Exact analytical solutions of the Schrödinger equation for a two dimensional purely sextic double-well potential

Abstract

Exact analytical solutions of the Schrödinger equation for a two-dimensional purely sextic double-well potential are proved to exist for a denumerably infinite set of the geometry parameter of the well. First, the geometry values which allow exact solutions are determined. Then, explicit wave functions and corresponding energies are calculated for the allowed geometry values. Concrete exact solutions are given for the principal quantum number n up to 10. Moreover, some interesting rules for the obtained exact analytical solutions are also given; particularly, the number of negative energy levels for a given geometry parameter is obtained. For analyzing the obtained exact solutions and for their classification by quantum number, we also use numerical calculations by the Feranchuk-Komarov operator method. Published by AIP Publishing. https://doi

A Theoretical Investigation of the Schrödinger Equation for a Modified Two Dimensional Purely Sextic Double-Well Potential

“…Let us present a brief review of time independent Schrödinger equation for a fermionic particle like electron of rest -43 -mass M and its energy E moving in 2Dsextic DWAO potential [1]:…”

“…is dimensionless and is referred to as a geometry parameter of the well. The wave function ) and the energy is equal to zero 0  E , the wave function for this case is [1]:…”

“…The double-well anharmonic oscillator (DWAO) holds a great importance because of its relation to many problems of quantum chemistry and field theory [1][2][3][4]. Dai-Nam Le et al was studied the Schrödinger equation for the purely sextic double-well potential problem in twodimensional space by wave function ansatz method for analytical approach and by Feranchuk-Komarov operator method for numerical approach [1].…”

“…The double-well anharmonic oscillator (DWAO) holds a great importance because of its relation to many problems of quantum chemistry and field theory [1][2][3][4]. Dai-Nam Le et al was studied the Schrödinger equation for the purely sextic double-well potential problem in twodimensional space by wave function ansatz method for analytical approach and by Feranchuk-Komarov operator method for numerical approach [1]. Over the past few years, many efforts have been produced to study quantum systems on based to the Dirac equation, Klein-Gordon equation and Schrödinger equation, which is undoubtedly the most widely studied equation of modern physics, for spherical and non spherical potentials that in ordinary space of quantum mechanics and its extension, which known by extended quantum mechanics.…”

Semiexact Solutions of the Razavy Potential

“…It should be recognized that almost all soluble potentials mentioned above belong to single well potentials. The double-well potentials have not been studied well due to their complications [8][9][10][11][12][13][14][15][16][17], in which many authors have been searching the solutions of the double-well potentials for a long history. This is because the double-well potentials could be used in the quantum theory of molecules to describe the motion of the particle in the presence of two centers of force, the heterostructures, Bose-Einstein condensates, superconducting circuits, etc.…”