Higher‐order semiclassical quantization of an asymmetric double minimum potential
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Although the semiclassical quantization condition of a general (asymmetric) double minimum potential has been known for some time it has only been applied to the simpler symmetric case. In this work, new versions of a method for evaluating higher-order semiclassical phase integrals are applied to semiclassical quantization of an asymmetric double minimum potential. The quantization condition is discussed and energy eigenvalues for a model potential are determined in the first- , third- , and fifth-order phase integral approximations. Agreement of three, five, and seven significant digits, respectively, where the exact quantum mechanical eigenenergies are obtained.Keywords:
Semiclassical physics
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Previously proposed quantum conditions are used to calculate the higher eigenvalues of a simple two-dimensional (2D) potential from classical trajectories. A new modification to our earlier work is introduced to deal with problems arising from the complicated behaviour of the caustics in certain cases. Results are in excellent agreement with quantum-mechanical calculations. Our methods are shown to be competitive with quantum calculations in their ease of application for these upper levels. However, it is not possible to calculate certain semiclassical eigenvalues near the escape energy because corresponding trajectories are ergodic.
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WKB approximation
Semiclassical physics
Eigenfunction
Brillouin zone
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