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.

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

1. http://www.scielo.org.mx/pdf/rmf/v54n6/v54n6a9.pdf G.F. Torres del Castillo, E. Navarro Morales. Bound states of the hydrogen atom in parabolic coordinates. REVISTAMEXICANADEF ́ISICA54(6)454–458.
2. http://mathworld.wolfram.com/ParabolicCoordinates.html
3. https://en.wikipedia.org/wiki/Parabolic_coordinates