# Hamiltonian for a Hydrogen atom

• FloridaGators

#### 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?

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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.

FloridaGators said:
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.

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

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