I How to discretize the Schrödinger equation with spin

[aaaa202]

So I have previously learned how to discretize the Schrödinger equation on the form:
(p^2/2m + V)ψ = Eψ
, where the second order derivative is approximated as:
(ψ_i+1+ψ_i-1-2ψ_i)/2Δx
Such that the whole equation can be translated into a matrix eigenvalue-equation.
The problem is that I am now studying systems with spin of the type shown on the picture, where the spatial terms p^2/2m, V etc. can also enter in the non-diagonal elements of 2x2 matrices.
What is the procedure for discretizing equations of this type, if there is any?

 

[DrClaude]

The wave function now has two components, corresponding to the two spin projections.

You can treat it as a system of two coupled equations, one for each spin component, but this is better suited for the time-dependent Schrödinger equation. For the time-independent case, you can write the discretized wave function as a long vector, containing for example all the values for spin-up at each grid point followed by all the values for spin-down at each grid point. You can also put them in order of grid point, with one element each for each spin component (thinking about, I guess the latter is better as it will give a banded matrix for the Hamiltonian, which is easier/faster to work with).