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So, I am trying to solve Schrödinger Equation for Dirac delta potential.

The Schrödinger equation: $$ -\frac{\hbar^2}{2m}\frac{d^2\Psi(x)}{dx^2} + V(x)\Psi(x) = E\Psi(x) $$ And, the potential looks like: $$ V(x) = \sum_{i=1}^P \sigma_i\delta(x-x_i) $$

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  • $\begingroup$ If the number of delta functions goes to infinity, this is a slightly generalized version of the Dirac Comb - that might help in your search. $\endgroup$ – J. Murray Feb 26 '18 at 16:12
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    $\begingroup$ This is essentially a boundary matching problem $\endgroup$ – By Symmetry Feb 26 '18 at 16:34
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    $\begingroup$ The double delta potential is treated in plenty of online resources and textbooks, and the tools to expand from two to $n$ deltas are identical to those used in an $n$-piece piecewise-constant potential. So: what exactly do you want to know? What is it about the procedure that is causing you trouble? $\endgroup$ – Emilio Pisanty Feb 26 '18 at 16:41
  • $\begingroup$ Related. $\endgroup$ – Cosmas Zachos Feb 26 '18 at 16:44
  • $\begingroup$ thank you for your responses. I am taking Quantum Mechanics for the first time and in book of Stephen Gasiorowicz, I couldn't find anything similar to this or even in wikipedia. My problem is that I don't know how to manipulate this V(x) potential to solve the differential equation. $\endgroup$ – Zulfidin Khodzhaev Feb 26 '18 at 17:09
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This is the Kronig-Penney model. Solve the Schrödinger equation between two Dirac peaks (easy: the particle is free) then write the continuity of the wavefunction. Integrate the Schrödinger equation around a Dirac peak (i.e. over $[x_i-\epsilon;x_i+\epsilon]$ to show that $\psi'$ is not continuous and that the discontinuity is proportional to $\sigma_i$. With the continuity of $\psi$, you now have enough equations to determine all coefficients.

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