Anharmonic oscillator: a solution

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 PLANCK CONSTANT OVER TWO PI 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 PLANCK CONSTANT OVER TWO PIg (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 PLANCK CONSTANT OVER TWO PI 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 similar to 10(-6) locally and unprecedented accuracy similar to 10(-9)-10(-10) in energy for any g (2) > 0. A generalization to the radial quartic oscillator is briefly discussed.

Automated summary

The study shows that in the one-dimensional quantum anharmonic oscillator system, perturbation theory and semiclassical expansion are consistent in terms of energy levels, and the dynamics in x-space and (gx)-space correspond to the same energy spectrum. By deriving the Riccati-Bloch and generalized Bloch equations, it is revealed that perturbation on the logarithmic derivative leads to PT for the RB equation, while the true semiclassical expansion reveals the GB equation, corresponding to a loop expansion in the path integral formalism.