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## The Harmonic Oscillator and the Quantum Field

The quantization of an electromagnetic field can be accomplished in numerous manners. The best place to begin such a derivation is in the classical world. A classical electromagnetic field is governed by Maxwell's equations. It will be assumed that this field exists in free space in which Maxwell's equations have the form:

These equations imply a wave equation for both the electric and magnetic fields. In a closed one dimensional cavity with reflectors at either end, the magnitude of the electric and magnetic fields for a single mode of a linearly polarized field has the form

where , k is the wave number, V is the volume of the cavity, and q(t) is a measure of the field amplitude.[11] The classical hamiltonian can be constructed by integrating the energy density of the electromagnetic field over the volume of the cavity

By substituting eq:q1 and eq:q2 into eq:q3 and integrating, the hamiltonian can be shown to have the form

eq:q4 should be recognized as the hamiltonian for a harmonic oscillator where position and momentum are analogous to E and B, respectively. Therefore, a single mode of an electromagnetic field in a one--dimensional cavity is merely a harmonic oscillator. This equality is the basis of all the theory of the quantum field.

An arbitrary eigenfunction for this system is written as |n>, where n is the eigenvalue for an arbitrary field state. n can also be thought of as the number of photons in the mode that has been selected; however, this analogy's validity depends on the definition of photon---we define a photon to be a quantum of energy of magnitude . n may be any number greater than or equal to zero---a quantum field has an infinite number of states.

A number of results can now be obtained by the analogy of harmonic oscillator and field. The and in the harmonic oscillator are usually redefined in terms of the destruction, , and the creation, , operators:

When and operate on |n>

eq:q5 and eq:q6 clearly show why the names creation and destruction operator are used. In a generic harmonic oscillator, these two operators are merely a way to raise and lower the state of the system. However, these operators have a very physical meaning for the quantum field. When operates on |n> it removes a photon from the mode, and when operates on |n> it adds a photon to the mode.

Using the commutator for these operators, , one can redefine the hamiltonian for this system as:

is an operator that one deals with for all harmonic oscillator systems. It is called the number operator and is usually denoted by , where . Thus, the energy for the field is:

When is plotted against the state number, the well--known simple harmonic oscillator energy level diagram is formed. Each state is equally spaced by the amount, , which is the energy of a single photon with frequency, . Each energy level corresponds to an energy of n photons plus . Therefore, state three which corresponds to a mode with three photons has an energy of .

One of the fundamental results of quantum theory is that at n=0 a harmonic oscillator has a non--zero energy. In quantum field theory, this result implies that an electromagnetic field exists even if there are no photons in the field. This energy is ignored in most field calculations because it is merely a constant; however, its existence is quite important, and we shall see in the next application how the field associated with the ground state gives rise to spontaneous emission.

Next: The Master Equation Up: Theory Previous: Introduction

Andy Antonelli
Wed May 17 14:34:24 EDT 1995