WKB prediction of the energy splitting in the symmetric double well potential

The accuracy of the WKB approximation when predicting the energy splitting of bound states in a double well potential is the main subject of this paper. The splitting of almost degenerate energy levels below the top of the barrier results from the tunneling and is thus supposed to be exponentially small. By using the standard WKB quantization we deduce an analytical formula for the energy splitting, which is the usual Landau formula with additional quantum corrections. We also examine the accuracy of our and Landau formula numerically for the case of the symmetric double well quartic potential. Apart from the perturbative and variational methods the semiclassical or WKB approximation is the most widely used approximation in quantum mechanics to obtain the analytic expressions. The semiclassics serves not only as an approximative tool but it also helps us to picture and to understand the global behaviour of eigen-functions and energy spectra of quantum systems which is particularly important in the context of quantum chaos 1), 2). Since the systematic study of the accuracy of semiclassical approximation is a difficult task, it has been attempted for simple systems, where in a few cases even exact solutions may be worked out 3), 4), 5), 6), 7). In this paper we analyze the energy splitting of doublets in a generic one– dimensional double well potential. The splitting due to the tunneling is supposed to be exponentially small and is thus lost a-priori in the semiclassical approximation by expanding the quantum phase in the ¯ h power series. By using the standard WKB technique we deduce an analytical formula for the energy splitting which is the usual Landau 8) formula with additional quantum corrections and can be formally written as ∆E = A exp − S ¯ h , (1. 1) where S is the usual classical action inside the classically forbidden region (between the two turning points) and A is called the tunneling amplitude, which can be written as an ¯ h power series. This formula is based on a linear approximation of the potential near the turning points and so is inapplicable in the case of the square potentials 9). * First we introduce the basic definitions, then we present the WKB splitting formula based on the higher order semiclassical WKB expansion and finally we study its validity for the quartic potential. We shall demonstrate that our formula is indeed a significant improvement …