Quantum Mechanics for Mathematicians

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zz→−ia†a

This is the normal ordered form, with eigenvalues−in.

With either choice, we get a number operator

N=

1

2

(a†a+aa†), or N=

1

2

:(a†a+aa†): =a†a

In both cases we have
[N,a] =−a, [N,a†] =a†

so
eiθNae−iθN=e−iθa, eiθNa†e−iθN=eiθa†

Either choice ofNwill give the same action on operators. However, on states
only the normal ordered one will have the desirable feature that

N| 0 〉= 0, eiNθ| 0 〉=| 0 〉

Since we now want to treat fields, adding together an infinite number of such
oscillator degrees of freedom, we will need the normal ordered version in order
to not get∞·^12 as the number eigenvalue for the vacuum state.
We now generalize as described in section 38.1 and get, in momentum space,
the expression

N̂=

∫+∞

−∞

a†(p)a(p)dp (38.2)

which is just the number operator already discussed in chapter 36. Recall from
section 36.3 that this sort of operator product requires some interpretation in
order to give it a well-defined meaning, either as a limit of a finite dimensional
definition, or by giving it a distributional interpretation.
Fourier transforming to position space, one can work withΨ(̂x),Ψ̂†(x) in-
stead ofa(p),a†(p) and find that

N̂=

∫+∞

−∞

Ψ̂†(x)Ψ(̂x)dx (38.3)

Ψ̂†(x)Ψ(̂x) can be interpreted as an operator-valued distribution, with the phys-
ical interpretation of measuring the number density atx. On field operators,N̂
satisfies
[N,̂Ψ] =̂ −Ψ̂, [N,̂Ψ̂†] =Ψ̂†

soΨ acts on states by reducing the eigenvalue of̂ N̂by one, whileΨ̂†acts on
states by increasing the eigenvalue ofN̂by one. Exponentiating gives

eiθ

N̂̂

Ψe−iθ

N̂

=e−iθΨ̂, eiθ

N̂̂

Ψ†e−iθ

N̂

=eiθΨ̂†

which are the quantized versions of theU(1) action on the phase space coordi-
nates (see equations 38.1) that we began our discussion with.