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College Physics

Estimating the energy levels of particles in the gravitational field:some approaches other than Schrdinger equations

Zhang Yu Fu

Published Jan 1, 2003• 0 citations
OpenAlex

Abstract

It is onerous work to solve the energy levels of particle in the gravitational field (or any other linear potential field) using the Schrdinger equation; still no accurate algebraic expression for the solution could be obtained. Some methods that are much more accessible are employed and carefully investigated to estimate the energy levels, which proves to be successful.

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It is onerous work to solve the energy levels of particle in the gravitational field (or any other linear potential field) using the Schrdinger equation; still no accurate algebraic expression for the solution could be obtained. Some methods that are much more accessible are employed and carefully investigated to estimate the energy levels, which proves to be successful.
Most of Schrdinger equations in quantum mechanics haven't analytic solutions except the problems of well with infinite depth,one-dimensional linear harmonic oscillator and Coulomb's potential as well as three-dimensional isotropic harmonic oscillator.These Schrdinger equations include the various equations in the non-spherical ring shaped oscillator potential that haven't analytic solutions. We can only solve these problems approximately in our research activities, but the solving process of these problems is often exceptional complex.The limitation handicaps us to lucubrate the idiographic problems.This paper adopts NU Method(A.F.Nikiforov-V.B.Uvarov Method) to solve the angular equation of the Schrdinger equation in the non-spherical ring shaped oscillator potential V(r,θ)=μω2r2/2+2α/(2μr2)+2βcosθ/(2μr2sin2θ).We decompose the Schrdinger equation in the non-spherical ring shaped oscillator potential step by step until we obtains the analytic solutions of the equation,thus we obtain the analytic solution of the angular eigenfunction,and simplify the process of solutions.Moreover,this paper adopts special functions in the radial equation at the same time to broaden our study in solving the Schrdinger equations.
We should try to establish the Schr ö dinger equation for gravitational potential energy and use it to describe macroscopic objects. Replacing the electromagnetic potential energy in the Hamiltonian operator with gravitational potential energy can achieve the goal. The successful application of the Schr ö dinger equation for gravitational potential energy can change many existing concepts in quantum mechanics and shake the conceptual system of quantum mechanics. This can promote the development of quantum mechanics and other fields of physics.