Prakash Parajuli
Konig-Penny model
Consider a periodic square well potential as shown in the fig 1.
Figure 1 Square w ell potential
The Schrodinger wave equation is,
Where
is the potential energy and E is the energy eigenvalue. is
In the region,
Where,
Now in the region
Where,
in which
.
, with
1
Prakash Parajuli
Konig-Penny model
According to Bloch theorem, the solution in the region solution in the region by following relationship.
Where
is connected with the
and defines the wavevector used as an index to label the solution.
Now applying boundary boundary conditions for
to be continuous at
Rearranging equations,
For non trivial solution,
2
.
Prakash Parajuli
Apply,
Apply,
Apply,
Konig-Penny model
Apply,
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Prakash Parajuli
Konig-Penny model
Now, reduced determinant determinant is,
Apply,
Apply,
Now, reduced determinant determinant is,
4
Prakash Parajuli
Konig-Penny model
Rearranging,
Taking determinant,
or,
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Prakash Parajuli
or,
or,
Konig-Penny model
or,
or,
or,
Finally rearranging,
6
Prakash Parajuli
Konig-Penny model
Now, to simplify the equation, let us consider consider potentials potentials to be periodic delta function by taking with
equation becomes,
being finite quantity. This makes makes
. The final
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5 1
Plot for
.
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