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, is the fine-structure constant, and is the energy of the particle. The solution – 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 to obtain angle-independent radial functions. Here. A single term of the expansion can be isolated by the scalar product with a specific spherical harmonic The equation for single partial wave can be obtained by rewriting the laplacian in the Coulomb wave equation in spherical coordinates and projecting the equation on a specific spherical harmonic The solutions are also called Coulomb wave functions or spherical Coulomb functions. Putting changes the Coulomb wave equation into the Whittaker equation, so Coulomb wave functions can be expressed in terms of Whittaker functions with imaginary arguments and. The latter can be expressed in terms of the confluent hypergeometric functions and. One defines the special solutions where is called the Coulomb phase shift. One also defines the real functions In particular one has The asymptotic behavior of the spherical Coulomb functions,, and at large is where The solutions correspond to incoming and outgoing spherical waves. The solutions and are real and are called the regular and irregular Coulomb wave functions. In particular one has the following partial wave expansion for the wave function
Properties of the Coulomb function
The radial parts for a given angular momentum are orthonormal. When normalized on the wave number scale, the continuum radial wave functions satisfy Other common normalizations of continuum wave functions are on the reduced wave number scale, and on the energy scale The radial wave functions defined in the previous section are normalized to as a consequence of the normalization The continuum Coulomb wave functions are also orthogonal to all Coulomb bound states due to being eigenstates of the same hermitian operator with different eigenvalues.
