130_notes.dvi

This one checks out OK.

Theother equation of Hamiltonis

∂H

∂Pk,α

= Q ̇k,α

Pk,α =

1

c

( ̇ck,α+ ̇c∗k,α)

−

iω

c

(ck,α−c∗k,α) =

1

c

(−iωck,α+iωc∗k,α)

−

iω

c

(ck,α−c∗k,α) = −

iω

c

(ck,α−c∗k,α)

This also checks out, sowe have identified the canonical coordinates and momentaof our
oscillators.

We have a collection of uncoupled oscillators with identified canonical coordinate and momentum.
The next step is to quantize the oscillators.

33.5 Quantization of the Oscillators

To summarize the result of the calculations of the last section we have theHamiltonian for the
radiation field.

H=

∑

k,α

(ω

c

) (^2) [
ck,αc∗k,α+c∗k,αck,α

]

Qk,α=

1

c

(ck,α+c∗k,α)

Pk,α=−

iω

c

(ck,α−c∗k,α)

H=

1

2

∑

k,α

[

Pk,α^2 +ω^2 Q^2 k,α

]

Soon after the development of non-relativistic quantum mechanics, Dirac proposed that the canonical
variables of the radiation oscillators be treated likepandxin the quantum mechanics we know. The
place to start is with the commutators. The coordinate and its corresponding momentum
do not commute.For example [px,x] = ̄hi. Coordinates and momenta that do not correspond, do
commute. For example [py,x] = 0. Different coordinates commute with each other as do different
momenta. We will impose the same rules here.

[Qk,α,Pk′,α′] = i ̄hδkk′δαα′

[Qk,α,Qk′,α′] = 0

[Pk,α,Pk′,α′] = 0

By now we know that if theQandPdo not commute, neither do thecandc∗so we should continue
to avoid commuting them.