Quantum Mechanics for Mathematicians

state spaceH=F, whereF is the space of holomorphic functions (satisfying
d
dzψ= 0) onCwith finite norm in the inner product

〈ψ 1 |ψ 2 〉=

1

π

∫

C

ψ 1 (z)ψ 2 (z)e−|z|

2

d^2 z (22.4)

whered^2 z=dRe(z)dIm(z). The spaceFis sometimes called “Fock space”. We
define the following two operators acting on this space:

a=

d

dz

, a†=z

Since

[a,a†]zn=

d

dz

(zzn)−z

d

dz

zn= (n+ 1−n)zn=zn

the commutator is the identity operator on polynomials

[a,a†] = 1

One finds

Theorem.The Bargmann-Fock representation has the following properties

• The elements
  zn
  √
  n!
  ofFforn= 0, 1 , 2 ,...are orthonormal.


• The operatorsaanda†are adjoints with respect to the given inner product
  onF.


• The basis
  zn
  √
  n!
  ofFforn= 0, 1 , 2 ,...is complete.

Proof.The proofs of the above statements are not difficult, in outline they are

• For orthonormality one can compute the integrals
  ∫

C

zmzne−|z|

2

d^2 z

in polar coordinates.

• To show thatzanddzd are adjoint operators, use integration by parts.