Phase-integral approach to quantum-mechanical tunneling

Abstract

The phase-integral method is applied to the problem of quantum-mechanical tunneling in one-dimensional double-well potentials. Thus the results for the energy splitting of the nth level Δn up to the seventh-order of phase-integral approximation in the closed form in terms of the complete elliptic integrals are obtained. Furthermore the asymptotic approximations of the complete elliptic integrals to the first-order phase-integral result are used to calculate the quantity W(0) which appears in the expression for Δn obtained applying the LSZ procedure to the instanton-dominated Green function with 2n legs. Thus it is confirmed that the instanton calculus gives a correct value for the energy splitting of the nth level.