On connection between perturbation theory and semiclassical expansion in quantum mechanics

A. V. Turbiner, E. Shuryak

Published 2021-02-09Version 1

It is shown that for one-dimensional anharmonic oscillator with potential $V(x)= a x^2+\ldots=\frac{1}{g^2}\,\hat{V}(gx)$ (and for perturbed Coulomb problem $V(r)=\frac{\alpha}{r} + \ldots = g\,\tilde{V}(gr)$) the Perturbation Theory in powers of coupling constant $g$ (weak coupling regime) and semiclassical expansion in powers of $\hbar^{1/2}$ for energies coincide. %The same is true for strong coupling regime expansion in inverse fractional powers in $g$ of energy. It is related to the fact that the dynamics developed in two spaces: $x\ (r)$-space and in $gx\ (gr)$ space, leads to the same energy spectra. The equations which govern dynamics in these two spaces, the Riccati-Bloch equation and the Generalized Bloch(GB) equation, respectively, are presented. It is shown that perturbation theory for logarithmic derivative of wave function in $gx\ (gr)$ space leads to true semiclassical expansion in powers of $\hbar^{1/2}$ and corresponds to flucton calculus for density matrix in path integral formalism in Euclidean (imaginary) time.