Hamiltonian being a function of either orbital and spin operators

[go quantum!]

Homework Statement

The title presents my problem. I know in principle how to find eigenvalues and eigenfunctions of the Hamiltonian if it depends only on orbital operators or in spin operators. On the other hand I have no clue how to solve it if there are both types of operators.

The Attempt at a Solution

The complete state space will be the cartesian (or direct?!) product of the orbital state space with the spin state space. Nevertheless, I have no idea how the hamiltonian will act on that complete space.

 

[diazona]

Are you talking about something like this?
$$H = \vec{L}\cdot\vec{S}$$
In that case, as you said, you represent the states as products of a spatial wavefunction with a spinor,
$$\vert\psi\rangle\vert\chi\rangle$$
The orbital angular momentum acts on only the spatial wavefunction $\vert\psi\rangle$ and the spin operator acts on only the spinor $\vert\chi\rangle$.
$$H\vert\psi\rangle\vert\chi\rangle = \bigl(\vec{L}\vert\psi\rangle\bigr)\cdot\bigl(\vec{S}\vert\chi\rangle\bigr)$$

If you have a particular example in mind, why don't you post it.