214 Progress of Theoretical Physics Supplement No. 139, 2000 Accuracy of the WKB Approximation: The Case of General Quartic Potential

We analyse the accuracy of the approximate WKB quantization for the case of gen-eral one-dimensional quartic potential. In particular, we are interested in the validity of semiclassically predicted energy eigenvalues when approaching the limit E → ∞, and in the accuracy of low lying energy levels below the potential barrier in the case of generally asymmetric double-well quartic potential. In the latter case, using the standard WKB quan-tization an unnatural localization of eigenstates due to the negligence of tunneling is implied and thus the validity of semiclassics is uncertain. In all computations the higher order cor-rections to the leading semiclassical approximation are included using the complex contour integration technique. We show that these corrections can improve accuracy of semiclassical approximation greatly by many orders of magnitude. The semiclassical approximation is a very useful approximative tool in quantum mechanics, not only because it often yields analytic expressions but also because it helps us to understand the correspondence between quantum features and classical mechanics, that is particularly important in the quantum chaos