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Approximate Solution of Hill's Equation

### Abstract

A modified WKB approximation, amenable to successive corrections, to the solution of the linear differential equation of second order having a periodic coefficient in normal form is presented. Considered as an application of the related equation method, it uses the free‐particle wave equation rather than, as in an alternative approach, Mathieu's equation. Particular attention is given the instances, where two simple turning points appear in the period and where there are no turning points. With respect to the one‐dimensional crystal, it is shown how the energy band structure can be gleaned directly from the given periodic potential.

© 1966 The American Institute of Physics

Received 01 February 1965
Published online 09 December 2004

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