130_notes.dvi

It waswith the Hamiltonian that we first quantized the non-relativistic motion of parti-
cles. Theposition and momentum became operators which did not commute. Lets define
ck,αto be the time dependent Fourier coefficient.

̈ck,α=−ω^2 ck,α

We can thensimplify our notation a bit.

H=

∑

k,α

(ω

c

) (^2) [
ck,αc∗k,α+c∗k,αck,α

]

This nowclearly looks like the Hamiltonian for a collection of uncoupled oscillators; one
oscillator for each wave vector and polarization.

We wish to write the Hamiltonian in terms of a coordinate for each oscillator and the conjugate
momenta. The coordinate should be real so it can be represented by a Hermitian operator and have
a physical meaning. The simplest choice for a real coordinates isc+c∗. With a little effort we can
identify the coordinate

Qk,α=

1

c

(ck,α+c∗k,α)

and itsconjugate momentumfor each oscillator,

Pk,α=−

iω

c

(ck,α−c∗k,α).

TheHamiltonian can be written in terms of these.

H =

1

2

∑

k,α

[

Pk,α^2 +ω^2 Q^2 k,α

]

=

1

2

∑

k,α

[

−

(ω

c

) 2

(ck,α−c∗k,α)^2 +

(ω

c

) 2

(ck,α+c∗k,α)^2

]

=

1

2

∑

k,α

(ω

c

) (^2) [
−(ck,α−c∗k,α)^2 + (ck,α+c∗k,α)^2

]

=

1

2

∑

k,α

(ω

c

) 2

2

[

ck,αc∗k,α+c∗k,αck,α

]

=

∑

k,α

(ω

c

) (^2) [
ck,αc∗k,α+c∗k,αck,α

]

This verifies that this choice gives the right Hamiltonian. We should alsocheck that this choice of
coordinates and momenta satisfy Hamilton’s equationsto identify them as the canonical
coordinates. The first equation is

∂H

∂Qk,α

= −P ̇k,α

ω^2 Qk,α =

iω

c

( ̇ck,α−c ̇∗k,α)

ω^2

c

(ck,α+c∗k,α) =

iω

c

(−iωck,α−iωc∗k,α)

ω^2

c

(ck,α+c∗k,α) =

ω^2

c

(ck,α+c∗k,α)