Trial Introduction of a Complex Potential Function for Perturbation That Causes Quantum Mechanical Transition in a One-Dimensional Harmonic Oscillator

摘要

Some results of computer simulation of the behavior of a one-dimensional quantum mechanical oscillator are reported in this article.This harmonic oscillator comprises a particle trapped within a hyperbolic potential V_o(x) = x~2.Further,a perturbation potential function V'(x,t) was superposed upon the hyperbolic potential in order to induce a quantum mechanical transition.This perturbation function V'(x,t) is a function of both of space and time variables,and is set to represent a wave packet that is enveloped by a Gaussian bell-shaped curve.A wave that probably has an appropriate wave number and angular frequency was inputted into the expression for the wave packet.In the initial phase,while the harmonic oscillator was allowed to oscillate almost freely,the wave packet was allowed to approach the harmonic oscillator.In the middle phase,the wave packet passes through the harmonic oscillator,affecting the shape of the quantum mechanical wave that represents the physical state of the system.In the last phase,when the wave packet left the system of the harmonic oscillator,the system settled onto an energetically stable state.The main objective of the simulation was to simulate the instance of a quantum mechanical transition from one eigenstate to another.After several trials,it was found that the perturbation function consisting of a complex function was,at least superficially,able to cause one desired transition,that is,a transition from one eigenstate to another eigenstate.By using such a complex perturbation function,a transition from the first excited state to the ground state was observed to occur.