Coulomb wave function

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 is

${\frac {d^{2}w}{d\rho ^{2}}}+\left(1-{\frac {2\eta }{\rho }}-{\frac {L(L+1)}{\rho ^{2}}}\right)w=0$

where L is usually a non-negative integer. The solutions are called Coulomb wave functions. Putting x = 2iρ changes the Coulomb wave equation into the Whittaker equation, so Coulomb wave functions can be expressed in terms of Whittaker functions with imaginary arguments. Two special solutions called the regular and irregular Coulomb wave functions are denoted by F_L(η,ρ) and G_L(η,ρ), and defined in terms of the confluent hypergeometric function by

$F_{L}(\eta ,\rho )={\frac {2^{L}e^{-\pi \eta /2}|\Gamma (L+1+i\eta )|}{(2L+2)!}}\rho ^{L+1}e^{-i\rho }M(L+1-i\eta ,2L+2,2i\rho )$

References

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