130_notes.dvi

33.9 The Time Development of Field Operators

The creation and annihilation operators are related to thetime dependent coefficientsin our
Fourier expansion of the radiation field.

ck,α(t) =

√

̄hc^2

2 ω

ak,α

c∗k,α(t) =

√

̄hc^2

2 ω

a†k,α

This means that the creation, annihilation, and other operators are time dependent operators as
we have studied the Heisenberg representation (See section 11.6). In particular, we derived the
canonical equation for the time dependence of an operator.

d

dt

B(t) =

i

̄h

[H,B(t)]

a ̇k,α =

i

̄h

[H,ak,α(t)] =

i

̄h

(− ̄hω)ak,α(t) =−iωak,α(t)

a ̇†k,α =

i

̄h

[H,a†k,α(t)] =iωa†k,α(t)

So the operators have thesame time dependence as did the coefficientsin the Fourier expan-
sion.

ak,α(t) = ak,α(0)e−iωt

a†k,α(t) = a†k,α(0)eiωt

We can now write the quantized radiation field in terms of the operators att= 0.

Aμ=

1

√

V

∑

kα

√

̄hc^2

2 ω

ǫ(μα)

(

ak,α(0)eikρxρ+a†k,α(0)e−ikρxρ

)

Again, the 4-vectorxρis aparameter of this field, not the location of a photon. Thefield
operator is Hermitianand the field itself is real.

33.10Uncertainty Relations and RMS Field Fluctuations

Since the fields are a sum of creation and annihilation operators, theydo not commute with the
occupation number operators
Nk,α=a†k,αak,α.

Observables corresponding to operators which do not commute have an uncertainty principle between
them. So wecan’t fix the number of photons and know the fieldsexactly. Fluctuations in
the field take place even in the vacuum state, where we know there are no photons.