170
6
Quantization of the Electromagnetic Field
where we have identified the energy
E
with the Hamiltonian
H
of the system of
infinitely many degrees of freedom, described by
Q
k
=
(
Q
i
k
)
, each labeled by a wave
number vector
k
and a polarization index
i
(as a consequence of the transversality
condition, not all these polarizations are independent, as we shall discuss below).
We can easily verify that
P
k
,
Q
k
are indeed the canonical variables corresponding
to the Hamiltonian
H
by showing that they satisfy Hamilton’s equations
3
:
˙
Q
i
k
=
∂
H
∂
P
i
k
=
P
i
k
,
˙
P
i
k
= −
∂
H
∂
Q
i
k
= −
ω
2
k
Q
i
k
,
where, as usual, the dot represents the time-derivative. These equations can also be
written in the second order form:
¨
Q
i
k
+
ω
2
k
Q
i
k
=
0
,
(6.32)
which, given the relation (
6.29
), are equivalent to the Maxwell equations (
6.12
) for
each component
A
k
. We realize that the above equation in the variable
Q
i
k
, for each
polarization component
i
and wave-number vector
k
, is the equation of motion of
a harmonic oscillator with angular frequency
ω
k
. Note now that the vectors
P
k
and
Q
k
are orthogonal to
k
in virtue of the transversality property
A
k
and
A
∗
k
:
k
·
P
k
=
k
·
Q
k
=
0
.
(6.33)
This allows us, for a given direction of propagation
n
k
, to decompose
P
k
and
Q
k
along an ortho-normal basis
u
k
,α
,
α
=
1
,
2 on the plane transverse to
n
:
P
k
=
2
1
P
k
α
u
k
,α
,
Q
k
=
2
1
Q
k
α
u
k
,α
.
The index
α
labels the two polarizations of the plane wave. Taking into account the
ortho-normality of the
(
u
k
,α
)
we can write the Hamiltonian as follows:
H
=
1
2
k
α
=
1
,
2
(
P
2
α
k
+
ω
2
k
Q
2
α
k
)
=
k
H
k
=
k
E
k
=
k
α
=
1
,
2
H
k
,α
,
(6.34)
which describes a system of infinitely many, decoupled, harmonic oscillators, each
described by the conjugate variables
P
α
k
,
Q
α
k
and Hamiltonian function
H
k
,α
.