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 $\hslash$ 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 $\hslash g^2$. 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 $\hslash$ 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 $g^2\ge 0$. A generalization to the radial quartic oscillator is briefly discussed.

arXiv comment12 pages, 3 figures, 1 table; extended version by adding a discussion on non-orthogonal polynomials and section on radial quartic anharmonic oscillator

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