# Finite Quantum Theory of the Harmonic Oscillator

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We apply the Segal process of group simplification to the linear harmonic oscillator. The result is afinite quantum theory with three quantum constants instead of the usual one. We compare the classical CLHO, quantum QLHO, and finite FLHO linear harmonic oscillators and their canonical or unitary groups. The FLHO is isomorphic to adipole rotator with N=ll+1 states where l is very large for physically interesting case. The position and momentum variables are quantized with uniform finite spectra. For fixed quantum constants and large N there are three broad classes of FLHO: soft, medium, and hard corresponding respectively to cases where ratio of the of potential energy to kinetic energy in the Hamiltonian is very small, almost equal to one, or very largeThe field oscillators responsible for infra-red andultraviolet divergences are soft and hard respectively. Medium oscillators approximate the QLHO. Theirlow-lying states have nearly the same zero-pointenergy and level spacing as the QLHO, and nearly obeying the Heisenberg uncertainty principle and the equipartition principle. The corresponding rotators are nearly polarized along the z-axis. The soft and hard FLHO-s have infinitesimal0-point energy and grossly violate equipartition and the Heisenberg uncertainty principle. They do not resemble the QLHO at all. Their low-lying energy states correspond to rotators polaroizd along x-axis or y-axis respectively. Soft oscillators havefrozen momentum, because their maximum potential energy is too small to produce one quantum of momentum. Hard oscillators have frozen position, because their maximum kinetic energy is too small to produce one quantum of momentum. Hard oscillators have frozen position, because their maximum kinetic energy is too small to excite one quantum of position.

Georgia Tech Theses and Dissertations - School of Physics Theses and Dissertations -

Author: ** Shiri-Garakani, Mohsen - - **

Source: https://smartech.gatech.edu/