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

is the Schrödinger equation with Coulomb potential

is the product of the charges of the particle and of the field source (in units of the elementary charge is the fine-structure constant 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

is the confluent hypergeometric function is the gamma function. The two boundary conditions used here are

-oriented plane-wave asymptotic states before or after are related to each other by the formula

Partial wave expansion

.

A single term of the expansion can be isolated by the scalar product with a specific spherical harmonic

can be obtained by rewriting the laplacian in the Coulomb wave equation in spherical coordinates and projecting the equation on a specific spherical harmonic

changes the Coulomb wave equation into the Whittaker equation. The latter can be expressed in terms of the confluent hypergeometric functions, one defines the special solutions

where

is called the Coulomb phase shift. One also defines the real functions

In particular one has

is

where

are proportional to Spherical Bessel functions are proportional to Spherical Hankel functions

and are normalized same as Spherical Bessel functions

and similar for other 3.

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

-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