On a Hidden Symmetry of Quantum Harmonic Oscillators

TL;DR

This paper explores a six-parameter family of quantum harmonic oscillator wave functions derived from symmetry transformations, visualizes phase space oscillations, and discusses implications for the Heisenberg Uncertainty Principle.

Contribution

It introduces a new class of wave functions obtained via symmetry group actions, beyond standard separation of variables, and provides visualizations of their phase space behavior.

Findings

New six-parameter family of wave functions identified

Visualizations of phase space oscillations created

Insights into the Heisenberg Uncertainty Principle presented

Abstract

We consider a six-parameter family of the square integrable wave functions for the simple harmonic oscillator, which cannot be obtained by the standard separation of variables. They are given by the action of the corresponding maximal kinematical invariance group on the standard solutions. In addition, the phase space oscillations of the electron position and linear momentum probability distributions are computer animated and some possible applications are briefly discussed. A visualization of the Heisenberg Uncertainty Principle is presented.