On The Harmonic Oscillator Group

TL;DR

This paper explores the symmetry group of the quantum harmonic oscillator, revealing a new family of wave functions and establishing isomorphism with the free particle's Schrödinger group, thus advancing understanding of oscillator invariance.

Contribution

It introduces a six-parameter family of wave functions not obtainable by standard methods and analyzes the oscillator's invariance group via an Ermakov-type system.

Findings

Identifies a new family of wave functions for the harmonic oscillator.

Shows the invariance group is isomorphic to the free particle's Schrödinger group.

Provides insights into the symmetry properties of driven harmonic oscillators.

Abstract

We discuss the maximum kinematical invariance group of the quantum harmonic oscillator from a view point of the Ermakov-type system. A six parameter family of the square integrable oscillator wave functions, which seems cannot be obtained by the standard separation of variables, is presented as an example. The invariance group of generalized driven harmonic oscillator is shown to be isomorphic to the corresponding Schroedinger group of the free particle.