p|»2 |»1 § p § »3 § · · · § »N + ···

+

p|»N |»1 § »2 § · · · § »N ’1 § p .

+ (2.7)

2 Weare here guided by the knowledge that a bra has the feature of an annihilation operator

and a ket has the feature of a creation operator, and the transition operators constitute the basis

of the measurement algebra of a quantum system, i.e. completeness of a basis in the state space

» |» »| = 1, and the set {|» » |}»,» is a

is expressed, in the »-representation, by the identity

basis in the dual space, the space of linear operators on the state space. For details see chapter 1

in reference [1].

2.1. Physical observables 35

Since by antisymmetrization

(’1)n’1 |p § »1 § · · · § ( no »n ) § · · · § »N

= |»1 § · · · § »n’1 § p § »n+1 § · · · § »N (2.8)

we have

N

ˆ (pp )

FN |»1 § · · · § »N p|»n (’1)n’1 |»1 § · · · (no »n instead p ) · · · § »N ,

=

n=1

(2.9)

but according to Eq. (1.77) this is the same state which is obtained when operating

with the operator a† ap so that

p

a† ap |»1 § · · · § »N .

ˆ (pp )

FN |»1 § · · · § »N = (2.10)

p

We have thus established how to implement a one-body operator onto the multi-

particle space so that its restriction to any N -particle subspace is the corresponding

N -particle operator. The implementation for bosons is identical to the above, as

usual the derivation is completely analogous, in fact simpler since no minus signs

occur.

There is of course nothing special about momentum labels; the formal machinery,

i.e. the combinatorics, works for any set of one-particle states, say labeled by μ,

so that corresponding to the one-particle operator |μ2 μ1 | corresponds the operator

F (1) in the multi-particle space

F (1) = a† 2 aμ1 . (2.11)

μ

An arbitrary one-particle operator has, in an arbitrary basis, the form

ˆ ˆ

|» »|f (1) |» »|

f (1) = (2.12)

»,»

and by linearity the corresponding operator F (1) in the multi-particle space is thus

»|f (1) |» a† a» .

ˆ

F (1) = (2.13)

»

»,»

ˆ

We note that if f (1) is hermitian in the one-particle state space, as is F (1) in the

multi-particle state space.

The total momentum operator P in the multi-particle space is thus

dp p a† ap = dx ψ † (x) ∇x ψ(x)

P= (2.14)

p

i

expressed in either the momentum or position representation of the ¬eld.

36 2. Operators on the multi-particle state space

Exercise 2.1. Show that the commutator of the total momentum operator and the

¬eld is

[ψ(x), P] = ∇x ψ(x) (2.15)

i

or equivalently

ψ(x) = e’ x·P ψ(0) e x·P .

i i

(2.16)

We now have the prescription for mapping any one-particle operator into the

corresponding operator on the multi-particle space. For a non-relativistic particle of

mass m in a potential V the Hamiltonian is

p2 p2

ˆ ˆ

ˆ

H= + V (ˆ , t) = + dx n(x) V (x, t) ,

ˆ (2.17)

x

2m 2m

where in the second equality we have introduced the probability density operator for

a particle, n(x) = δ(ˆ ’ x) (recall Section 1.2.4). In the position representation the

ˆ x

Hamiltonian has the matrix elements

2

p2 ‚2

ˆ 1

+ V (ˆ , t) |x δ(x ’ x )

= + V (x, t) (2.18)

x| x

‚x2

2m 2m i

and according to Eq. (2.13) the corresponding Hamiltonian on the multi-particle

space becomes

2

‚2

1

†

H= dx ψ (x) + V (x, t) ψ(x) . (2.19)

‚x2

2m i

We note that the single particle properties can be expressed in terms of the

occupation number operators. For the case of energy, the energy eigenstates should be

used, recall Eq. (1.147) and see Exercise 2.2, and of course for the case of momentum,

Eq. (2.14), the reference states should be the momentum states.

Exercise 2.2. Show that the kinetic energy operator for an assembly of non-relativistic

free identical particles of mass m

2

‚2

1

†

H= dx ψ (x) ψ(x) (2.20)

‚x2

2m i

in the momentum and energy representation has the form

a† ap =

H= np , (2.21)

p p

p

p p

where p = p2 /2m is the kinetic energy of the free particle with momentum p.

The sum over momenta occurs, one momentum state per momentum volume ”p =

(2π )3 /V in three dimensions, as the particles are assumed enclosed in a box of

volume V .

2.2. Probability density and number operators 37

Exercise 2.3. Show that the average value of the kinetic energy operator for an

electron gas consisting of N electrons in the ground state, i.e. the energy of N free

electrons in the ground state, is

3 p2

a† F

ap = N, (2.22)

p p

5 2m

p

where pF is the Fermi momentum, the radius of the sphere of occupied momentum

states (in three dimensions pF = (3π 2 n)1/3 , where n = N/V is the density of the

electrons).

Exercise 2.4. Show that the vacuum state is non-degenerate and uniquely charac-

terized by all the eigenvalues of the state number operators np being zero.

Exercise 2.5. For the quantities discussed so far, a possible (say) spin degree of free-

dom of the particles did not have its two spin states discriminated, and its presence

was left implicit in the notation. To consider a situation where spin states needs to be

speci¬ed explicitly, consider (say) electrons interacting with the magnetic moments

of impurities. The interaction of an electron interacting with the magnetic moments

of impurities is

(sf)

u(x ’ xa ) Sa · σ ±,±

V±,± (x) = (2.23)

a

where xa is the location of a magnetic impurity with spin Sa and σ represents the

electron spin. In the multi-particle space, the interaction of the impurity spins and

the electrons thus becomes

†

dx u(x ’ xa ) ψ± (x) Sa · σ ±,± ψ± (x) .

Vsf = (2.24)

a

Show it can be rewritten in the form

† †

S ’ ψ‘ (x)ψ“ (x) + S + ψ“ (x)ψ‘ (x)

dx u(x ’ xa )

Vsf =

a

† †

S z ψ‘ (x)ψ‘ (x) ’ ψ“ (x)ψ“ (x)

+ , (2.25)

where S ± = S x ± iS y .

Density and current density operators are important as well as their coupling

to external ¬elds, and we now turn to their construction in the multi-particle state

space.

2.2 Probability density and number operators

The one-particle probability density operator (recall Section 1.2.4)

n(x) = δ(ˆ ’ x) = |x x|

ˆ (2.26)

x

38 2. Operators on the multi-particle state space

maps according to the general prescription, Eq. (2.13), to the operator on the multi-

particle space

dx1 dx2 x1 |ˆ (x)|x2 ψ † (x1 ) ψ(x2 )

n(x) = n (2.27)

and therefore the probability density operator in the multi-particle space is

n(x) = ψ † (x) ψ(x) . (2.28)

By construction this operator reduces in each N -particle subspace to the N -particle

density operator

N

δ(ˆ i ’ x) .

n(x) =

ˆ (2.29)

x

i=1

The identity operator in the one-particle state space3

ˆ dx |x x| =

I= dx n(x)

ˆ (2.30)

becomes in the multi-particle space according to the general prescription, Eq. (2.13),

the operator

dx ψ † (x) ψ(x) = dp a† ap ,

N= (2.31)

p

which is the operator that counts the total number of particles in each N -particle

state, the total number operator. For example, according to the equation analogous

to Eq. (1.77) but in the position representation

dx ψ † (x) ψ(x) |x1 § x2

N |x1 § x2 =

dx ( x|x1 |x § x2 ’ x|x2 |x § x1 )

=

2 |x1 § x2 .

= (2.32)

Or more e¬ciently by just using the basic anti-commutation or commutation relation

for the ¬elds, we obtain by consecutively anti-commuting or commuting, depending

on the particles being fermions or bosons, the ψ(x)-operator in the number operator

to the right and eventually killing the vacuum, that for the basis state Eq. (1.115)

N ¦x1 x2 ... xn = n ¦x1 x2 ... xn . (2.33)

For the case of the vacuum state the eigenvalue is zero, there are no particles

in the vacuum. In non-relativistic quantum mechanics, the total number operator

for each set of species is always conserved; however, this is of course not the case

in relativistic quantum theory. We note that the vacuum state has zero energy and

momentum (and of course, as noted, zero number of particles).

3 The

physical interpretation of the identity operator in the one-particle state space is the number

operator, counting the particle number in any one-particle state, I |ψ = 1 |ψ .

ˆ

2.2. Probability density and number operators 39

Exercise 2.6. Show that if |ψn represents a state with n particles, N |ψn = n |ψn ,

then

N ψ † (x) |ψn = (n + 1) ψ † (x) |ψn (2.34)

i.e. ψ † (x) |ψn is a state with (n + 1) particles.

Since ψ(x) removes a particle from any state, the relationship

ψ(x) f (N ) = f (N + 1) ψ(x) (2.35)

is valid for an arbitrary function, f , of the number operator. In particular we have

e’±N ψ(x) e±N = e± ψ(x) . (2.36)

Exercise 2.7. Show that the number operator for electrons in the momentum rep-

resentation takes the form

dp a† apσ .

N= (2.37)

pσ

σ

Exercise 2.8. The state considered in Exercise 1.7 on page 20 is the famous BCS-

state, which describes remarkably well the ground state properties of many s-wave

superconductors as realized by J. Bardeen, L. N. Cooper and J. R. Schrie¬er (in

1957). Note its total disrespect for the sacred conservation law of non-relativistic

Fermi systems, the conservation of the number of particles or, equivalently, we can

say that the state corresponds to a situation with broken global gauge invariance.

For the reader interested in BCS-ology (which is further investigated in Section

8.1), verify for the average of the number operator

≡ BCS|N |BCS |vp |2

N =2 (2.38)

p

and for the variance

(N ’ N )2 = N2 ’ N 2

u2 vp .

2

=4 (2.39)

p

p

Show that a† |BCS and a’p“ |BCS represent the same state and that they are

p‘

orthogonal to the state |BCS . The BCS-pairing state consists of linear superpositions

of particle and hole states. Show that, as a consequence, anomalous moments are

non-vanishing in the state |BCS , for example

BCS|ap‘ a’p“ |BCS = ’u— vp . (2.40)

p

40 2. Operators on the multi-particle state space

2.3 Probability current density operator

For a single particle, the probability current density operator is, according to Eq.

(1.58),

1

ˆ

j(x) = {ˆ , n(x)} .

pˆ (2.41)

m

For particles carrying electric charge, e, the electric current density operator or charge

current density operator in the presence of a vector potential, A(x, t), is speci¬ed in

terms of the kinematic momentum operator (ˆ can being the operator satisfying the

p

canonical commutation relation, Eq. (1.8))

pkin = pcan ’ eA(ˆ , t)

ˆt ˆ (2.42)

x

and the charge current density operator is

e

ˆt (x) = {ˆ kin , n(x)} .

ˆ (2.43)

j p

2m t

The current density operator then has two distinct parts

ˆt (x) = ˆp (x) + ˆd (x) (2.44)

j j jt

consisting of the so-called paramagnetic current density operator (or simply the cur-

rent density operator in the absence of a vector potential)

ˆp (x) = e {ˆ can , n(x)}

ˆ (2.45)

j p

2m

and in the present case a time-dependent so-called diamagnetic current density op-

erator

2 2

ˆd (x) = ’ e {A(ˆ , t), n(x)} = ’ e n(x) A(ˆ , t) ,

ˆ ˆ (2.46)

jt x x

2m m

the last equality sign following from the fact that the two operators commute.

For particles carrying electric charge e, the electric current density operator on

the multi-particle space is therefore, according to Eq. (2.12),

jA(t) (x, t) = ψ † (x) ˆA(t) (x, t) ψ(x) , (2.47)

j

where

2

ˆA(t) (x, t) = ˆ(1) (x, t) ’ e A(x, t) (2.48)

j j

m

is the one-particle current density operator in the position representation in the

presence of an external vector potential A(x, t) and

’ ←

e ‚ ‚

ˆ(1) ’

(x, t) = , (2.49)

j

2mi ‚x ‚x

the arrows indicating on which ¬eld, to the left or right, the di¬erential operator

operates.

2.3. Probability current density operator 41

The interaction between an electron and an electromagnetic ¬eld represented by

a vector potential, A(x, t), can be written in terms of the current density operator

and the density operator

e2

= ’ dx ˆt (x) · A(x, t) ’

ˆ dx n(x) A2 (x, t)

HA(t) ˆ