Derivation of the free-field Hamiltonian

A classical representation of a single-mode field satisfying Maxwell's equations is given by

(3.40)

where

(3.41)

V is the effective volume of a cavity, k is the wave number and q(t) is a time-dependant factor having the dimension of length.

The magnetic field inside the cavity is given by

(3.42)

where

(3.43)

that is the canonical momentum for a ‘article’ of unit mass. The Hamiltonian H, or the classical field energy, is given by

(3.44)

where

(3.45)

In the same way, it can be shown that the free-field Hamiltonian in operator form is given by

(3.46)

where the operators of p and q satisfy the canonical commutation relation

(3.47)

It is now convenient to introduce

(3.48)

which are respectively the non-Hermitian annihilation and creation operators. From these equations it can be shown that

(3.49)

and therefore

(3.50)

As a result, the free-field Hamiltonian operator takes the form

(3.51)

because by definition

(3.52)

and because the creation and annihilation operator satisfy the commutation relation

(3.53)

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