I'd like to write a small program for solving the 1D Schroedinger equation with Numerov's algorithm in the general case. I've ran into a perplexity though: seeing how the standard bisection convergence criterion for Numerov's algorithm relies on changes of sign of the deviation of the wavefunction from the boundary value, what happens if there is a, say, doubly degenerate eigenvalue? Shouldn't that mean that there are two sign changes for the same E value, leading to the eigenvalue being undetectable? If that's not the case, how can I obtain the two separate eigenfunctions as a solution rather than an arbitrary linear combination of the two? Is there any change/improvement I could use to still make use of Numerov's algorithm in these situations, rather than having to use a basis set and matrix diagonalization approach?
Oh, this is interesting. I would have thought that with, say, a double well symmetric potential, there would be two ground states reflecting the two distinct minima. Does it mean these would actually be the same state?
Interesting, thanks a lot! I have to check out this Messiah theorem. I guess I can add a symmetry detection feature, and make use of the symmetry if needed. Maybe I can relate the 'accuracy' of the symmetry to the step with which I should check for eigenvalues too.
Indeed the is nothing to add to comments made by Giovanni Garberoglio only one simple addition. Besides a double well there may also be a triple well etc. up to a periodic potential complicating things a bid more.
I give you the good advise to have a look at http://www.aip.org/cip/pdf/vol_11/iss_5/514_1.pdf
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