Comment on 'Application of nonlinear deformation algebra to a physical system with Poschl-Teller potential'
Comment on 'Application of nonlinear deformation algebra to a physical system with Poschl-Teller potential'
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DOI：
10.1088/0305-4470/32/38/401
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发表时间：
1999-11
期刊：
影响因子：
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通讯作者：
C. Quesne
中科院分区：
文献类型：
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作者：
C. Quesne
本文评述了Chen，Liu，Ge（J.Phys.A：Math.Gen.31（1998）6473）的一篇论文，其中在具有Poschl-Teller势的精确可解量子力学问题中实现了su（1，1）的包含两个变形函数的非线性变形，并利用它导出了该问题的著名su（1，1）谱生成代数.我们证明了作者提出的非线性代数的定义关系之一只在无限深方阱的极限情况下成立，并确定了一般情况下的正确关系.我们也用它来建立与su（1，1）的正确联系，以及提供本征函数归一化常数的代数推导。简短标题：非线性变形代数的应用PACS：02.10.Tq，03.65.Fd Directeur de recherches FNRS电子邮件地址：cquesne@ulb.ac.be 1在一篇有趣的论文（以下简称为I，其方程将以其前面的编号引用）中，Chen、Liu和Ge [1]最近指出，Delbecq和Quesne [2]引入的具有两个变形函数f（J 0）和g（J 0）的su（2）和su（1，1）李代数的非线性变形可以在量子力学中找到一些有用的应用。他们确实声称已经证明了这样的代数之一可以在具有Poschl-Teller势的物理系统中实现，Poschl-Teller势是精确可解的一维量子力学势之一。从Nieto和Simmons [3]的“自然”量子算符X，P出发，构造了相互伴随的降算符和升算符B，B，它们与哈密顿量H一起生成了一个具有两个变形函数f（H）和g（H）的非线性代数。他们还用这个代数代替解薛定谔方程，得到了H的本征值和（未归一化的）本征函数，并指出了它与著名的Poschl-Teller势的su（1，1）对称性的关系（见[4]和其中引用的参考文献）。在本评论中，我们想表明，在I中给出的非线性代数的一个定义关系并不完全正确，实际上应该包含一个附加项，它只在ν → 1极限时消失，对应于一个无限方阱。为了支持修正后的关系，我们将证明它允许我们代数地导出已知的本征函数归一化常数[5]。最后，我们将建立非线性代数与su（1，1）之间的正确关系。设H，B，B如I中定义为H = p 2 m + V（x）V（x）= V0 cos 2（kx）V0 = hkv（v − 1）n = hk 2 2 m（1）B = 1 2 n [
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: cquesne@ulb.ac.be 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 [