How to Derive the Hydrogen Atom Hamiltonian in Spherical Coordinates?

[FloridaGators]

The Hamiltonian for a Hydrogen atom in Cartesian Coordinates (is this right?):
\hat{H} = - \frac{\bar{h}^2}{2m_p}\nabla ^2_p - \frac{\bar{h}^2}{2m_e}\nabla ^2_e - \frac{e^2}{4\pi\epsilon _0r}
In Spherical Coordinates do I just use:
x=r sin θ cos φ, y = r sin θ sin φ, and z = r cos θ?

 

[nickjer]

It is \hbar instead of h. But that is essentially correct. You might want to convert it to the center of mass reference frame before you do any work on it though. There are tons of sites out there that solve it as well and show all the work.

 

[gabbagabbahey]

The Hamiltonian for a Hydrogen atom in Cartesian Coordinates (is this right?):
\hat{H} = - \frac{h^2}{2m_p}\nabla ^2_p - \frac{h^2}{2m_e}\nabla ^2_e - \frac{e^2}{4\pi\epsilon _0r}

First, this form has no explicit reference to Cartesian coordinates.

Second, this is only correct if you define r to be the separation between the proton and the electron; not the distance from the origin.

In Spherical Coordinates do I just use:
x=r sin θ cos φ, y = r sin θ sin φ, and z = r cos θ?

There are many sites and texts that derive expressions for \nabla^2 in Spherical coordinates.

 

[nickjer]

Just google it. But here is a site that explains things in a lot of detail.

http://www.indiana.edu/~ssiweb/C561/PDFfiles/HAtom2008.pdf

 

[FloridaGators]

Thank you for helping. Do you mind my asking what your search inquiry in google was to find that?