Wave equations in momentum space

Abstract

The non-relativistic and relativistic wave equations of a particle are solved in momentum space for different kinds of fields. It is assumed that the potential, when defined in co-ordinate space, is central; but the method applies also to more general forms of momentum space potentials. First, the general integral equations which correspond, in momentum space, to Schrödinger and Dirac equations are derived, and the angular variables separated. Then, a general method of solution of these integral equations is given, which lies mainly on a transformation into a four-dimensional hyperspace. These results are (relativistically and non-relativistically) applied rigorously to the Coulombian field (hydrogen atom) and to the scalar Yukawa potential. A more general case is further investigated. In this, the momentum space potential can be expanded in a series of powers of the modulus of the momentum (this last application involves a cut-off procedure). A mathematical appendix deals with some properties of the hyperspherical harmonics (especially the integral equation of these functions) and with the Fourier transform of some kinds of central potentials.