The Kronig–Penney model for an electron in a one-dimensional crystal can be solved in closed form for a sinusoidal potential of wavelength
. Consider the Schrödinger equation (in atomic units
)
.
This can be put in the form of Mathieu's differential equation
,
with the parametrization
,
,
.
The periodic solutions of Mathieu's equation are the even functions
(MathieuC[a,q,z]) with the characteristic values
(MathieuCharacteristicA[n,q]), with
and the odd functions
(MathieuS[b,q,z]) with the characteristic values
(MathieuCharacteristicB[n,q]), with
.
The normalized even-parity solutions to the Schrödinger equation are then given by
,
, 
while the odd-parity solutions are
,
,
.
Click the button "wavefunctions" to see the eigenfunctions for selected values of
. The potential energy
is shown as a dashed curve (not to scale).
Click the button "energy bands" to see the patterns of allowed energy values, colored gray, for selected values of
and
. The red vertical line indicates the value of
. For a given value of
, the allowed values lie between the curves for
and
. For energy values outside these regions, the corresponding wavefunctions are not real periodic functions of
, and thus not acceptable. We have thereby a rudimentary simulation of the band structure of crystals.
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