Conformal Transformation of the Schr\"{o}dinger Equation for the Harmonic Oscillator into a Simpler Form

TL;DR

This paper demonstrates how conformal transformations can simplify the Schrödinger equation for the harmonic oscillator and explores the implications of introducing harmonic interactions as an imaginary time component.

Contribution

It introduces a conformal transformation approach to simplify the harmonic oscillator's Schrödinger equation and relates it to the Bargmann representation, also proposing a novel way to incorporate harmonic interactions.

Findings

Simplification of the Schrödinger equation via conformal transformation

Connection to the Bargmann representation

Harmonic interactions as imaginary time components

Abstract

The Schr\"{o}dinger equation and ladder operators for the harmonic oscillator are shown to simplify through the use of an isometric conformal transformation. These results are discussed in relation to the Bargmann representation. It is further demonstrated that harmonic interactions can be introduced into quantum mechanics as an imaginary component of time equivalent to adding the oscillator potential into the hamiltonian for the confined particle.