Phase‐integral approach to quantum‐mechanical tunneling

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 Δ n 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 Δ n 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.