# Group theory, coherent states, and the N-dimensional oscillator

## TL;DR

This paper explores the symmetry group of the N-dimensional harmonic oscillator, deriving energy spectra, raising/lowering operators, and constructing coherent states using group theory, simplifying wave function calculations.

## Contribution

It introduces a group theoretical framework for the N-dimensional oscillator, including new raising/lowering operators and a method for constructing coherent states.

## Key findings

- Energy spectrum determined by O(2,1) x O(N) symmetry
- Explicit raising and lowering operators for each angular momentum
- Wave functions derived from first-order differential equations

## Abstract

The isotropic harmonic oscillator in N dimensions is shown to have an underlying symmetry group O(2,1)X O(N)which implies a unique result for the energy spectrum of the system. Raising and lowering operators analogous to those of the one-dimensional oscillator are given for each value of the angular momentum parameter. This allows the construction of an infinite number of coherent states to be carried out. In the N=1 case there is a twofold family of coherent states, a particular linear combination of which coincides with the single set already well known for that case. Wave functions are readily derived which require only the solution of a first order differential equation, an attribute generally characteristic of group theoretical approaches.