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African Journal of Mathematics and Computer Science Research

Approximation formulae for phase shifts of swave Schrdinger equation due to binomial potential and their applications

Hemant Kumar, Vimal Pratap Singh

Published Apr 30, 2011• 0 citations
Publisher

Abstract

In this paper, we derive the approximation formulae for phase shifts of s-wave Schrodinger equation on introducing binomial potential function and then make their applications to study the fluctuations in the phase shift difference from the s-wave with respect to different values of the arbitrary parameter occurring in a given binomial potential function.   Key words: S-wave Schrodinger equation, phase shifts, binomial potential function.

DOI
10.5897/ajmcsr.9000045

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วธการทางคณตศาสตรสำหรบสมการชเรอดงเงอร และวธประมาณคาแบบดบเบลยเคบ

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In this paper, we derive the approximation formulae for phase shifts of s-wave Schrodinger equation on introducing binomial potential function and then make their applications to study the fluctuations in the phase shift difference from the s-wave with respect to different values of the arbitrary parameter occurring in a given binomial potential function.
 

Key words: S-wave Schrodinger equation, phase shifts, binomial potential function.
1. INTRODUCTION In this chapter we shall apply the method of invariant imbedding to two types of problems that occur often in quantum mechanical studies. The first general class is frequently referred to as the “phase shift” problem, while the second is a particular form of the eigenvalue problem. In this latter case the method of the previous chapter is, within certain technical limits, applicable. However, the way in which we shall now make use of the imbedding scheme gives rise to a completely different characterization of the eigenvalues than the one we have been discussing. It seems particularly desirable to include a presentation of this method because the results illustrate so strikingly the way in which different kinds of imbeddings can yield completely different formulas. Although the phase shift problem is of less interest in itself, since our results can be derived in a rather more elementary way, we include it since once again, it demonstrates the versality of the method (see [1,2]).
The phase-shift equations are derived in a form that is convenient for the analysis of both the radial Schr\"odinger equation and the problem of wave propagation in nonhomogeneous media. In the case of bound quantum states, the conservation condition is discussed for the sum of the classical action (or eikonal) and the phase. The expression for the reflection coefficient describes exactly two opposite limiting cases: the Fresnel reflection and the short-wavelength limit. In the case of wave propagation in symmetric media, both transmission and reflection coefficients are given as elementary functions of the phase shifts only.
It is shown that a solution of the classical Kepler problem with a binomial potential explains the stationary orbits of the hydrogen atom, its discrete spectrum, and trajectories of scattered electrons with energies from several electron-volts to relativistic values. The Schrodinger equation with the binomial potential takes the form of a classical wave equation. Hence, the binomial potential is a more appropriate model for understanding atomic phenomena than the model based on the Coulomb potential.
For potentials with attractive tails, as occur in typical atomic interactions, we present a simple formula for the sss-wave phase shift ${\ensuremath{\delta}}_{0}$. It exposes a universal dependence of ${\ensuremath{\delta}}_{0}(E)$ on the potential tail and the influence of effects specific to a given potential, which enter via the scattering length aaa, or equivalently, the noninteger part ${\ensuremath{\Delta}}_{\mathrm{th}}$ of the threshold quantum number nth{n}_{\mathrm{th}}nth​. The formula accurately reproduces ${\ensuremath{\delta}}_{0}(E)$ from threshold up to the semiclassical regime, far beyond the validity of the effective-range expansion. We derive the tail functions occurring in the formula for ${\ensuremath{\delta}}_{0}(E)$ and demonstrate the validity of the formula for attractive potential tails proportional to 1/r61/{r}^{6}1/r6 or to 1/r41/{r}^{4}1/r4, and also for a mixed potential tail consisting of a 1/r41/{r}^{4}1/r4 term together with a non-negligible 1/r61/{r}^{6}1/r6 contribution.
The Schrödinger equation is a fundamental equation in physics for describing quantum mechanical behavior. It is a partial differential equation that shows how a wave function of a physical system evolves over time. In addition, it is associated with the kinetic energy and the potential energy, both of which contribute towards the total energy. This paper explains mathematical aspect of the Schrödinger equation. The concept of the WKB approximation has also been reviewed, which is an important method in finding the approximate solutions for the wave function. In addition, the paper also shows how to derive the transmission and reflection probabilities in relation to a one-dimensional quantum problem, called a “tunneling problem”, which occurs when a classical particle passes through a region where total energy is less than its potential energy. The WKB approximation can be used as a basis for formally writing down the exact solutions. Furthermore, it gives a high accuracy for the propagation of waves with high frequency or short wavelength.