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Extended Coherent States

Z. M. McIntyre, A. Kasman, R. Milson

TL;DR

This work extends the concept of coherent states to rational extensions of the harmonic oscillator by combining Maya-diagram combinatorics, Schur-function technology, and intertwiners of SUSYQM. The authors define extended coherent states (ECS) as Barut–Girardello-type joint eigenfunctions of a commuting annihilator algebra generated by special ladder operators, and show that ECS are exact time-dependent solutions of the corresponding Schrödinger equation. A Miwa-shift–based generating function for bound states is derived, allowing a compact description of ECS via a reduced tau-function framework and its Schur-function expansion. The key result is that ECS asymptotically minimize position–momentum uncertainty, paralleling canonical coherent states, with an explicit example demonstrating the construction and spectral properties of a particular rational extension.

Abstract

Using the formalism of Maya diagrams and ladder operators, we describe the algebra of annihilating operators for the class of rational extensions of the harmonic oscillator. This allows us to construct the corresponding coherent state in the sense of Barut and Girardello. The resulting time-dependent function is an exact solution of the time-dependent Schrödinger equation and a joint eigenfunction of the algebra of annihilators. Using an argument based on Schur functions, we also show that the newly exhibited coherent states asymptotically minimize position-momentum uncertainty.

Extended Coherent States

TL;DR

This work extends the concept of coherent states to rational extensions of the harmonic oscillator by combining Maya-diagram combinatorics, Schur-function technology, and intertwiners of SUSYQM. The authors define extended coherent states (ECS) as Barut–Girardello-type joint eigenfunctions of a commuting annihilator algebra generated by special ladder operators, and show that ECS are exact time-dependent solutions of the corresponding Schrödinger equation. A Miwa-shift–based generating function for bound states is derived, allowing a compact description of ECS via a reduced tau-function framework and its Schur-function expansion. The key result is that ECS asymptotically minimize position–momentum uncertainty, paralleling canonical coherent states, with an explicit example demonstrating the construction and spectral properties of a particular rational extension.

Abstract

Using the formalism of Maya diagrams and ladder operators, we describe the algebra of annihilating operators for the class of rational extensions of the harmonic oscillator. This allows us to construct the corresponding coherent state in the sense of Barut and Girardello. The resulting time-dependent function is an exact solution of the time-dependent Schrödinger equation and a joint eigenfunction of the algebra of annihilators. Using an argument based on Schur functions, we also show that the newly exhibited coherent states asymptotically minimize position-momentum uncertainty.

Paper Structure

This paper contains 15 sections, 4 theorems, 83 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Partitions and Maya diagrams
  4. Vertex operators and Schur functions
  5. The harmonic oscillator and its rational extensions
  6. Hermite polynomials
  7. The canonical Hamiltonian and coherent state
  8. Hermite Pseudo-Wronskians
  9. Rational Extensions of the Harmonic Oscillator
  10. Extended coherent states
  11. Ladder Operators
  12. The annihilator algebra
  13. Definition of the extended coherent states
  14. Minimized uncertainty
  15. Example

Key Result

Proposition 2.1

For every partition $\lambda$ of length $\ell$, we have where $1$ is the Schur function corresponding to the trivial partition.

Figures (4)

  • Figure 1: The Young diagram and corresponding hooklengths for the partition $(5,5,4,2,2)$.
  • Figure 2: The bent Maya diagram with index set $K=\{-5,-4,-1,1,3,4 \}$ is the rim of the Young diagram of the corresponding partition $\lambda=(5,5,4,2,2)$.
  • Figure 3: Top: The Maya diagram $M$ corresponding to index set $K=\{2,3\}$. The corresponding partition and index are $\lambda=(2,2)$ and $\sigma_M=2$, respectively, while the threshold critical degree is $q_c = 4$. Middle: $M+3$. Bottom: $M+4$. Note that $4$ is a critical degree since $M\subset M+4$. However, $3$ fails to be a critical degree since $3\in M$ but $3\notin M+3$.
  • Figure 4: The position-momentum uncertainty value $E_\lambda(\Delta x)^2 E_\lambda(\Delta p)^2$ as a function of time for the ECS with $\lambda=(2,2)$ and $\alpha=4$ (blue), $8$ (red), and $16$ (yellow).

Theorems & Definitions (5)

  • Proposition 2.1
  • Proposition 2.2
  • Theorem 2.3
  • Proposition 3.1
  • proof