The Wave Function of the Lyman-Alpha Photon Part I. The Wave Function in the Angular and Linear Momentum Representations

To the best of the writer‘s knowledge no one has given the wave function of a photon emitted in an atomic, molecular, or nuclear transition. In the present paper we derive the wave function in the angular momentum and linear momentum representations for the photon emitted by a non-relativistic hydrogen atom, when the electron of the atom falls from the first excited state to the ground state. This is the simplest transition which produces a photon. A two level model for the atom is used, in which the lower level (the ground state energy) is associated with a non-degenerate wave function and the upper level (the energy of the first excited state) is associated with wave functions corresponding to the four-fold degeneracy of that state. We use a generalization of Dirac‘s method for finding the eigenfunctions in resonance scattering. We find the exact solution of the two-level problem using the exact matrix elements of the interaction. The calculations are finite without renormalization. In the next paper we shall introduce the (x)\vec] \vec{x} -representation and thereby obtain the "position", "shape", and "trajectory" of the photon.