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SOLVING LINEAR POTENTIAL USING THE VARIATION METHOD

Mudassir Umar Ali, Abubakar Sani Garba, Nazifi Usman, Uttar Pradesh

Published Jan 1, 2015• 0 citations
OpenAlex

Abstract

We examine the variation of 1 - dimentional quantum mechanics system in linear potential (constant force), using variation wave functions for the ground state up to the second order shift. This approach use simple wave functions to describe the exact Airy so lution. The exact solution is compare with solutions obtain from the variation function.

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We examine the variation of 1 - dimentional quantum mechanics system in linear potential (constant force), using variation wave functions for the ground state up to the second order shift. This approach use simple wave functions to describe the exact Airy so lution. The exact solution is compare with solutions obtain from the variation function.
Variational calculations play an important role in quantum mechanics, particularly in determining the ground-state energy of physical systems. Didactic examples in which one can see how the exact energy and wave function are approached by using a family of trial functions are not very numerous. An example was given in a recent book [M. Moshinsky, The Harmonic Oscillator in Modern Physics: From Atoms to Quarks (Gordon and Breach, New York, 1969)] which discusses the ground state of the hydrogen atom using as trial wave function a linear combination of harmonic-oscillator states. In the present paper we carry out a similar analysis for the ground state of a particle of mass m in a three-dimensional square-well potential. We discuss not only the approach to the exact energy when we vary the frequency and the number of oscillator states but also analyze the overlap of the exact and variational wave functions and compare the exact and approximate form factors.
If the potential is not closely related to a potential that we already know how to solve exactly, the energies and states can still be approximated via the Variational Method. This method is simpler to use than to understand. A simple five-step procedure is given and then illustrated via an example. The principle, in a nutshell, is that if you guess at the ground state and are wrong, the energy you calculate will be higher than the ground state energy, so if you keep guessing and keep getting smaller energies, then you must be getting closer to the ground state wave function. This minimization procedure is called the Variational Method, and it was known prior to quantum mechanics.
In this paper I have drawn out the steps to be followed in order to derive the exact Schr\"odinger wave function for a particle in a general one-dimensional time-dependent linear potential. To this end I have used the so-called Lewis and Riesenfeld invariant method, which is based on finding an exact quantum-mechanical invariant in whose eigenstates the exact quantum states are found. In particular, I have obtained the wave functions of a particle in the linear potential well, driven by a monochromatic electric field.
A general solution of the Schrödinger equation in the potential representation has been obtained in the form of integral equations. In this representation, the wave function for positive and negative energies or bound states can be expressed as a product of the unperturbed solution for model potential and the function which depends on the additional potential or potential perturbation. Here we have proved that this method is equivalent to the method of variation of constants for negative energies. The linearly independent solutions of Schrödinger equation for harmonic oscillator potential have been obtained for derivation of integral equations, which are used for finding eigenfunctions and eigenvalues for Woods–Saxon potential. Eigenvalues obtained by numerical iterations of these integral equations are in good agreement with results obtained by the discretization method. The kernels of the obtained integral equations are proportional to the perturbation or difference of Woods–Saxon and harmonic oscillator potentials.