Phy5645/Square Wave Potential Problem

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Let us once again confine our attention to the region, The wave function for this region is given by

where and

By Bloch's theorem, the full wave function must have the form,

Continuity of and at requires that

and

The periodicity of and continuity of and at gives us

and

We have thus obtained four linear equations in and To derive the condition under which these equations have a nontrivial solution, we first eliminate and and then determine when the resulting system has nontrivial solutions; this yields the condition,

where is the width of the "barrier" parts of the potential. This, along with the equation,

yields the energy spectrum of the system.

If we take the limit and in such a way as to keep finite, then we can obtain:

In this limit,

and

Our equations then reduce to, noting that in this limit,

This is just the equation that we obtained for the Dirac comb potential; note that, here, stands for the in the Dirac comb problem described earlier.

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