Electromagnetic Field and Photons

Abstract

The probabilistic treatment of the electromagnetic field in vacuum would have to be based on the results of Chapter 7 dealing with random fields. But the definition of a random field there proceeds, as should be the case in probabilistics, from the notion of a continuous concrete object regarded as the carrier of a field. This notion is, however, irrelevant to the case under consideration. Hence, it would be illogical to apply the results of Chapter 7 to the electromagnetic field, even though it obeys Maxwell equations, which allow one to think of it as a Hamiltonian random field. The equations of the electromagnetic field in vacuum can be reduced to the form given in Eqn. (7.46). This leads, when making use of the quantum approach, to the expressions of the form given in Eqn. (7.77) for the field energy values, which, in view of Eqn. (7.78),\(\alpha = \hbar ,{\text{ and }}{\lambda _j} = \omega _j^2{\text{ (}}{\omega _j} \) (ω j is the circular frequency of the corresponding plane electromagnetic wave), yields $$E = \mathop \sum \limits_j ({n_j} + \frac{1}{2})\hbar {\omega _j}. $$ (12.1) .