Manyb o dy phenomena in condensed matter and atomic physics Last modied September

Lecture Second Quantization Bosons

In this lecture we discuss second quantization a formalism that is commonly used to

analyze manyb o dy problems The key ideas of this metho d were develop ed starting

from the initial work of Dirac most notably by Fo ck and Jordan In this approach one

thinks of multiparticle states of b osons or fermions as single particle states each lled

with a certain numb er of identical particles The language of second quantization often

allows to reduce the manybo dy problem to a single particle problem dened in terms of

quasiparticles ie particles dressed by interactions

The Fo ck space

The manyb o dy problem is dened for N particles here b osons describ ed by the sum

of singleparticle Hamiltonians and the twob o dy interaction Hamiltonian

N

X X

(1) (2)

H H x H x x

b a a

a�1

a6�b

2

h

(1) (2) 0 (2) 0 2

H x H x x U x x U x r

x

m

where x are particle co ordinates In some rare cases eg for nuclear particles one

a

also has to include the threeparticle and higher order multiparticle interactions such as

P

(3)

H x x x etc

a b c

abc

The system is describ ed by the manyb o dy wavefunction x x x symmetric

2 1 N

with resp ect to the p ermutations of co ordinates x The symmetry requirement fol

a

lows from pareticles indestinguishability and Bose statistics ie the wav efunction invari

ance under p ermutations of the particles The wavefunction x x x ob eys the

2 1 N

h H Since the numb er of particles in typical situations of Schrodinger equation i

t

interest is extremely large solving this equation directly presents a very hard problem

There are several approaches however that allow to gain insight The metho d of sec

ond quatization historically the rst manybo dy technique develop ed in the s will b e

discussed here

We shall start with dening the Fo ck space sometimes also called the big many

particle space

N

o n

M O

(1)

V V

sy mm

N

(1)

the sum of the N th symmetric p owers of the single particle Hilb ert space V It

describ es the states of a system containing any numb er of particles N

One can cho ose the basis in the big space V in the form of symmetrized pro ducts of

single particle wavefunctions x drawn from an orthonormal complete set of states in

p

(1)

V

12

X

m m

1 2

x x x x x x

m 1 2 N m p 1 p 2 p N

1 2 1 2

N

N

P

with the sum taken over all p ermutations of the states x The numb ers m indicate

p p

how many times the function x app ears in the pro duct The numb er of p ermutations

p

P

in the sum is equal to the numb er of ways to combine N elements into groups containg

P

m m etc elements each m m N This combinatorial factor equal to

1 2 1 2

N m m denes the normalization factor in Eq One can check that the states

1 2

are orthogonal and form a complete set in V

As an example consider free Bose particles in a b ox L L L in which case the

single particle states can b e chosen as eigenstates of the single particle problem E x

2

h�

2

r Assuming p erio dic b oundary conditions we have eigenstates of a plane wave

2m

form

p

r exp ik r k n n n n n

n n n 1 2 3

L

V

2

�h

3 2

with integer n and V L The energies of these states are E k The space V

123 n

n

2m

is then spanned by the functions

0 0 0

p

r r r r r r r

n n m m n n n

corresp onding to the noparticle state one particle two particles etc The energies of

these states are

E E E E

n n m n

Note that the structure of the twoparticle functions dep ends on whether the participating

singleparticle states are dierent or the same

To make progress one can intro duce the socalled numb er representation Allow any

total particle numb er N and fo cus on the dep endence of the state on the occupation

numbers m This dep endence is captured most vividly by the representation in which an

i

auxiliary oscillator along with the creation and annihilation op erators is asso ciated with

each single particle state The o ccupation numb ers are interpreted in this representation

as numb er of quanta in eachy oscillator The corresp oning Fock states in the numb er

representation have the form

1

Y

m

i

+

p

jm m i a ji

1 2

i

m

i

i�1

P

where ji is the noparticle state and m N This representation accounts correctly

i

for the symmetry prop erties of the states due to Bose statistics

Secondquantized op erators

In the numb er representation the manyb o dy Hamiltonian is represented by a p oly

+

nomial in the op erators a a

i

i

X X

(1)

ij

+ (2) + +

H H a a H a a a a

j k m

ij

i k m i j

ij

ij k m

(1) (2) ij

with the quantities H H b eing the one and twoparticle matrix elements

ij

k m

Z

(1)

(1) (1)

H h xjH xj xi xH x xdx

i j i j

ij

Z Z

ij

(2) 0 (2) 0 0 0 (2) 0 0 0

H h x x jH x x j x x i x x H x x x x dxdx

i j k m i j k m

k m

One can prove the equivalence of this representation to the original manyb o dy problem

formulated in terms of manyparticle wavefunction x x x by directly evalu

2 1 N

ating the matric elements of the Hamiltonian b etween all pairs of many b o dy states and

showing that in b oth representations the results agree The combinatorics involved in

this pro of is comb ersom alb eit completely straightforward Instead of reviewing it here

we refer to the b o ok by J R Schrieer The Theory of Sup erconductivity that contains

an App endix describing the analysis in some detail Another pro of can b e devised using

functional integral and we shall talk ab out it later on

The expressions are true for any orthogonal set of functions x In the case

i

when these functions are chosen to b e the eigenstates of the singleparticle problem the

(1)

vanish b etween dierent states matrix elements of H

(1)

h xjH xj xi E

i j i ij

and the oneparticle part of the Hamiltonian simplies to

X

(1) +

H E a a

i i

i

i

+

Since n a a is nothing but the numb er op erator the eigenvalues of the oneparticle

i i

i

Hamiltonian corresp onding to the numb er states are

X

(1)

hn n jH E jn n i E n

n 1 2 n 1 2 i i

1 2

i

In the ab ove example of free b osons in a b ox the states are lab eled by discrete momenta

and the expression b ecomes

X

E n E

n k k n

1 2

k

(2) 0

If the b osons are in U r r from Eqs teracting via a twobo dy p otential U

we obtain the twoparticle Hamiltonian of the form

X

(2) (2) + +

H hk k jU jk k ia a a a

k k 1 2 3 4

3 4

k k

1 2

k k k k

1 2 3 4

where the matrix element in evaluated on the plane wave states has the form

Z Z

0 0

(2) ;ik r;ik r +ik r+ik r 0 3 3 0

1 2 3 4

hk k jU jk k i e U r rd r d r

1 2 3 4

2

V

0

This expression can b e simplied and evaluated by choosing a r r as an integration

0

variable instead of r after which the integral in factors as

Z Z

i(k ;k )a 3 ;ik r;i r+ik r+ik r 3 k

4 2 1 2 3 4

e U ad a e d r U k k

+k �k +k 2 4 k

1 2 3 4

R

;ikr 3

where U k e U rd r is the Fourier transform of the interaction p otential and

V k k k k

1 2 3 4

+k �k +k k

1 2 3 4

k k k k

1 2 3 4

Finally the twobo dy Hamiltonian takes the form

X

(2) + +

H U k k a a a a

2 4 k k

3 4

k k

1 2

V

+k �k +k k

1 2 3 4

where the sum is taken over all integers parameterizing the plane wave states sub ject to

the constraint k k k k This constraint as it is clear from the calculation arises

1 2 3 4

due to translational invariance of the system Physically it expresses the conservation of

momentum in two particle scattering

The secondquantized interaction Hamiltonian is written in terms of the op erators a

k

+

a which remove or add particles One may thus b e worried by apparent particle non

k

conservation After lo oking at it closer however and taking into account the commutation

+ +

with the numb er op erator n a a relations of a a

+ +

na an na a n

P

+

one can show that the total number of particles N a a commutes with the Hamil

k

k

k

tonian and is thus conserved

Field op erator

A very useful representation of the secondquantized manybo dy hamiltonian is provided

by the eld operator rst introduced by Jordan

X X

+ +

x a x x a x

k k k

k

k k

where x lab els conguration space eg x r in D The states x in Eq can

k

b e the eigenstates of a singleparticle problem such as eg the plane waves of the ab ove

section or any other convenient orthonormal basis set

The op erators ob ey commutation relations

0 + + 0 + 0 0

x x x x x x x x

+

which can b e proven by using the commutation relations of a and a along with the

k

k

orthogonality of the states x For example

k

X X

+ 0 0 + 0

0 0 0

x x x x a a x x

0

k k k k k kk

k

0 0

kk kk

X

0 0

x x x x

k k

k

We note that although some particular basis set x was employed to contruct the

k

eld op erators their prop erties such as the commutation relations are invariant

with resp ect to the choice of basis

Using the eld op erators the secondquantized problem can b e expressed as

a p olynomial

Z Z Z

2

h

+ 2 + + 0 0 0 0

H x r U x xdx x x U x x xx dxdx

x

m

in which the quadratic and the quartic parts describ e noninteracting particles and their

interaction resp ectively

It is sometimes helpful to think of x as a quantized wavefunction In the eld

op erator representation the many b o dy problem starts lo oking very much like a single

particle problem Of course this simplicity is only apparent since we still have a quartic

term in the Hamiltonian expressing the interactions and leading to nonlinear dynamics

Not just the Hamiltonian but many other quantities also take a simple form in terms

P

of the eld op erator For example particle density nx x x b ecomes

a

a

X

0 0 0 + +

h x j x x j x ia a xx n x

i j j

i

ij

R

0 0 0 0

due to x x x x dx x x Similarly the particle current op erator is

i j i j

h

+ +

j x xr x r x x

x x

mi

The density and current ob ey continuity relation written as an op erator equation

n rj

t