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Solutions to the following equation (from Kronig-Penney)?

  • Thread starter gomboc
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  • #1
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I'm looking to find solutions for K to the following equation which arises in the Kronig-Penney model of a periodic potential:

[tex] \frac{P}{Ka}\sin(Ka) + \cos(Ka) = -1 [/tex]

The equation is such that for a given value of P, there will be a range of K values having a real solution, and a range of K values having no real solution. The idea is to find the boundaries of these ranges. (knowing of course that the trivial solution of K = pi/a is one of them, I'm looking for the other).

I can't think of a good way to do this analytically. If I expand the trig functions into series, I can find the maximum value that can be added to K and still give a solution, but that is not an exact method.

Any ideas would be much appreciated.
 

Answers and Replies

  • #2
39
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Aha.

Upon further study, I've discovered that this is, in fact, a transcendental equation, and thus unsolvable analytically. I shall revert to numerical/graphical methods!
 

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