
In
mathematics, a Coulomb wave function is a solution of the Coulomb wave equation, named after
Charles-Augustin de Coulomb
Charles-Augustin de Coulomb (; ; 14 June 1736 – 23 August 1806) was a French officer, engineer, and physicist. He is best known as the eponymous discoverer of what is now called Coulomb's law, the description of the electrostatic force of att ...
. They are used to describe the behavior of
charged particle
In physics, a charged particle is a particle with an electric charge. It may be an ion, such as a molecule or atom with a surplus or deficit of electrons relative to protons. It can also be an electron or a proton, or another elementary particle ...
s in a
Coulomb potential
The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as the amount of work energy needed to move a unit of electric charge from a reference point to the specific point in ...
and can be written in terms of
confluent hypergeometric function
In mathematics, a confluent hypergeometric function is a solution of a confluent hypergeometric equation, which is a degenerate form of a hypergeometric differential equation where two of the three regular singularities merge into an irregular ...
s or
Whittaker function
In mathematics, a Whittaker function is a special solution of Whittaker's equation, a modified form of the confluent hypergeometric equation introduced by to make the formulas involving the solutions more symmetric. More generally, introduced W ...
s of imaginary argument.
Coulomb wave equation
The Coulomb wave equation for a single charged particle of mass
is the
Schrödinger equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...
with
Coulomb potential
The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as the amount of work energy needed to move a unit of electric charge from a reference point to the specific point in ...
:
where
is the product of the charges of the particle and of the field source (in units of the
elementary charge
The elementary charge, usually denoted by is the electric charge carried by a single proton or, equivalently, the magnitude of the negative electric charge carried by a single electron, which has charge −1 . This elementary charge is a funda ...
,
for the hydrogen atom),
is the
fine-structure constant
In physics, the fine-structure constant, also known as the Sommerfeld constant, commonly denoted by (the Greek letter ''alpha''), is a fundamental physical constant which quantifies the strength of the electromagnetic interaction between ele ...
, 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
In mathematics, a confluent hypergeometric function is a solution of a confluent hypergeometric equation, which is a degenerate form of a hypergeometric differential equation where two of the three regular singularities merge into an irregular ...
,
and
is the
gamma function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except th ...
. 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
In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields.
Since the spherical harmonics for ...
:
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
In mathematics, a confluent hypergeometric function is a solution of a confluent hypergeometric equation, which is a degenerate form of a hypergeometric differential equation where two of the three regular singularities merge into an irregular ...
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
In mathematics, a self-adjoint operator on an infinite-dimensional complex vector space ''V'' with inner product \langle\cdot,\cdot\rangle (equivalently, a Hermitian operator in the finite-dimensional case) is a linear map ''A'' (from ''V'' to it ...
(the
hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
) with different eigenvalues.
Further reading
*.
*
*.
References
{{Reflist
Special hypergeometric functions