2.2 Methods of Quantum Mechanics in Electromagnetism

We now revisit Maxwell s equations in the form (2.15), (2.16), (2.17), (2.18) and apply the formalism of Hamiltonian quantum mechanics to characterize their solutions. We start by recapitulating the properties of Hermitian Hamiltonians. In the quantum mechanical description of a single particle in an external potential U( r), the two major unknowns are the particle energy E and a particle scalar ?( r) function. The ? function defines the probability density ?( r) of finding a particle at a certain point in space as ?( r) = ?( r) ². The particle energy and its density function are related by a differential equation, known as Shr dinger equation:

Equation (2.26) is said to define an eigenvalue problem, with an eigenvalue E (also called a conserved property of a state) and an eigenfunction ?( r) (also called a mode or a state). The operator in square brackets is said to define a Hamiltonian:

while the ? function is said to define a state ? _E ?. In this notation (also called Dirac notation) (2.26) takes a more concise form:

In the following, we assume that operator ? is linear (does contain ?( r) explicitly in its definition). We define the dot product between the two states as:

where the star means complex conjugation, and integration is performed over the whole volume...