Coulomb wave function
In mathematics, a Coulomb wave function is a solution of the Coulomb wave equation, named after Charles-Augustin de Coulomb. They are used to describe the behavior of charged particles in a Coulomb potential and can be written in terms of confluent hypergeometric functions or Whittaker functions of imaginary argument. Coulomb wave equation The Coulomb wave equation for a single charged particle of mass is the Schrödinger equation with Coulomb potential where is the product of the charges of the particle and of the field source (in units of the elementary charge, for the hydrogen atom), is the fine-structure constant, and is the energy of the particle. The solution, which is the Coulomb wave function, can be found by solving this equation in parabolic coordinates Depending on the boundary conditions chosen, the solution has different forms. Two of the solutions are where is the confluent hypergeometric function, and is the gamma function. The two boundary conditions used here are which correspond to -oriented plane-wave asymptotic states before or after its approach of the field source at the origin, respectively. The functions are related to each other by the formula Partial wave expansion The wave function can be expanded into partial waves