Parabolic coordinates

Parabolic coordinates allow separation of variables in the Schroedinger equation for the hydrogen atom.

In the simplest form, relation of parabolic coordinates \(u,v\) with Cartesian coordinates \(\rho, z\) can be expressed with the following relation:

\(\displaystyle \rho=\sqrt{uv}\)

\(\displaystyle z=\frac{u\!-\!v}{2}\)

The straightforward generalisation to the three-dimensional case, with cartesian coordinates \(x,y,z\) can be expressed with relation

\(\displaystyle x=\rho \cos(\phi)\)

\(\displaystyle y=\rho \sin(\phi)\)

where \(\phi\) is additional, third coordinate. Then \(u,v,\phi\) are interpreted as parabolic coordinates.

Laplacian
Laplacian in parabolic coordinates can be written as follows:

\(\displaystyle \nabla^2= \Delta = \frac{4}{u+v} \Big( \partial_u u \partial_u + \partial_v v \partial_v \Big) + \frac{1}{uv} \partial_\phi^{\,2} \)

This can be verified, transforming the operator in the cylindrical coordinates,

\(\displaystyle \nabla^2= \frac{1}{r} \partial_r r \partial_r + \partial_z^2 + \frac{1}{r^2}\partial_\phi^2\)

The calculus can be done with the Mathematica code below:

Rq = Sqrt[r^2 + z^2]

Simplify[ReplaceAll[ 1/r D[r D[F[Rq+z, Rq-z], r], r] + D[D[F[Rq+z, Rq-z], z], z], {r -> Sqrt[u v], z -> u/2-v/2}], {u>0, v>0}]

that does

\(\frac{4 \left(F^{(0,1)}(u,v)+v F^{(0,2)}(u,v)+F^{(1,0)}(u,v)+u F^{(2,0)}(u,v)\right)}{u+v}\)

Notations
Some sites use different notations; \(u^2\) and \(v^2\) are treated as parabolic coordinates \(u\) and \(v\); and such a notation seems to be more usual .

In such a way, term Parabolic coordinates should be provided at the beginning of each article.

Radius
In the application for atomic physics, the important is coordinate

\(\displaystyle r=\sqrt{x^2+y^2+z^2}\)

In parabolic coordinates, it can be expressed as follows:

\(\displaystyle r=\sqrt{\rho^2+z^2}=\sqrt{\big(\sqrt{uv}\big)^2+\frac{1}{4}(u\!-\!v)^2}=\frac{1}{2}u +\frac{1}{2}v\)

It is assumed, that \(u\!>\!0\) and \(v\!>\!0\).

Hydrogen atom
In the dimensionless variables, the Stationary Schroedinger equation can be written as follows:

\(\displaystyle - \Delta \psi - \frac{2}{r} \psi = \mathcal E \psi \)

The scale of physical coordinates is determined by the Bohr radius

\(\displaystyle \mathrm{BohrRadius}=\frac{\hbar^2}{e^2 M}\approx 5.2917720859 \times 10^{-11}\, \mathrm{Meter} \)

and the scale of physical energy is determined by the Bohr energy

\(\displaystyle \mathrm{BohrEnedry}=\frac{e^4 M}{2\hbar^2}\approx 2.17987197 \times 10^{-18}\, \mathrm{Joule}\)

In parabolic coordinates, the Stationary Schroedinger equation appears as follows:

Referebces
https://en.wikipedia.org/wiki/Parabolic_coordinates

Keywords
Atomic optics, Hydrogen, Quantum mechanics, Schroedinger equation