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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:
(ψi+1+ψi-1-2ψi)/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?
(p^2/2m + V)ψ = Eψ
, where the second order derivative is approximated as:
(ψi+1+ψi-1-2ψi)/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?