130_notes.dvi

constants.

ak,α|nk,α〉 =

√

nk,α|nk,α− 1 〉

a†k,α|nk,α〉 =

√

nk,α+ 1|nk,α+ 1〉

Thenk,αcan only take oninteger valuesas with the harmonic oscillator we know.

As with the 1D harmonic oscillator, we also can define thenumber operator.

H =

(

a†k,αak,α+

1

2

)

̄hω=

(

Nk,α+

1

2

)

̄hω

Nk,α = a†k,αak,α

The last step is tocompute the raising and lowering operators in terms of the original
coefficients.

ak,α =

1

√

2 ̄hω

(ωQk,α+iPk,α)

Qk,α =

1

c

(ck,α+c∗k,α)

Pk,α = −

iω

c

(ck,α−c∗k,α)

ak,α =

1

√

2 ̄hω

(ω

1

c

(ck,α+c∗k,α)−i

iω

c

(ck,α−c∗k,α))

=

1

√

2 ̄hω

ω

c

((ck,α+c∗k,α) + (ck,α−c∗k,α))

=

1

√

2 ̄hω

ω

c

(ck,α+c∗k,α+ck,α−c∗k,α)

=

√

ω

2 ̄hc^2

(2ck,α)

=

√

2 ω

̄hc^2

ck,α

ck,α=

√

̄hc^2

2 ω

ak,α

Similarly we can compute that

c∗k,α=

√

̄hc^2

2 ω

a†k,α

Since we now have the coefficients in our decomposition of the field equal to a constant times
the raising or lowering operator, it is clear thatthese coefficients have themselves become
operators.