Numerical solution of Schrodinger equation

[arroy_0205]

Suppose for some specific problem (symmetric potential well) the Schroedinger equation is expected to give certain discrete bound states and corresponding eigenfunctions. Now I am trying to obtain the eigenfunctions by numerically solving the equation and plotting the solutions by randomly assigning energy value. In general most of the times my guess of energy value will be wrong but the program (say Mathematica) will give some result (plot for the wave function). What will be interpretation for those results?

The answer may be those are simply wrong eigenfunctions so no interpretation is needed. However I was doing this stupid job and was getting divergent wave functions almost always. But I do not understand why Mathematica was giving divergent plot rather than some arbitrary finite plot. Do you have any clue?

 

[Avodyne]

There is a solution (actually, two linearly independent solutions) of the time-indpendent Schrödinger equation in one dimension for every value of the energy. For bound states, these are only allowed if they are normalizable, and this only happens for certain precise values of the energy (and these are the energy eigenvalues).

 

[quantumlaser]

Suppose for some specific problem (symmetric potential well) the Schroedinger equation is expected to give certain discrete bound states and corresponding eigenfunctions. Now I am trying to obtain the eigenfunctions by numerically solving the equation and plotting the solutions by randomly assigning energy value. In general most of the times my guess of energy value will be wrong but the program (say Mathematica) will give some result (plot for the wave function). What will be interpretation for those results?

The answer may be those are simply wrong eigenfunctions so no interpretation is needed. However I was doing this stupid job and was getting divergent wave functions almost always. But I do not understand why Mathematica was giving divergent plot rather than some arbitrary finite plot. Do you have any clue?

I don't really understand what you are trying to ask. Have you written a program to calculate the eigenvalues and eigenfunctions or are you using someone else's program? The normal technique for numerically solving the 1-D Schrödinger Equation is the Numerov method, is your algorithm based on that technique? If not, you might want to google 'Numerov method' and look at the theory behind the method and maybe even some sample code and check it against your algorithm.