The Hagedorn--Hermite Correspondence

TL;DR

This paper establishes a relationship between Hagedorn wave packets and Hermite functions through ladder operators, revealing their algebraic structure and applications to uncertainty principles and generating functions.

Contribution

It introduces a Hagedorn--Hermite correspondence that unifies and simplifies the understanding of Hagedorn wave packets and their algebraic properties.

Findings

Hagedorn's ladder operators are derived from position and momentum operators via the symplectic group.

The algebraic structure of Hagedorn wave packets is clarified and related to Hermite functions.

Existence of minimal uncertainty products for multi-dimensional Hagedorn wave packets is demonstrated.

Abstract

We investigate the relationship between the semiclassical wave packets of Hagedorn and the Hermite functions by establishing a relationship between their ladder operators. This Hagedorn--Hermite correspondence provides a unified view as well as simple proofs of some essential results on the Hagedorn wave packets. Particularly, we show that Hagedorn's ladder operators are a natural set of ladder operators obtained from the position and momentum operators using the symplectic group. This construction reveals an algebraic structure of the Hagedorn wave packets, and explains the relative simplicity of Hagedorn's parametrization compared to the rather intricate construction of the generalized squeezed states. We apply our formulation to show the existence of minimal uncertainty products for the Hagedorn wave packets, generalizing Hagedorn's one-dimensional result to multi-dimensions. The Hagedorn--Hermite correspondence also leads to an alternative derivation of the generating function for the Hagedorn wave packets based on the generating function for the Hermite functions. This result, in turn, reveals the relationship between the Hagedorn polynomials and the Hermite polynomials.