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CRC Press eBooks

Free Particle and Harmonic Oscillator

Lukong Cornelius Fai

Published Feb 19, 2021• 0 citations
Publisher

Abstract

We find the eigenfunctions and eigenenergy values via the Feynman path integration with the help of the transition amplitude Κ for the free particle as well as the harmonic oscillator. This will be done without solving the Schrödinger equation and implies not solving the wave equation directly but everything determined from the transition amplitude Κ.

DOI
10.1201/9781003145554-9

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CRC Press eBooks•Oct 5, 2022•0 citations
We find the eigenfunctions and eigenenergy values via the Feynman path integration with the help of the transition amplitude Κ for the free particle as well as the harmonic oscillator. This will be done without solving the Schrödinger equation and implies not solving the wave equation directly but everything determined from the transition amplitude Κ.
Quantum mechanics is one of the branches of physics. In this mechanics, physical problems are solved by algebraic and analytic methods. By applying a simple procedure we can find the general solutions of Schrodinger’s time - independent wave equation of one dimensional quantum mechanical harmonic oscillator without making any approximation. In this paper, we will discuss the Eigenenergy values and Eigenfunctions of one of the most important physical models of quantum mechanics, namely the one-dimensional Quantum mechanical Harmonic Oscillator by modifying the Hermite differential equation.
The conventional approach to quantum mechanics begins with the classical Hamiltonian and changes observables to non-commuting operators. Generally, the quantum mechanics based on the Schrodinger equation is preferred for a system of particles. But path integral formulation of quantum mechanics finds applications in more complicated situations, particularly, in quantum field theory. The purpose of solving the Schrodinger equation lies in finding the time evolution operator which is able to generate the time translation of the given system. Richard Feynman noticed that it was necessary to perform integrals over all space variables at every instant of time to find the transition amplitudes. Dissipative phenomena with a quantum origin manifest themselves in many different situations and hence the study of dissipative systems and their quantization is of a great theoretical and practical value. Many dissipative systems can be modelled as a linearly damped harmonic oscillator.