Derivation free-field Hamiltonian9
A classical representation of a single-mode field satisfying Maxwell's equations is given by
where
The magnetic field inside the cavity is given by
where
that is the canonical momentum for a ‘article’ of unit mass. The Hamiltonian H, or the classical field energy, is given by
where
In the same way, it can be shown that the free-field Hamiltonian in operator form is given by
where the operators of p and q satisfy the canonical commutation relation
It is now convenient to introduce
which are respectively the non-Hermitian annihilation and creation operators. From these equations it can be shown that
and therefore
As a result, the free-field Hamiltonian operator takes the form
because by definition
and because the creation and annihilation operator satisfy the commutation relation
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