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I'm trying to solve the 1D Schrodinger equation for an arbritary potential, to calculate Franck Condon factors for absorption and emmision spectra. I can do this using iterative techniques (e.g. the Numerov method), but I can't seem to get it to work by discretrizing the hamiltonian, and then finding the eigen vectors/values. Here's what I'm doing, hopefully someone can help.

initial hamiltonian : [tex]H=\frac{1}{2 I}\frac{d^2}{dx^2} + W(x) [/tex]

Split the space up into n points.

use the standard finite difference relations for a second derivative leads to:

[tex]\frac{1}{2I}\frac{v_{i-1}-2v_{i}+v_{i+1}}{\delta^2}+W_i v_i = E v_i[/tex]

and the relevent forward and backward differences for the endpoints.

I then set up a matrix for the above linear equations, and then just solve the system using either Matlab or Mathematica eigenvalue and eigenvector routines.

The problem is that the resulting eigenvalues ( energies ) are completely wrong, and the wavefunctions ( individual eigenvectors ) are weird. It seems that the eigenvectors oscillate around around y=0. It's difficult to explain, but if the desired wavefunction is phi, then at even grid points (n=2,4,6...) , I get the correct phi(x) value, but for odd grid points (n=1,3,5,7...), I get -phi(x). This is easily corrected, but why is it happening?

Is it possible to calculate the wavefunction using this method? I may have made a coding error, but I'm interested to know if the method above should work, and if not, why not.

Thanks in advance

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# Solution of 1D Schrodinger using discretized hamiltonian

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