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Journal of Mathematical Chemistry

Analysis of solutions of time-dependent Schrödinger equation of a particle trapped in a spherical box

Debraj Nath, Ramon Carbó‐Dorca

Published Apr 23, 2022• 9 citations
Publisher
DOI
10.1007/s10910-022-01350-x

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•Mar 16, 2019•0 citations
We consider a Dirac particle in a spherical box with a time-dependent radius. Analytical and numerical solutions of the time-dependent Dirac equation with time-dependent boundary (Dirichlet) conditions are obtained. Using the obtained solutions, physically observable characteristics of the dynamical confinement, such as the average kinetic energy (as a function of time) and average quantum force acting on the particle by the moving wall, are calculated. The trembling motion is analysed by computing the average coordinate of the Dirac particle as a function of time. The absence of the geometric phase is shown by direct calculation.
The solution of radial Schrödinger equation for hydrogen like atoms is complex. One needs simple solution the behavior of nano particle needs also to be described. Nano spherical particles are isolated non interacting particles that obey quantum laws. Thus one expects their behavior to resemble that of particles in a box, or hydrogen like atoms. The spherical nano shape requires using spherical coordinate to solve Schrödinger Equation. The solution found that the wave function is diminishing outside the sphere, which agrees with observations. This solution is simpler than the conventional one, where it is a sum of sine and cosine function.