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The Kronig-Penney model is an exactly solvable quantum mechanics model showing how a periodic potential produces energy bands and band gaps. R. de L. Kronig and W. G. Penney began with rectangular potential barriers and then took a narrow-barrier limit. This makes the relation between repeated quantum scattering and electronic band structure explicit.^[ref]

In the delta potential version, a particle of mass $m$ moves inThe lattice spacing $a$ has units of length; the strength $V_0$ has units of energy times length. The wavefunction is continuous, but integrating the Schrödinger equation across a barrier givesBetween barriers the wavefunction is a combination of plane waves. Bloch's theorem requires translation by one cell to multiply it by a phase $e^{iqa}$, where $q$ is the crystal wavenumber. Matching across one cell yields the Floquet discriminant of the delta-comb Kronig-Penney model, which decides whether a real $q$ is possible. With $V_0=0$ the band gaps close and the free particle dispersion is recovered; finite barriers open forbidden intervals through coherent repeated scattering.

By R. de L. Kronig and W. G. Penney. Proceedings of the Royal Society A 130, 499–513. Section 1, pp. 500–503, derives the periodic rectangular-barrier model, its narrow-barrier limit and the allowed/forbidden energy intervals; the printed received date is 13 November 1930.

Original paper scan: www.physics.umd.edu/courses/Phys402/AnlageFall21/Kronig-Penney%20Model%201931-Quantum%20mechanics%20of%20electrons%20in%20crystal.pdf

R. de L. Kronig, with W. G. Penney, constructed the Kronig-Penney model for electron motion through a periodic crystal potential. The model turns the general periodicity argument behind Bloch's theorem into an explicit calculation of allowed energies.^[ref]

A periodic potential can exclude intervals of energy even when an electron moves freely inside each simplified region. This links wave matching to a solid's band gaps, while retaining an approximation to the many-electron crystal problem.

W. G. Penney developed a solvable model of electrons in a periodic lattice with R. de L. Kronig. Their 1931 calculation replaced the complicated crystal environment by a repeating one-dimensional potential, exposing allowed and forbidden electron-energy intervals.^[ref]

The Kronig-Penney model makes energy bands concrete: matching wave functions across repeated cells constrains the possible energies. Its value lies in isolating the effect of periodicity, not reproducing every interaction in a real solid.