Coulomb Wave Equation

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 $m$ is the Schrödinger equation with Coulomb potential
:$\left(-\hbar^2\frac+\frac\right) \psi_(\vec) = \frac \psi_(\vec) \,,$
where $Z=Z_1 Z_2$ is the product of the charges of the particle and of the field source (in units of the elementary charge, $Z=-1$ for the hydrogen atom), $\alpha$ is the fine-structure constant, and $\hbar^2k^2/(2m)$ is the energy of the particle. The solution – Coulomb wave function – can be found by solving this equation in parabolic coordinates
:$\xi= r + \vec\cdot\hat, \quad \zeta= r - \vec\cdot\hat \qquad (\hat = \vec/k) \,.$
Depending on the boundary conditions chosen, the solution has different forms. Two of the solutions are
:$\psi_^(\vec) = \Gamma(1\pm i\eta) e^ e^ M(\mp i\eta, 1, \pm ikr - i\vec\cdot\vec) \,,$
where $M(a,b,z) \equiv _1\!F_1(a;b;z)$ is the confluent hypergeometric function, $\eta = Zmc\alpha/(\hbar k)$ and $\Gamma(z)$ is the gamma function. The two boundary conditions used here are
:$\psi_^(\vec) \rightarrow e^ \qquad (\vec\cdot\vec \rightarrow \pm\infty) \,,$
which correspond to $\vec$-oriented plane-wave asymptotic states ''before'' or ''after'' its approach of the field source at the origin, respectively. The functions $\psi_^$ are related to each other by the formula
:$\psi_^ = \psi_^ \,.$

Partial wave expansion

The wave function $\psi_(\vec)$ can be expanded into partial waves (i.e. with respect to the angular basis) to obtain angle-independent radial functions $w_\ell(\eta,\rho)$. Here $\rho=kr$.
:$\psi_(\vec) = \frac \sum_^\infty \sum_^\ell i^\ell w_(\eta,\rho) Y_\ell^m (\hat) Y_^ (\hat) \,.$
A single term of the expansion can be isolated by the scalar product with a specific spherical harmonic
:$\psi_(\vec) = \int \psi_(\vec) Y_\ell^m (\hat) d\hat = R_(r) Y_\ell^m(\hat), \qquad R_(r) = 4\pi i^\ell w_\ell(\eta,\rho)/r.$
The equation for single partial wave $w_\ell(\eta,\rho)$ can be obtained by rewriting the laplacian in the Coulomb wave equation in spherical coordinates and projecting the equation on a specific spherical harmonic $Y_\ell^m(\hat)$
:$\frac+\left(1-\frac-\frac\right)w_\ell=0 \,.$
The solutions are also called Coulomb (partial) wave functions or spherical Coulomb functions. Putting $z=-2i\rho$ changes the Coulomb wave equation into the Whittaker equation, so Coulomb wave functions can be expressed in terms of Whittaker functions with imaginary arguments $M_(-2i\rho)$ and $W_(-2i\rho)$. The latter can be expressed in terms of the confluent hypergeometric functions $M$ and $U$. For $\ell\in\mathbb$, one defines the special solutions
:$H_\ell^(\eta,\rho) = \mp 2i(-2)^e^ e^\rho^e^U(\ell+1\pm i\eta,2\ell+2,\mp 2i\rho) \,,$
where
:$\sigma_\ell = \arg \Gamma(\ell+1+i \eta)$
is called the Coulomb phase shift. One also defines the real functions
:$F_\ell(\eta,\rho) = \frac \left(H_\ell^(\eta,\rho)-H_\ell^(\eta,\rho) \right) \,,$
:$G_\ell(\eta,\rho) = \frac \left(H_\ell^(\eta,\rho)+H_\ell^(\eta,\rho) \right) \,.$
In particular one has
:$F_\ell(\eta,\rho) = \frac\rho^e^M(\ell+1+i\eta,2\ell+2,-2i\rho) \,.$
The asymptotic behavior of the spherical Coulomb functions $H_\ell^(\eta,\rho)$, $F_\ell(\eta,\rho)$, and $G_\ell(\eta,\rho)$ at large $\rho$ is
:$H_\ell^(\eta,\rho) \sim e^ \,,$
:$F_\ell(\eta,\rho) \sim \sin \theta_\ell(\rho) \,,$
:$G_\ell(\eta,\rho) \sim \cos \theta_\ell(\rho) \,,$
where
:$\theta_\ell(\rho) = \rho - \eta \log(2\rho) -\frac \ell \pi + \sigma_\ell \,.$
The solutions $H_\ell^(\eta,\rho)$ correspond to incoming and outgoing spherical waves. The solutions $F_\ell(\eta,\rho)$ and $G_\ell(\eta,\rho)$ 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 $\psi_^(\vec)$
:$\psi_^(\vec) = \frac \sum_^\infty \sum_^\ell i^\ell e^ F_\ell(\eta,\rho) Y_\ell^m (\hat) Y_^ (\hat) \,,$

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
:$\int_0^\infty R_^\ast(r) R_(r) r^2 dr = \delta(k-k')$
Other common normalizations of continuum wave functions are on the reduced wave number scale ($k/2\pi$-scale),
:$\int_0^\infty R_^\ast(r) R_(r) r^2 dr = 2\pi \delta(k-k') \,,$
and on the energy scale
:$\int_0^\infty R_^\ast(r) R_(r) r^2 dr = \delta(E-E') \,.$
The radial wave functions defined in the previous section are normalized to
:$\int_0^\infty R_^\ast(r) R_(r) r^2 dr = \frac \delta(k-k')$
as a consequence of the normalization
:$\int \psi^_(\vec) \psi_(\vec) d^3r = (2\pi)^3 \delta(\vec-\vec') \,.$

The continuum (or scattering) Coulomb wave functions are also orthogonal to all Coulomb bound states
:$\int_0^\infty R_^\ast(r) R_(r) r^2 dr = 0$
due to being eigenstates of the same hermitian operator (the hamiltonian) with different eigenvalues.

Further reading

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References

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Special hypergeometric functions