Discrete delta-potential method
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The discrete delta potential method is a combination of both numerical and analytic method used to solve the Schrödinger equation the main feature of this method is to obtain first a discrete approximation of the potential in the form:
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V(r) = ∑ V(ai))δ(r − ai) i
where the i index runs over several discrete chosen points of the continuous potential, with this we can solve the SE in the form:
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Φ(s) = ∑ G(s,ai)V(ai)Φ(ai) + γ(s) i
Where G(s, r) is the Green function associated to the free particle Hamiltonian H0, setting s=ai for every i would give a system of linear equation.
For a better approach if potential V(x) is weak we can treat the continuous part of the potential and perform perturbation theory to obtain a better wave function.
If the points are equidistant : | ai + 1 − ai | = h the wave function will depend on h so performing the solution to several h and extrapolating to the value h=0 we could obtain the solution of SE (Schrodinger equation).
This method is valuable when we cannot perform perturbation theory and need some analytic approach to the solution of SE:HΦ = EnΦ