# 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 ) .
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.

### PERMANENT CITATION

 Share: Embed Interactive Demonstration New! Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details » Download Demonstration as CDF » Download Author Code »(preview ») Files require Wolfram CDF Player or Mathematica.

• Kronig-Penney Model with Dirac CombS. M. Blinder
• Topological Winding Number in 1D Su-Schrieffer-Heeger ModelJessica Alfonsi
• Self-Similar Qubistic Plot of the S_z=0 Half-Filled Hubbard Model Basis StatesJessica Alfonsi
• Electron in a Nanocrystal Modeled by a Quantum Particle in a SphereYezhi Jin, Kyle Smola, Rahil Ukani
• Hydrogen Atom Radial FunctionsPorscha McRobbie and Eitan Geva
• Plots of Quantum Probability Density Functions in the Hydrogen AtomCarlos Rodríguez Fernández and Andrés Santos
• Wigner Function of Harmonic OscillatorSamira Bahrami
• Wigner Function of Two-Dimensional Isotropic Harmonic OscillatorJens Peder Dahl and Wolfgang P. Schleich
• Wigner Function of a Canonical Ensemble of Harmonic Oscillators at a Given TemperatureSamira Bahrami
• Zero-Energy Limit of Coulomb WavefunctionsS. M. Blinder