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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