arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 2211.05751math-phmath.MPquant-ph
收藏

源自自由哈密顿量约化的量子广义 Calogero-Moser 系统

Quantum generalized Calogero-Moser systems from free Hamiltonian reduction

Katarzyna Kowalczyk-Murynka, Marek Kuś

备注：
20 pages, 2 figures
阅读 PDF
arXiv 页面 ↗HTML 原文 ↗

AI总结：

本文通过自由量子系统的哈密顿量约化构造广义量子 Calogero-Moser 模型，揭示正交情形特有的吸引项，并给出 N=2、3 的谱、波函数及一般 N 的部分对角化。

摘要：

具有 $1/x^2$ 排斥势的一维粒子系统被称为 Calogero-Moser 系统。其经典版本可以通过用附加自由度替代耦合常数来推广，这些附加自由度关于 Poisson 括号张成 $\mathfrak{so}(N)$ 或 $\mathfrak{su}(N)$ 代数。我们提出该广义模型的量子版本。由于经典推广是通过对自由系统进行辛约化得到的，我们沿类似思路给出一种获得量子系统的方法。对自由量子系统的约化产生一个哈密顿量，该哈密顿量保留了经典系统依赖于底层正交或幺正对称群的动力学差异。已知正交系统的排斥性弱于幺正系统，而约化后的自由量子哈密顿量通过一个额外的吸引项 $\sum_{i<j}\frac{-\hbar^2}{(x_i-x_j)^2}$ 体现这一特征；当对所考虑系统直接进行 Dirac 量子化时，该项并不存在。我们给出广义量子 Calogero-Moser 哈密顿量详细而严格的推导，求出粒子数 $N=2,3$ 时的谱和波函数，并对一般 $N$ 值部分对角化该哈密顿量。

英文摘要：

The one-dimensional system of particles with a $1/x^2$ repulsive potential is known as the Calogero-Moser system. Its classical version can be generalised by substituting the coupling constants with additional degrees of freedom, which span the $\mathfrak{so}(N)$ or $\mathfrak{su}(N)$ algebra with respect to Poisson brackets. We present the quantum version of this generalized model. As the classical generalization is obtained by a symplectic reduction of a free system, we present a method of obtaining a quantum system along similar lines. The reduction of a free quantum system results in a Hamiltonian, which preserves the differences in dynamics of the classical system depending on the underlying, orthogonal or unitary, symmetry group. The orthogonal system is known to be less repulsive than the unitary one, and the reduced free quantum Hamiltonian manifests this trait through an additional attractive term $\sum_{i<j}\frac{-\hbar^2}{(x_i-x_j)^2}$, which is absent when one performs the straightforward Dirac quantization of the considered system. We present a detailed and rigorous derivation of the generalized quantum Calogero-Moser Hamiltonian, we find the spectra and wavefunctions for the number of particles $N=2,3$, and we diagonalize the Hamiltonian partially for a general value of $N$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

↑