﻿ The Formulation of a Hamiltonian Model of the Interaction of a Particle Field with an Electromagnetic Field
San José State University

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Thayer Watkins
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The Formulation of a Hamiltonian Model
of the Interaction of a Particle Field
with an Electromagnetic Field

The system being considered is two quantized field and their interaction. The Hamiltonian H is then composed of three parts; i.e,,

#### H = HEM + Hpart + HInter

Since the objects are fields their Hamiltonians are the integrals over all space of their Hamiltonian densities, which in this case are same as their energy densities. In the case of the electromagnetic field the energy density at a point in space is given by

#### (E² + B²)/8π

where E is the electric field intensity at that point in space and B is the magnetic field intensity at the same point.

Thus

#### HEM = ∫dx³(E² + B²)/8π

For a particle in a potential V the Hamiltonian operator is

#### −(h²/(2m))∇² + V

where m is the mass in the particle field.

If φ is the wave function for the particle field and φ* is its conjugate then the Hamiltonian for the particle field is

#### Hpart = ∫dx³φ*[−(h²/(2m))∇² + V]φ

The interaction of the EM field and the particle field is best described in terms of the vector potential of the EM field. The vector potential A is such that

#### B = ∇×A E = −(∂A/∂t)

This is known as the Coulomb gauge for the EM field.

The Hamiltonian operator for the interaction is then given by

and hence

#### HInter = ∫dx³φ*[−(eh/(imc))A·∇ + (e²/(2mc²))A·A]φ

where e is the charge of the particle field, i is the square root of negative one and c is the speed of light.

The interaction term is derived from replacing (h/i) in the Hamiltonian operator with (h/i)−(e/c)A which gives the Hamiltonian as

#### H = ∫dx³φ*[(1/2m)|(h/i)−(e/c)A|² + V]φ + ∫dx³(E²+B²)/8π

When the term |(h/i)−(e/c)A|² is expanded the terms involving A become part of the expression for the interaction Hamiltonian and the others become part of the field particle Hamiltonian.