Note on the Interaction of an Electron and a Lattice Oscillator

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Abstract
The interaction of an electron and a lattice oscillator is studied for an interaction energy of a special type linear in the oscillator coordinates and momenta. The energy values and eigenfunctions for arbitrary coupling strength are found by solving a three-term recurrence relation. A plot of energy vs total momentum of electron plus oscillator reveals the role of degeneracies of states involving different numbers of quanta in the oscillator. As the frequency of the oscillator tends to zero, one finds the bandlike spectrum characteristic of an electron moving in a periodic potential. With increasing total momentum the electron makes Bragg reflections, transferring quanta of energy and momentum to the oscillator, and remaining bounded in velocity. For strong coupling the state of minimum energy is one of nonzero total momentum. For sufficiently strong coupling, regions of small effective electron mass cease to exist.

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