130_notes.dvi

Pk,α=−

iω

c

(ck,α−c∗k,α)

for the harmonic oscillator at each frequency. We assume that a coordinate and its conjugate
momentum have the same commutator as in wave mechanics and thatcoordinates from different
oscillators commute.

[Qk,α,Pk′,α′] = i ̄hδkk′δαα′

[Qk,α,Qk′,α′] = 0

[Pk,α,Pk′,α′] = 0

As was done for the 1D harmonic oscillator, we write the Hamiltonian in terms of raising and
lowering operators that have the same commutation relations as in the 1D harmonic oscillator.

ak,α =

1

√

2 ̄hω

(ωQk,α+iPk,α)

a†k,α =

1

√

2 ̄hω

(ωQk,α−iPk,α)

H =

(

a†k,αak,α+

1

2

)

̄hω

[

ak,α,a†k′,α′

]

= δkk′δαα′

This means everything we know about the raising and lowering operators applies here. Energies are
in steps of ̄hωand there must be a ground state. The states can be labeled by a quantum number
nk,α.

H =

(

a†k,αak,α+

1

2

)

̄hω=

(

Nk,α+

1

2

)

̄hω

Nk,α = a†k,αak,α

The Fourier coefficients can now be written in terms of the raising andlowering operators for the
field.

ck,α =

√

̄hc^2

2 ω

ak,α

c∗k,α =

√

̄hc^2

2 ω

a†k,α

Aμ =

1

√

V

∑

kα

√

̄hc^2

2 ω

ǫ(μα)

(

ak,α(t)ei

~k·~x

+a†k,α(t)e−i

~k·~x)

H =

1

2

∑

k,α

̄hω

[

ak,αa†k,α+a†k,αak,α

]

=

∑

k,α

̄hω

(

Nk,α+

1

2

)