Turning now from classical mechanics to quantum mechanics, a free photon can be represented by a wave function , which satisfies the equation

(7.97) |

A plane-wave solution to this equation is

(7.98) |

where is the four-momentum and is the unit polarization four-vector of the photon. We will discuss the polarization and normalization presently.

On substituting into the equation, we fine satisfies

(7.99) |

The polarization vector has four components and yet it describes a spin-1 particle. The Lorentz condition , gives

(7.100) |

and this reduces the number of independent components of to three. Moreover, we have to explore the consequences of the additional gauge freedom. Choose a gauge parameter

(7.101) |

with constant so that is satisfied. This, together with the solution for shows that the physics is unchanged by the replacement

(7.102) |

In other words, two polarization vectors which differ by a multiple of describe the same photon. We may use this freedom to ensure that the time component of vanishes, ; and then the Lorentz condition reduces to

(7.103) |

This (noncovariant) choice of gauge is known as the Coulomb gauge.

We see that there are only two independent polarization vectors and that they are both transverse to the three-momentum of the photon. For example, for a photon traveling along the -axis, we may take

(7.104) |

These are referred to as linear polarization vectors. The linear combinations

(7.105) | |||

(7.106) |

are called circular polarization vectors. A free photon is thus described by its momentum and a polarization vector . Since transforms as a vector, we anticipate that it is associated with a particle of spin-1. If were along , it would be associated with a helicity-zero photon. This state is missing because of the transversability condition . It can only be absent because the photon is massless.

In a special Lorentz frame is pure space-like, , with . In an arbitrary Lorentz frame is space-like and normalized to . The normalization constant of the plane wave for a photon is chosen such that the energy in the wave is just (ie. for a single photon). To verify this, we compute

(7.107) |

This if for the Heaviside-Lorentz system of units. In the Gaussian system of units the constant in front of the integral would be . Since ( in the Lorentz gauge)

(7.108) |

(7.109) |

and

(7.110) |

Thus and

(7.111) |

where we have taken the time average.

The completeness relationship for polarization vectors is

(7.112) |

2004-03-18