Technical Report No. 2013-6 : Coefficients of Transformations in Multidimensional Quantum Harmonic Oscillators (AU-CAS-MathStats)

When one represents a physical system of harmonic oscillators, it is possible to represent the system in many ways. For a system with three degrees of freedom, one could represent a single particle in three dimensions which would be equivalent to three particles in one dimension. The main idea here is that in each case, the total degrees of freedom must be equal for different representations of a given system. There are many other ways that someone can represent a physical system in three dimensions. The system can be represented in Cartesian coordinates, cylindrical coordinates, or spherical coordinates, just to name a few. Then within each of these representations, one could imagine rotating the coordinate systems or scaling them differently. As a result, one specific state of the multidimensional quantum harmonic oscillator can be represented in many different ways. The purpose of this research project is to calculate different coefficients for translating from one representation to another.