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 (i.e. with respect to the angular basis) 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 (partial) 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 . For , 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 (k-scale), the continuum radial wave functions satisfy 

Other common normalizations of continuum wave functions are on the reduced wave number scale (-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 (or scattering) Coulomb wave functions are also orthogonal to all Coulomb bound states

due to being eigenstates of the same hermitian operator (the hamiltonian) with different eigenvalues.

Further reading
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References

Special hypergeometric functions