QuantumPhysics.dvi

19.4 Momentum in terms of radiation oscillators

The classical electro-magnetic field carries momentum, which may be expressed in terms of the
Poynting vector,

P=

∫

d^2 xE(t,x)×B(t,x) (19.43)

This qunatity may be similarly expressed in terms of the radiation oscillatorsaα, in a manner
analogous to energy, and we have

P=i

∫

d^3 k

(2π)^3

∂tC(t,k)×

(

k×C(t,k)

)

(19.44)

ExpressingCin terms ofaαanda∗α, using the algebraic identity

A×(B×C) =B(A·C)−C(A·B) (19.45)

applied to the problem at hand yields

εα(k)×

(

k×εβ(−k)

)

=kεα(k)·εβ(−k) (19.46)

The integral of (19.44) contains a part ine−^2 iωt, one ine^2 iωt, and finally one independent oft. The
first one of these is proportional to
∑

α,β

kεα(k)·εβ(−k)αα(k)αβ(−k) (19.47)

which is odd ink, and whose integral overkthus vanishes. The second term similarly vanishes.
Thet-independent term is the only one that contributes, and we have

P=

∫ d (^3) k
(2π)^3
k
∑
α
αα(k)∗αα(k) (19.48)

19.5 Canonical quantization of electro-magnetic fields

From the above classical discussion of the electro-magnetic fields, it is manifest that their dynamics
is equivalent to that of an infinite number of harmonic oscillators

aα(k) aα(k)∗ α= 1, 2 k∈R^3 (19.49)

We shall nowquantize the electro-magnetic fieldby canonically quantizing each of the harmonic
oscillators which build up the field. Thus,aα(k) is to be viewed as an operator on (an until now
undetermined) Hilbert space. Together with its adjointa†α(k), the oscillators satisfy the following
canonical commutation relations,

[aα(k),aβ(k′)] = 0

[a†α(k),a†β(k′)] = 0

[aα(k),a†β(k′)] = (2π)^3 δαβδ(3)(k−k′) (19.50)