Kronig-Penney Model with Mathieu Functions
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 )[more]
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.[less]