Energy levels of a quantal particle enclosed in N identical, mirror symmetric wells of a periodic potential. Numerical test of phase-integral formulas

Abstract

An interesting structure prevails for the energy levels of a quantal particle in a periodic potential with N (⩾2) mirror symmetric wells separated by N−1 mirror symmetric barriers, when the logarithmic derivative of the wave function is given at corresponding (periodically and mirror symmetrically situated) points in the barrier to the left of the first well and in the barrier to the right of the Nth well. It is shown that the quantization conditions that one obtains for these energy levels by means of a careful and rigorous phase-integral treatment are capable of giving extremely accurate results. The accuracy obtainable is demonstrated for N=3 by comparison with numerically exact results, which were obtained by means of the extended version of the phase-amplitude method presented in an Appendix. In the concluding section we summarize the results and point out unexpected features of the energy spectrum and the wave functions. Two different boundary conditions, commonly used in the theory of crystals, and closely related to the present investigation, are also discussed there.