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The Hagedorn--Hermite Correspondence

Tomoki Ohsawa

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.

The Hagedorn--Hermite Correspondence

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.

Paper Structure

This paper contains 25 sections, 13 theorems, 199 equations.

Table of Contents

  1. Introduction
  2. The Hagedorn Wave Packets and Generalized Squeezed States
  3. Main Results
  4. Outline
  5. Canonical Commutation Relations and Ladder Operators
  6. Linear Transformations of Position & Momentum Operators
  7. Symplectic Group $\mathsf{Sp}(2d,\mathbb{R})$ and Ladder Operators
  8. Transformations under the Heisenberg--Weyl and Metaplectic Operators
  9. Ladder Operators and Semiclassical Wave Packets of Hagedorn
  10. The Hagedorn Ladder Operators
  11. Symplectic Covariance of the Hagedorn Ladder Operators
  12. The Hagedorn Wave Packets and the Hermite Functions
  13. Minimal Uncertainty Products for Ground State Hagedorn Wave Packet
  14. Symplectic Rotation of Position & Momentum Operators
  15. Minimal Uncertainty Products for Ground State Hagedorn Wave Packet
  16. ...and 10 more sections

Key Result

Proposition 2.2

Let $\mathcal{X} \in \mathsf{M}_{2d}(\mathbb{R})$ and $\hat{z}$ be $2d$ symmetric operators on $L^{2}(\mathbb{R}^{d})$ that satisfy the canonical commutation relations on $\mathscr{S}(\mathbb{R}^{d})$. Then the set of $2d$ operators $\rho(\mathcal{X}; \hat{z})$ defined in eq:rho are also symmetric o

Theorems & Definitions (41)

  • Remark 2.1
  • Proposition 2.2: Li1986; see also Wo1974
  • proof
  • Remark 2.3
  • Definition 2.4: Ladder operators on $\mathscr{S}(\mathbb{R}^{d})$
  • Example 2.5: Ladder operators for harmonic oscillator
  • Theorem 2.6
  • proof
  • Remark 2.7
  • Proposition 2.8
  • ...and 31 more