So I have previously learned how to discretize the Schrödinger equation on the form:(adsbygoogle = window.adsbygoogle || []).push({});

(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?

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# I How to discretize the Schrödinger equation with spin

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