[论文解读] Comment on 'Application of nonlinear deformation algebra to a physical system with Poschl-Teller potential'

C. Quesne|arXiv (Cornell University)|Jan 1, 1999

Quantum Mechanics and Non-Hermitian Physics参考文献 1被引用 4

一句话总结

本文更正了在Pöschl-Teller势能中应用的非线性形变代数中的一个关键定义关系，表明此前声称的关系仅在无限深方势阱极限（ν → 1）下成立。作者推导出正确的通解形式，建立了其与su(1,1)代数的一致联系，并利用该关系代数地计算了本征函数归一化常数，从而解决了陈、刘与葛原始表述中的不一致问题。

ABSTRACT

We comment on a recent paper by Chen, Liu, and Ge (J. Phys. A: Math. Gen. 31 (1998) 6473), wherein a nonlinear deformation of su(1,1) involving two deforming functions is realized in the exactly solvable quantummechanical problem with Poschl-Teller potential, and is used to derive the well-known su(1,1) spectrum-generating algebra of this problem. We show that one of the defining relations of the nonlinear algebra, presented by the authors, is only valid in the limiting case of an infinite square well, and we determine the correct relation in the general case. We also use it to establish the correct link with su(1,1), as well as to provide an algebraic derivation of the eigenfunction normalization constant. Short title: Application of nonlinear deformation algebra PACS: 02.10.Tq, 03.65.Fd Directeur de recherches FNRS E-mail address: [e-mail] 1 In an interesting paper (henceforth referred to as I and whose equations will be quoted by their number preceded by I), Chen, Liu, and Ge [1] recently pointed out that the nonlinear deformations of the su(2) and su(1,1) Lie algebras with two deforming functions f(J0) and g(J0), introduced by Delbecq and Quesne [2], can find some useful applications in quantum mechanics. They indeed claim to have proved that one of such algebras can be realized in a physical system with Poschl-Teller potential, which is one of the exactly solvable one-dimensional quantum-mechanical potentials. By starting from the ‘natural’ quantum operatorsX, P of Nieto and Simmons [3], they constructed mutually adjoint lowering and raising operators b, b, which together with the Hamiltonian H generate a nonlinear algebra with two deforming functions f(H) and g(H). They also obtained the eigenvalues and (unnormalized) eigenfunctions ofH by using this algebra instead of solving the Schrodinger equation, and pointed out a relation with the well-known su(1,1) symmetry of the Poschl-Teller potential (see [4] ans references quoted therein). In the present comment, we want to show that one of the defining relations of the nonlinear algebra, as given in I, is not entirely correct, and should actually contain an additional term, which only disappears in the ν → 1 limit, corresponding to an infinite square well. In support of the amended relation, we will prove that it allows us to algebraically derive the known eigenfunction normalization constant [5]. Finally, we will establish the correct relation between the nonlinear algebra and su(1,1). Let H , b, b be defined as in I by H = p 2m + V (x) V (x) = V0 cos2(kx) V0 = ǫν(ν − 1) ǫ = hk2 2m (1) b = 1 2ǫ [

研究动机与目标

• 识别并更正用于Pöschl-Teller势能的非线性形变代数中一个错误的定义关系。
• 在一般情况下，建立非线性形变代数与标准su(1,1)代数之间正确的代数联系。
• 利用更正后的代数结构代数地推导本征函数归一化常数。
• 证明陈、刘与葛原始关系仅在无限深方势阱极限（ν → 1）下成立。

提出的方法

• 通过分析Pöschl-Teller势能中降算符和升算符b与b†的结构，推导出非线性代数定义关系的正确形式。
• 利用更正后的代数关系，通过代数方法重构已知的本征函数归一化常数。
• 将陈等人的原始关系与更正后的关系进行比较，表明附加项仅在ν → 1时消失。
• 在一般情况下，建立非线性形变代数与标准su(1,1)代数之间的一致联系。
• 验证更正后的关系保持了推导能谱所必需的代数结构，而无需求解薛定谔方程。

实验结果

研究问题

• RQ1 原始论文中非线性形变代数的定义关系是否对所有势参数ν值都成立，还是仅在极限情况下成立？
• RQ2 在一般Pöschl-Teller势能情况下，非线性代数定义关系的正确形式是什么？
• RQ3 如何利用更正后的非线性代数代数地推导本征函数归一化常数？
• RQ4 该体系中非线性形变代数与标准su(1,1)代数之间的精确代数关系是什么？

主要发现

• 陈、刘与葛提出的非线性代数定义关系在一般情况下是错误的，其中包含了未被考虑的附加项。
• 正确关系仅在极限情况ν → 1时退化为原始关系，对应于无限深方势阱势能。
• 更正后的代数结构允许对已知本征函数归一化常数进行一致且代数化的推导。
• 通过更正后的关系，非线性形变代数与标准su(1,1)代数建立了恰当联系，保持了物理意义的完整性。