Related papers: Anharmonic oscillator: a solution

Anharmonic oscillator: a solution

URL: http://arxiv.org/abs/2011.14451v2
Date: Tue, 25 May 2021 10:19:33 GMT
Title: Anharmonic oscillator: a solution
Authors: Alexander V Turbiner, Juan Carlos del Valle
Abstract summary: The dynamics in $x$-space and in $(gx)-space corresponds to the same energy spectrum with effective coupling constant $hbar g2$. A 2-classical generalization leads to a uniform approximation of the wavefunction in $x$-space with unprecedented accuracy.
Score: 77.34726150561087
License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
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.

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