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Kirkuk journal of science

Solving Quantum Mechanics Problems Via Integral Rohit Transform

Rohit Gupta, Rahul Gupta, Dinesh Verma

Published Dec 30, 2024• 0 citations

Abstract

This solves quantum physics problems using the integral Rohit transform, including the scattering of low energy particles by a completely rigid sphere and particle behavior in a one-dimensional infinitely high potential box. The standard calculus approach is typically used to solve these quantum mechanics issues. To solve these quantum physics challenges, this research presents a new method: the integral Rohit transform. When compared to other approaches that are currently in the literature, the obtained solutions show how accurate the suggested method is. This demonstrates the potential and efficacy of the suggested approach to overcoming quantum mechanical issues, such as low-energy particle scattering by a completely rigid sphere and particle behavior in a one-dimensional infinitely high potential box. In this paper, the successful application of the integral Rohit Transform has been demonstrated in solving the one-dimensional time-independent Schrodinger's equation. This application has yielded results that include the determination of eigen energy values and eigen functions for a particle confined within an infinitely high potential well, as well as the calculation of the total scattering cross-section for low energy particles interacting with a perfectly rigid sphere. In the case of low energy limit, the total scattering cross-section for low energy particles due to a perfectly rigid sphere, as determined through quantum mechanics, is equivalent to the geometrical cross-section of said sphere. Additionally, the energy values that the particle can possess within a 1d infinitely high potential well demonstrate that the energy of said particle, when confined within this potential well, is quantized.

DOI: 10.32894/kujss.2024.151830.1164

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The wave nature of particles plays an important role in their physical properties that, for example, particles confined into a small bounded area can have only particular discrete energies. This chapter explains how to solve the time-independent Schrodinger equation. It considers four cases of a one-dimensional motion of particles confined in potentials rapidly changing with x : infinite potential quantum well; a potential step; square-well potential; and tunneling through a potential barrier. The chapter helps to learn the characteristic properties of solutions to this equation. It shows the differences between the predictions of quantum mechanics and classical physics. Quantum tunneling is important in the understanding of a number of physical phenomena such as thermonuclear reactions and conduction in metals and semiconductors. The most advanced application of quantum tunneling is the scanning tunneling microscope. The details of the calculations are left for the readers as a tutorial problem.

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A simple procedure has been found for the general solution of the time-independent Schrödinger equation (SE) with the help of quantization of potential area in one dimension without making any approximation. Energy values are not dependent on wave functions. So, to find the energy values, it is enough to find the classic turning points of the potential function. Two different solutions were obtained, namely, symmetric and antisymmetric in bound states. These normalized wave functions are always periodic. It is enough to take the integral of the square root of the potential energy function to find the normalized wave functions. If these calculations cannot be made analytically, they should then be performed by numerical methods. The SE has been solved for a particle in many one-dimension and the spherical symmetric central potential wells as examples. Their energies and normalized wave functions were found as examples. These solutions were also applied to the theories of scattering and alpha decay. The results obtained with the experimental values were compared with the calculated values. One was seen to be very fit.

The quantum model of finite potential well is a most subject in the book of quantum mechanics. The solutions maybe derive from one dimension Schrödinger’s equation for describe a bound state energy of confined particle in the finite potential well. Sometime, the problem of confined particle in the finite potential well does not have an exact solution yet, but there are in fact exact solution.In this work a numerical technique is adopted to find the exact solution for the quantum model of finite potential well. To achieve this numerical solution, I use Fortran code computer program by the Newton-Raphson method for calculating the Eigen state energy for a particle (electron) confined in the finite potential well. Five values of potential width and depth have studied, the results showed that large potential width and large potential depth led me to more bound states, while smaller potential width and depth led me to fewer bound states.A comparison between infinite potential and finite potential has been presented. The results bound state energy and penetration. The results of comparison showed that the wave function of particle confined in the infinite potential cannot penetrate in the well, while the wave function of particle in the finite potential was penetrate in the well; the energy levels of an infinite well are much higher than that the corresponding energy levels for finite potential well. For confined particle in 2 or 3-dimension, we can reduce from result of 1-dimension.

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Synopsis: Students are introduced to the ideas of particle scattering in quantum mechanics in a one-dimensional context. The integral Schrödinger equation is deduced using a Wronskian technique, and reflection and transmission amplitudes are defined. Several simple scattering problems are solved, and the Born series for the energy-wave function is explained. A three-dimensional spherical basis is introduced, which allows the integral Schrödinger equation in this context to be formulated. The relation between plane waves and spherical waves is found from a solution to the unit-source Helmholtz equation, again using the Wronskian technique. Partial waves, phase shifts, and cross sections are defined, and finite range scattering is considered. The chapter concludes with a consideration of identical particle scattering and an application to the proton–proton case, where a dramatic manifestation of the underlying substructure emerges at high energies.