Anharmonic oscillator: a solution

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 ℏ 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 ℏg 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 ℏ for the GB equation, which corresponds to a loop expansion for the density matrix in the path integral formalism. A two-parametric interpolation of these two expansions leads to a uniform approximation of the wavefunction in x-space with unprecedented accuracy ∼10−6 locally and unprecedented accuracy ∼10−9–10−10 in energy for any g 2 ⩾ 0. A generalization to the radial quartic oscillator is briefly discussed.