Interpretation of the photon: wave-particle duality

A simple model is provided to obtain the space time probability-distribution function of a photon emitted without recoil by an excited system (atom, nucleus, ...) in one dimension. A three-dimensional formulation is not needed for our discussion. A quantum mechanical calculation, using the Heitler method, is employed to obtain the solution. The space time probability-distribution function is not the photon wavefunction. In fact, the area under the space time probability-distribution function is time dependent. It obtains its final value only as t -> infinity. The frequency composition of the photon is found and its time dependence determined to be in accord with the time energy uncertainty principle. In the wave picture, the coherence length of a photon is found to be equal to the distance from the maximum probability-density position in the photon back toward the source to a position where the probability density has decreased to e(-1) of its maximum value. The concept of the coherence length is applied to understand the exponential lifetime curve in the wave picture. This latter measurement is usually explained by saying, in the particle picture, that the photon can appear immediately after formation of the excited state or at a variety of later times according to an exponential probability distribution.