Quantum mechanics: Suppose that there is a particle with orbital angular momentum $|L|$. But the particle also has spin quantity $|S|$. The question is, how do I reflect this into Schrodinger equation? I do know how Schrodinger equation becomes for each case - when a particle has particular orbital angular momentum and when a particle has some spin, but not when both occur.
The Schroedinger equation does not describe spin. If you need to describe spin as well, you should use the Pauli equation or the Dirac equation (for spin 1/2).
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I think we can talk about spin and spin interactions with the standard Schrodinger. Start with spin orbit coupling or LS Coupling
Next see the Zeeman effect, and especially Paschen Bach
You need perturbation theory to pick up on spin effects given standard Schrodinger model of the atom as seen on wikipedia: