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ANALYZING SCHRODINGER EQUATION FOR A PARTICLE IMPINGING ON VERTICAL POTENTIAL STEP BY MATRIX METHOD

Rohit Gupta, Rahul Gupta, Dinesh Verma

Published Mar 10, 2023• 0 citations
Publisher

Abstract

Quantum mechanics has been perceived as a requisite ingredient in the curricula of Physics, Chemistry, and Electrical Engineering. By applying Matrix Method, we will analyse Schrodinger’s equation for a particle impinging on the vertical potential step which is generally done by different algebraic and analytical methods. The matrix method has been applied successfully in science and engineering problems. This paper demonstrates the use of the Matrix Method for analyzing Schrodinger’s equation to find the reflection and transmis

DOI
10.5281/zenodo.7714506

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Quantum mechanics has been perceived as a requisite ingredient in the curricula of Physics, Chemistry, and Electrical Engineering. By applying Matrix Method, we will analyse Schrodinger’s equation for a particle impinging on the vertical potential step which is generally done by different algebraic and analytical methods. The matrix method has been applied successfully in science and engineering problems. This paper demonstrates the use of the Matrix Method for analyzing Schrodinger’s equation to find the reflection and transmis
In this paper, we illustrate an application of the Laplace transformation for finding the quantum mechanical Reflection and Transmission coefficients for a particle through a one-dimensional vertical step potential. Quantum mechanics is one of the branches of physics in which the physical problems are solved by algebraic and analytic methods. By applying the Laplace transformation, we can find the quantum mechanical Reflection and Transmission coefficients for a particle through a one-dimensional vertical step potential. Generally, the Laplace transformation has been applied in different areas of science and engineering and makes it easier to solve the problems inengineering applications. It is a mathematical tool which has been put to use for solving the differential equations without finding their general solutions. It has applications in nearly all science and engineering disciplines like analysis of electrical circuits, heat and mass transfer, fluid dynamics, nuclear physics, process controls, quantum mechanical problems,etc.
Based on the similarity between time-independent Schrodinger equation and Helmholtz equation, the analytical transfer matrix method was utilized to analyze macroscopic quantum tunneling effect, and the potential filed and eigenvalue were corresponded to the refractive index and propagation constant, respectively.Without solving the Schrodinger equation and any approximation,the exact analytic expressions of reflection coefficient and reflection probability of the particle were obtained.Initial results show the equations obtained here are exact and have clear physical meanings.It is expected to be applied in the researches of many basic quantum phenomena,such as quantum tunneling effect,quantum reflection, quantum particles and tunneling resonance.
This paper adds up how the matrix method can be used for solving the one-dimensional time-independent Schrodinger's equation for some specific potential energy variation like half-infinite potential well.The matrix method is illustrated to obtain solution of the time-independent Schrodinger's equation for half-infinite potential well, which is generally done by ordinary algebraic and analytical methods.The transcendental equation determining the discrete eigenvalues for bound state and the corresponding eigenwave functions are obtained by the time-independent Schrodinger's equation for half-infinite potential well via the application of matrix method.
The reflection and transmission of wave functions at a potential step is a well-known issue in a textbook of quantum mechanics. We studied the reflection and transmission characteristics analytically when the potential step is moving at a constant velocity vvv in the same direction as an incident wave function by means of solving the time-dependent Schrödinger equation. As for an infinite potential step, it is known that group velocity is the same as the moving velocity of the potential step. We found two interesting results when the potential step has a finite height of V0V_0V0​. The transmission occurs when the kinetic energy of incident wave function is larger than the effective potential hight of (m2v+V0)2(\sqrt{\frac{m}{2}}v + \sqrt{V_0})^2(2m​​v+V0​​)2. The other result is that the reflectivity depends on xxx, which derives from the interference between the incident and the reflected wave functions.