Anharmonic oscillator: a solution

TL;DR

This paper demonstrates that perturbation theory and semiclassical expansion for the quantum anharmonic oscillator are equivalent and introduces a highly accurate uniform approximation of the wavefunction by interpolating these expansions.

Contribution

It derives Riccati-Bloch and Generalized Bloch equations and develops a novel interpolation method for highly precise wavefunction and energy approximations.

Findings

Achieves accuracy of ~10^{-6} in wavefunction locally

Achieves accuracy of ~10^{-10} in energy for all g^2 ≥ 0

Establishes equivalence of perturbation and semiclassical expansions

Abstract

It is shown that for the one-dimensional quantum anharmonic oscillator with potential $V(x)= x^2+g^2 x^4$ the Perturbation Theory (PT) in powers of $g^2$ (weak coupling regime) and the semiclassical expansion in powers of $\hbar$ for energies coincide. It is related to the fact that the dynamics in $x$-space and in $(gx)$-space corresponds to the same energy spectrum with effective coupling constant $\hbar g^2$. Two equations, which govern the dynamics in those two spaces, the Riccati-Bloch (RB) and the Generalized Bloch (GB) equations, respectively, are derived. The PT in $g^2$ for the logarithmic derivative of wave function leads to PT (with polynomial in $x$ coefficients) for the RB equation and to the true semiclassical expansion in powers of $\hbar$ for the GB equation, which corresponds to a loop expansion for the density matrix in the path integral formalism. A 2-parametric interpolation of these two expansions leads to a uniform approximation of the wavefunction in $x$-space with unprecedented accuracy $\sim 10^{-6}$ locally and unprecedented accuracy $\sim 10^{-9}-10^{-10}$ in energy for any $g^2 \geq 0$. A generalization to the radial quartic oscillator is briefly discussed.