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]

Same thing you posted at https://www.physicsforums.com/showthread.php?t=366702 ...

 

[kerimek]

I can understand the confusion and difficulty in solving for the eigenvalues and eigenfunctions of a Hamiltonian that depends on both orbital and spin operators. This is a complex problem that requires a deep understanding of quantum mechanics and mathematical techniques.

One approach to solving this problem could be to use a technique called the "direct product rule," which allows us to combine the orbital and spin state spaces into a larger, combined state space. From there, we can use traditional methods for finding eigenvalues and eigenfunctions.

Another approach could be to use perturbation theory, which can help us approximate solutions for complex Hamiltonians that cannot be solved exactly. This method involves expanding the Hamiltonian in a series and solving for the eigenvalues and eigenfunctions at each successive order.

Ultimately, successfully solving this problem will require a combination of knowledge, mathematical skills, and creativity. It may also be helpful to consult with colleagues or experts in the field for additional insights and approaches. Good luck with your research!