Quantum Photonics by Thomas P. Pearsall

Author:Thomas P. Pearsall
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

5.1 Introduction

The time-independent Schrödinger equation for a particle in the presence of a one-dimensional potential is written:

(5.1)

The total energy is shared between kinetic and potential energies. In Chap. 3, we examined the properties of a particle confined by a constant potential (particle in a well). In this environment, the allowed quantized total energy scales according to the square of the quantum number:

(5.2)

where is the width of the potential well. The dependence of the allowed energies on the quantum number is a function of the geometry of the potential.

In this chapter we will study the behavior of a particle confined by a potential well, the width of which varies as a function of distance.

A simple example of a harmonic oscillator is the movement of a mass connected to a spring. When displaced from equilibrium the spring exerts a restoring force that is proportional to the displacement: . The corresponding potential energy stored in the spring is: . When released, the mass begins to move, and energy oscillates between reservoirs of kinetic and potential contributions with a frequency: .

The vibration of atoms in a crystalline solid is an important example in physics which can be treated accurately by harmonic oscillator analysis.

Maxwell’s equations describe the propagation of a photon as the periodic exchange of energy stored in its electric and magnetic field. We will argue in Sect. 5.4 that photon behavior can be described mathematically by the formalism of the harmonic oscillator. This important principle enables the description of single photon; that is, quantum photon behavior and the quantization of the electromagnetic field. This is the subject of this chapter.

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