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Periodic potential 
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#1
Jan713, 02:04 PM

P: 1,005

Okay so Im solving the SE numerically for different potentials. Amongst those I am trying to find the low energy wave functions for a periodic potential of the form:
V=V0cos(x) Now recall that for a numerical solution, at least the type I am doing, you somehow have to assume that the wave functions tends to zero for large lxl. This is obviously the case for any bound states, which I have been looking at so far. But this one I am not quite sure  I mean yes surely to be at infinity a particle would have to cross an infinite number of potential barriers, so it's intuitive from that perspective that the wave functions are indeed finite. On the other hand, these potential barriers are only finite so I am not quite sure. Can anybody, who have a bit more experience with the solutions to the Schrödinger equation tell me what is correct assume? :) 


#2
Jan713, 03:51 PM

PF Gold
P: 1,165

You are right in both respects; in the minimum of the potential curve one expects that normalizable wave functions may be present, but your potential is periodic, so one expects periodic eigenfunctions (that are not normalizable) as well. I am not sure about this, without actually solving the equation, but both kinds of eigenfunctions seem possible. It may be that localized wave functions will correspond to certain range of energies, and the infinite trains will correspond to the remaining range.



#3
Jan713, 03:59 PM

Sci Advisor
P: 3,593

Strictly speaking you are right that the functions in the Hilbert space should be normalizable.
However, a periodic potential only has a continuous spectrum and therefore it has no normalizable eigenfunctions in the strict sense. Maye you find the wikipedia article on "Bloch waves" helpful. 


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