Effects of Finite Boundaries on a One-Dimensional Harmonic Oscillator

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Abstract
In the study of the effects of finite boundaries on the magnetic properties of a solid, one encounters the problem of finding the energy eigenvalues of a one-dimensional linear harmonic oscillator located in a potential enclosure. The WKB method is employed to solve this problem for an oscillator whose center is fixed at some arbitrary position inside a potential box. Both numerical and analytical approximations of the WKB method are employed to find the energy eigenvalues over a wide range of the parameters of the problem. Numerical methods are also used on the exact series solutions of a bound oscillator to find the exact eigenvalues for the first few quantum states. A comparison of the two methods show that, in general, the WKB eigenvalues are accurate to much better than 1% except when the classical turning points are near the wall. Here the difference is of the order of 10% for the worst possible cases. The eigenvalues of a bound oscillator are shown to reduce to unbound-oscillator energy eigenvalues if the classical turning points are inside the potential enclosure and not near the walls. At the other extreme the eigenvalues are shown to become plane wave box eigenvalues when the separation of the classical turning points is large compared to the size of the enclosure. Also included is the relationship of the WKB solution to the Bohr-Sommerfeld quantization rule for the bound oscillator.

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