, we find

i

∂t

c

(r) ij (t) =

k

e

i(E

(0) j −E

(0) k )t/ℏ

ψ

(0) j

∣H

1 (t)

∣ψ (0) k

c

(r−1) ik (t). (G.2.6)

The explicit formulas for the first two terms obtained through iteration are

c

(1) ij (t) = −

i

∫^ t

t 0

ψ

(0) j

∣H

1 (t 1 )

∣ψ (0) i

e

i(E

(0) j −E

(0) i )t^1 /ℏdt 1 ,^ (G.2.7)

and

c

(2) ij (t) =

i

) 2 ∫t

t 0

dt 1

∫^ t^1

t 0

dt 2

k

ψ

(0) j

∣H

1 (t 1 )

∣ψ (0) k

e

i(E

(0) j −E

(0) k )t^1 /ℏ

×

ψ

(0) k

∣H

1 (t 2 )

∣ψ (0) i

e

i(E (0) k −E (0) i )t 2 /ℏ (G.2.8)

In the interaction picture the time dependence of an arbitrary operator O

is given by

Oˆ(t) = eiH^0 t/ℏOe−iH^0 t/ℏ^. (G.2.9)

Using this form for the Hamiltonian, which may have an intrinsic time de-

pendence as well, the first two coefficients c

(n) ij can be written in terms of the

operators

ˆ H 1 (t) = e

iH 0 t/ℏ H 1 (t)e

−iH 0 t/ℏ (G.2.10)

as

c

(1) ij (t) =^ −^

i

∫^ t

t 0

ψ

(0) j

H 1 (t 1 )

ψ

(0) i

dt 1 (G.2.11)

and

c

(2) ij (t) =

i

) 2 ∫t

t 0

dt 1

t 1 ∫

t 0

dt 2

k

ψ

(0) j

∣ Hˆ

1 (t 1 )

∣ψ (0) k

×

ψ

(0) k

∣ Hˆ

1 (t 2 )

∣ψ (0) i

(G.2.12)

Since the intermediate states

ψ

(0) k

constitute a complete set, the previous

formula simplifies to

586 G Quantum Mechanical Perturbation Theory

c

(2) ij (t) =

i

) 2 ∫t

t 0

dt 1

∫^ t^1

t 0

dt 2

ψ

(0) j

H 1 (t 1 )

H 1 (t 2 )

ψ

(0) i

. (G.2.13)

The same result is obtained when the double integral on the t 1 , t 2 plane is

evaluated in reverse order:

c

(2) ij (t) =

i

2 ∫t

t 0

dt 2

∫^ t

t 2

dt 1

ψ

(0) j

H 1 (t 1 )

H 1 (t 2 )

ψ

(0) i

, (G.2.14)

or by swapping the notation of the two time variables:

c

(2) ij (t) =

i

) 2 ∫t

t 0

dt 1

∫^ t

t 1

dt 2

ψ

(0) j

∣ Hˆ

1 (t 2 )

1 (t 1 )

∣ψ (0) i

. (G.2.15)

Using these two formulas, the coefficient can also be written as

c

(2) ij (t) =

i

∫^ t

t 0

dt 1

∫^ t

t 0

dt 2

ψ

(0) j

T

H 1 (t 1 )

H 1 (t 2 )

ψ

(0) i

, (G.2.16)

where T is the time-ordering operator, which orders the operators in a product

in descending order of their time argument. Its action can be written in terms

of the Heaviside step function as

T

H 1 (t 1 )

H 1 (t 2 )

= θ(t 1 − t 2 )

H 1 (t 1 )

H 1 (t 2 ) + θ(t 2 − t 1 )

H 1 (t 2 )

H 1 (t 1 ).

(G.2.17)

Generalizing this to arbitrary orders, and setting λ = 1,

cij (t) = δji+

∞ ∑

n=

i

)n ∫t

t 0

dt 1

∫^ t^1

t 0

dt 2...

tn− 1 ∫

t 0

dtn

×

ψ

(0) j

∣ Hˆ

1 (t 1 )

1 (t 2 )^...^

1 (tn)

∣ψ (0) i

(G.2.18)

or, in time-ordered form,

cij (t) = δji+

∞ ∑

n=

n!

i

)n ∫t

t 0

dt 1

∫^ t

t 0

dt 2...

∫^ t

t 0

dtn

×

ψ

(0) j

∣T

1 (t 1 )

1 (t 2 )^...^

1 (tn)

∣ψ (0) i

(G.2.19)

The time evolution of the wavefunction between times t 0 and t is therefore

governed by the operator S(t, t 0 ):

∣ ∣ ψ(t)

= S(t, t 0 )

ψ(t 0 )

, (G.2.20)

588 G Quantum Mechanical Perturbation Theory

in the interaction picture. Using the ground state

of energy E 0 of the

unperturbed system, the energy correction due to the perturbation is

ΔE =

∣H

1 S(0,^ −∞)

0

S(0, −∞)

〉 , (G.2.28)

and the wavefunction is

S(0, −∞)

∣S(0, −∞)

0

〉 . (G.2.29)

As J. Goldstone (1957) pointed out, the same result may be formu-

lated in a slightly different way. Considering a many-particle system with a

nondegenerate ground state, the contribution of each term in the perturba-

tion expansion can be represented by time-ordered diagrams that show the

intermediate states through which the system gets back to the ground state.

This representation contains terms in which some of the particles participat-

ing in the intermediate processes are in no way connected to the incoming

and outgoing particles. It can be demonstrated that the contributions of the

disconnected parts are exactly canceled by the denominator in (G.2.28) and

(G.2.29), so

ΔE =

∞ ∑

n=

∣H 1

E 0 − H 0

H 1

)n ∣ ∣ ∣Ψ 0

con

∞ ∑

n=

E 0 − H 0

H 1

)n ∣ ∣ Ψ 0

con

(G.2.30)

where the label “con” indicates that only the contribution of connected di-

agrams need to be taken into account. It should be noted that instead of

Goldstone’s time-ordered diagrams, the perturbation series for the ground-

state energy can also be represented in terms of Feynman diagrams, which

are more commonly used in the many-body problem. Only connected dia-

grams need to be considered in that representation, too.

Reference

  1. C. Cohen-Tannoudji, B. Diu and F. Laloë, Quantum Mechanics, John

Wiley & Sons, New York (1977).

H

Second Quantization

The quantum mechanical wavefunction is most often considered as the func-

tion of the space and time variables when the solutions of the Schrödinger

equation are sought. In principle, this approach is applicable even when the

system is made up of a large number of interacting particles. However, it is

then much more convenient to use the occupation-number representation for

the wavefunction. We shall introduce the creation and annihilation operators,

and express the Hamiltonian in terms of them, too.

H.1 Occupation-Number Representation

It was mentioned in Chapter 12 on the quantum mechanical treatment of

lattice vibrations that the eigenstates of the harmonic oscillator can be char-

acterized by the quantum number n that can take nonnegative integer values.

Using the linear combinations of the position variable x and its conjugate

momentum, it is possible to construct operators a

† and a that increase and

decrease this quantum number. We may say that when these ladder operators

are applied to an eigenstate, they create an additional quantum or annihi-

late an existing one. Consequently, these operators are called the creation and

annihilation operators of the elementary quantum or excitation. States can

be characterized by the number of quanta they contain – that is, by the oc-

cupation number. Using Dirac’s notation, the state ψn of quantum number

n – which can be constructed from the ground state of the oscillator by the

n-fold application of the creation operator a † , and thus contains n quanta –

will henceforth be denoted by |n〉. The requirement that such states should

also be normalized to unity leads to

a|n〉 =

n |n − 1 〉 , a

† |n〉 =

n + 1 |n + 1〉. (H.1.1)

The operators a and a

† of the quantum mechanical oscillator satisfy the

bosonic commutation relation.

H.1 Occupation-Number Representation 591

We shall now define the creation and annihilation operators that act in

the Fock space and increase and decrease the occupation number by one:

a

† k |n 1 , n 2 ,... , nk,... 〉 = fc|n 1 , n 2 ,... , nk + 1,... 〉 ,

a k |n 1 , n 2 ,... , nk,... 〉 = fa|n 1 , n 2 ,... , nk − 1 ,... 〉.

(H.1.5)

If the normalization factors for bosons are chosen the same way as for harmonic

oscillators, that is,

a

† k |n 1 , n 2 ,... , nk,... 〉 =

nk + 1|n 1 , n 2 ,... , nk + 1,... 〉 ,

a k |n 1 , n 2 ,... , nk,... 〉 =

nk|n 1 , n 2 ,... , nk − 1 ,... 〉 ,

(H.1.6)

then nˆk = a

† k a k is the number operator that gives the occupation number of

the state of index k, since

a

† k a k |n 1 , n 2 ,... , nk,... 〉 = nk|n 1 , n 2 ,... , nk,... 〉 , (H.1.7)

and the commutation relations are the usual ones for bosons:

a k a

† k′^ − a

† k′^ a k = δkk′. (H.1.8)

Any state |n 1 , n 2 ,... , nk,... 〉 can be constructed from the vacuum by

means of creation operators:

Φn 1 ,n 2 ,...,nk ,... =

n 1 !n 2!... nk!...

a

† 1

)n 1

a

† 2

)n 2

...

a

† k

)n k

... | 0 〉. (H.1.9)

As has been mentioned, these states make up a complete set, and the wave-

functions of interacting many-particle system can be expressed as linear com-

binations of them.

Fermions

A similar approach can be adopted for fermions, however, the many-particle

wavefunction has to be chosen as

Φn 1 ,n 2 ,...,nk ,... =

N!

P

P φp 1 (ξ 1 )φp 2 (ξ 2 )... φpN (ξN) (H.1.10)

to meet the requirement of complete antisymmetrization. This is equivalent

to building a Slater determinant from the one-particle wavefunctions:

Φn 1 ,n 2 ,...,nk ,... =

N!

φp 1 (ξ 1 ) φp 2 (ξ 1 )... φpN (ξ 1 )

φp 1 (ξ 2 ) φp 2 (ξ 2 )... φpN (ξ 2 )

. . .

φp 1 (ξN) φp 2 (ξN)... φp N (ξN)

. (H.1.11)

592 H Second Quantization

Since each one-particle state can occur at most once, when the states are

ordered in some arbitrary way, the product in (H.1.10) with indices

p 1 < p 2 < · · · < pN (H.1.12)

is chosen with positive sign, and the signs for other configurations follow from

the parity of the permutation.

The occupation-number representation can be used for fermions as well.

The wavefunction is then written in Fock space as

Φn 1 ,n 2 ,...,n k ,...^ ≡ |n 1 , n 2 ,... , nk,... 〉 , (H.1.13)

where ni can be either 0 or 1. The creation and annihilation operators must

be introduced in such a way that the equations

a

† k |n 1 , n 2 ,... , nk,... 〉 = 0 , if nk = 1 ,

a k |n 1 , n 2 ,... , nk,... 〉 = 0 , if nk = 0

(H.1.14)

be satisfied. After ordering the one-particle states, the normalization of the

states obtained by the application of the creation and annihilation operators

are chosen as

a

† k |n 1 , n 2 ,... , nk,... 〉 =

1 − nk(−1)

Sk |n 1 , n 2 ,... , nk + 1,... 〉 ,

a k |n 1 , n 2 ,... , nk,... 〉 =

nk(−1)

Sk |n 1 , n 2 ,... , nk − 1 ,... 〉 ,

(H.1.15)

where

Sk =

i<k

ni. (H.1.16)

With this choice nˆk = a

† k a k is the number operator for fermions as well,

since when it acts on the state |n 1 , n 2 ,... , nk,... 〉

a

† k a k |n 1 , n 2 ,... , nk,... 〉 = nk|n 1 , n 2 ,... , nk,... 〉. (H.1.17)

In reverse order, however,

a k a

† k |n 1 , n 2 ,... , nk,... 〉 =

(1 + nk)(1 − nk)|n 1 , n 2 ,... , nk,... 〉. (H.1.18)

Since nk can only take the values 0 and 1, the eigenvalue of a k a

† k is 1 − nk,

and thus the operator identity

a k a

† k

  • a

† k a k

= 1 (H.1.19)

holds. By taking states of different quantum numbers, if the state of quantum

number k precedes the state with quantum number k

′ in the order, we have

a k a

† k′^ |n 1 , n 2 ,... , nk,... , nk′^ ,... 〉 (H.1.20)

Sk (−1)

Sk′

nk

1 − nk′^ |n 1 , n 2 ,... , nk − 1 ,... , nk′^ + 1,... 〉,

594 H Second Quantization

∗ n 1 ,n 2 ,...,nk ,...

N ∑

i=

f (ξi)Φn 1 ,n 2 ,...,nk ,...^ dξ 1 dξ 2... dξN

= N

n 1 !n 2!... nk!...

N!

l

φ

∗ l (ξ)f (ξ)φ l (ξ) dξ (H.2.2)

×

P ′

φ

∗ p 1 (ξ 2 )^... φ

∗ pN (ξN )φp 1 (ξ 2 )^... φpN (ξN ) dξ 2...^ dξN.

To calculate the factor that remains after the separation of the matrix element

of the state l, only those states need to be considered in the permutation P

that contain the state of label l only nl − 1 times. Owing to the orthonormality

of the one-particle states, the value of the previous formula is

l

nl

φ

∗ l (ξ)f (ξ)φ l (ξ) dξ. (H.2.3)

In the off-diagonal terms nonzero matrix elements are obtained between

those states Φn 1 ,n 2 ,...,nk ,...,nl,... and Φn 1 ,n 2 ,...,nk +1,...,nl− 1 ,... for which the oc-

cupation numbers of two one-particle states differ by one unit each. Then

∗ n 1 ,n 2 ,...,nk +1,...,nl− 1 ,...

i

f (ξi)Φn 1 ,n 2 ,...,nk ,...,nl,...^ dξ 1 dξ 2... dξN.

(H.2.4)

Because of the normalization factors of the two wavefunctions the matrix

element is proportional to

I =

n 1 !n 2!... (nk + 1)!... (nl − 1)!

N!

n 1 !n 2!... nk!... nl!

N!

(H.2.5)

Since each particle contributes by the same amount, the matrix element is

∗ n 1 ,n 2 ,...,nk +1,...,nl− 1 ,...

i

f (ξi)Φn 1 ,n 2 ,...,nk ,...,nl,... dξ 1 dξ 2... dξN

= N I

kl

φ

∗ k (ξ)f (ξ)φ l (ξ) dξ (H.2.6)

×

P ′

φ

∗ p 1 (ξ^2 )^... φ

∗ pN (ξN^ )φp 1 (ξ^2 )^... φpN (ξN^ ) dξ^2...^ dξN^.

After the separation of the integral for the selected particle, the remaining

terms correspond to a state that contains N − 1 particles, with occupation

numbers n 1 , n 2 ,... , nk,... , nl − 1 ,.... Since there are

(N − 1)!

n 1 !n 2!... nk!... (nl − 1)!...

(H.2.7)

H.2 Second-Quantized Form of Operators 595

such states, the separation of the ξ-integral leaves behind a factor

nk + 1

nl,

so the matrix element is

nk + 1

nl

φ

∗ k (ξ)f (ξ)φ l (ξ) dξ. (H.2.8)

The same expressions are obtained for the diagonal and off-diagonal matrix

elements if the states are specified in occupation-number representation, the

operator F

(1) is chosen as

F

(1)

kl

a

† k fkla l

, (H.2.9)

where

fkl =

φ

∗ k (ξ)f (ξ)φ l (ξ) dξ , (H.2.10)

and the previously obtained relations for the action of the creation and annihi-

lation operators are used in the calculation of the matrix element. Therefore

the operator given in (H.2.9), which acts in the Fock space, is the second-

quantized form of one-particle operators for bosons. Note that while the sum

is over N particles in the first-quantized formula (H.2.1) of the one-particle

operator, it is over the quantum numbers of the one-particle states in the

second-quantized formula.

The intermediate steps are slightly different for fermions, since a Slater

determinant wavefunction is specified in terms of the occupation numbers,

and the normalization factors are also different – nevertheless the final result

is the same: the one-particle operators for fermions can again be represented

as (H.2.9) in terms of creation and annihilation operators.

H.2.2 Second-Quantized Form of Two-Particle Operators

This approach can be extended to the two-body interaction term in the Hamil-

tonian and similar operators that are the sums of terms containing the coor-

dinates of two particles:

F

(2)

ij

f (ξi, ξj ). (H.2.11)

Since the variables of two particles appear in each term, such an operator has

a nonvanishing matrix element only between states for which the occupation

numbers of at most four one-particle states change: two decrease and two

others increase by one. The matrix element to be evaluated is thus

∗ n 1 ,...,nk +1,...,nl+1,...,nm− 1 ,...,nn− 1 ,...

ij

f (ξi, ξj )

Φn 1 ,...,nk ,...,nl,...,nm,...,nn,...^ dξ 1 dξ 2... dξN.

(H.2.12)

H.2 Second-Quantized Form of Operators 597

[

ψ(ξ),

ψ(ξ

′ )

]

[

ψ

† (ξ),

ψ

† (ξ

′ )

]

= 0. (H.2.18)

When the spin variable is separated,

[

ψα(r),

ψ

† β (r

′ )

]

= δαβ δ(r − r

′ ) , (H.2.19)

and [ ˆ ψα(r),

ψ β (r

′ )

]

[

ψ

† α(r),^

ψ

† β (r

′ )

]

= 0. (H.2.20)

The creation and annihilation operators a

† k and a k change the occupation

of a one-particle state of a given quantum number. Below we shall see that

field operators can be interpreted as operators that create and annihilate a

particle at ξ. In other words, the state

ψ

† σ (r)|^0 〉^ contains a particle of spin

quantum number σ at point r of the real space. Similarly,

ψ

† σ (r)

ψσ (r) is the

density operator of spin-σ particles at r.

The one- and two-particle operators can then be rewritten as

F

(1)

dξ ψˆ

† (ξ)f

(1) (ξ) ψˆ(ξ) , (H.2.21)

and

F

(2)

dξ 1

dξ 2 ψˆ

† (ξ 1 ) ψˆ

† (ξ 2 )f

(2) (ξ 1 , ξ 2 ) ψˆ(ξ 2 ) ψˆ(ξ 1 ). (H.2.22)

In this representation the one-particle operator has exactly the same form as

the expectation value of the one-particle operator in first quantization, except

that the wavefunction and it complex conjugate are replaced by the field

operator and its Hermitian adjoint – hence the name second quantization.

H.2.4 Second-Quantized Form of the Electronic Hamiltonian

Among the terms of the Hamiltonian, the kinetic energy and the potential are

one-particle operators, and the pair interaction is a two-particle operator. For

electrons, when the potential and the pair interaction are spin independent,

H =

i

H

(1) i +^

1 2

ij

U

(2) (ri, rj ) , (H.2.23)

where the one-particle part contains

H

(1) i

2

2 me

2 i

  • U (ri). (H.2.24)

By taking a complete set of one-particle states, and denoting, as customary

for electrons, the creation and annihilation operators of a particle in state

φk(ξ) by c

† k and c k instead of a

† k and a k , we have

598 H Second Quantization

H =

kl

Hklc

† k c l

1 2

klmn

U

(2) klmn c

† k c

† l c m c n

, (H.2.25)

where

Hkl =

φ

∗ k(ξ)

2

2 me

2

  • U (r)

φl (ξ) dξ , (H.2.26)

and

U

(2) klmn

dξ dξ

′ φ

∗ k(ξ)φ

∗ l (ξ

′ )U

(2) (r, r

′ )φm(ξ

′ )φn(ξ). (H.2.27)

In general, the states are chosen in such a way that the one-particle part be

diagonal. This is the case when the Bloch functions determined in the presence

of a periodic potential are used as a complete basis set. However, this is not

the only option. In the Hubbard model the Wannier states are used, and so the

one-particle term in the Hamiltonian that describes the hopping of electrons

between lattice points is not diagonal.

Using the field operators instead of the creation and annihilation operators,

H =

ψ

† (ξ)

2

2 me

2

  • U (r)

ψ(ξ) dξ

dξ dξ

ψ

† (ξ)

ψ

† (ξ

′ )U

(2) (r, r

′ )

ψ(ξ

′ )

ψ(ξ).

(H.2.28)

Writing out the spin variable explicitly, the spin independence of the potential

and of the interaction implies

H =

σ

ψ

† σ (r)

2

2 me

2

  • U (r)

ψσ (r) dr

σσ′

dr dr

ψ

† σ (r)

ψ

† σ′^ (r

′ )U

(2) (r, r

′ )

ψ σ′^ (r

′ )

ψσ (r).

(H.2.29)

The description is highly simplified by choosing the plane waves as the

complete set. The one-particle states are then characterized by the wave vector

k and the spin quantum number σ. The usual formula

Hkin = −

i

2

2 me

2

∂r 2 i

(H.2.30)

for the kinetic energy can be rewritten in second-quantized form as

Hkin =

kk′σσ′

c

† kσ Hσσ′ (k, k

′ )c k′σ′^

, (H.2.31)

where