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