Quantum Mechanics for Mathematicians

Ψ(̂ψ)P+(Ψ(ψ 1 )⊗···⊗Ψ(ψn)) =

1

√

n

∑n

j=1

〈ψ,ψj〉P+(Ψ(ψ 1 )⊗···⊗Ψ(̂ψj)⊗···⊗Ψ(ψn)) (37.5)

(theΨ(̂ψj) means omit that term in the tensor product, andP+is the sym-
metrization operator defined in section 9.6). This gives a representation of the
Lie algebra relations 37.2, satisfying

[Ψ(̂ψ 1 ),Ψ(̂ψ 2 )] = [Ψ̂†(ψ 1 ),Ψ̂†(ψ 2 )] = 0, [Ψ(̂ψ 1 ),Ψ̂†(ψ 2 )] =〈ψ 1 ,ψ 2 〉

Conventional multi-particle wavefunctions in position space have the same
relation to symmetric tensor products as in the momentum space case of section
36.4. Given an arbitrary state|ψ〉in the multi-particle state space, the position
space wavefunction component with particle numberNcan be expressed as

ψN(x 1 ,x 2 ,···,xN) =〈 0 |Ψ(̂x 1 )Ψ(̂x 2 )···Ψ(̂xN)|ψ〉

37.2 Quadratic operators and dynamics

Other observables can be defined simply in terms of the field operators. These
include (note that in all cases these formulas require interpretation as limits of
finite sums in the finite cutoff theory):

• The number operatorN̂. A number density operator can be defined by

n̂(x) =Ψ̂†(x)Ψ(̂x)

and integrated to get an operator with eigenvalues the total number of

particles in a state

N̂=

∫∞

−∞

n̂(x)dx

=

∫∞

−∞

∫∞

−∞

∫∞

−∞

1

√

2 π

e−ip

′x

a†(p′)

1

√

2 π

eipxa(p)dpdp′dx

=

∫∞

−∞

∫∞

−∞

δ(p−p′)a†(p′)a(p)dpdp′

=

∫∞

−∞

a†(p)a(p)dp

• The total momentum operatorP̂. This can be defined in terms of field