Modeling of Double-Well Potentials for the Schrödinger Equation

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Abstract

A new method is proposed for determining level splitting Δ in a double-well 1D potential. Two “partner” functions (one symmetric Ψ+ and the other antisymmetric Ψ) are determined. From these functions, potentials V+(x) and V(x) and energies E+0">E+0 and E1">E1 corresponding to them are determined from the Schrödinger equation. A unique property of Ψ+ and Ψ is identity E+0">E+0 = E1">E1, which makes it possible to determine Δ from the perturbation theory in parameter V+(x) – V(x). For a double-well oscillator potential, the expression for the level splitting, which connects the instanton and single-well limits, is obtained. These results can be employed in the field theory, for which the possibility of obtaining instanton solutions from perturbation theory has been discussed more than once. A number of potentials are considered, for which the value of Δ can be determined without using the semiclassical approximation. Singular potentials of the funnel type are analyzed. The value of Δ determined in this study is compared with the results of numerical solution of the Schrödinger equation for the instanton potential.

About the authors

A. M. Dyugaev

Landau Institute of Theoretical Physics, Russian Academy of Sciences

Email: dyugaev@itp.ac.ru
Chernogolovka, Moscow oblast, 142432 Russia

P. D. Grigor'ev

Landau Institute of Theoretical Physics; National University of Science and Technology MISIS

Author for correspondence.
Email: grigorev@itp.ac.ru
Moscow region, 142432 Russia; Moscow, 119049 Russia

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