arXiv math-ph/04090032004-09-01math-phhep-thmath.MPquant-ph

量子力学中的超对称性

Supersymmetry in Quantum Mechanics

Avinash Khare

备注：: Latex file, 4 figures, Lecture Notes presented at EALF 2004, Will be published in the AIP Conference Proceedings AIP
DOI：: 10.1063/1.1853201

摘要：

本文对量子力学中的超对称性问题作了初步的介绍。我们用明确的例子证明，给定一个具有n个束缚态的量子力学中的可解问题，可以构造新的精确可解的n个哈密顿量，它们具有n-1，n-2，.，0绑定状态导出了超对称伙伴势的本征值、本征函数与散射矩阵之间的关系，并构造了一类无反射势。我们将求解一维谐振子问题的算子方法推广到一类形状不变的势。进一步，我们证明了，给定任何至少有一个束缚态的势，可以很容易地构造一个连续的具有相同特征值和s-矩阵的势参数族。本文还讨论了超对称激发的WKB近似（SWKB），结果表明，与通常的WKB不同，最低阶SWKB近似对形状不变势是精确的。最后，我们还利用超对称量子力学构造了新的精确可解周期势。

英文摘要：

An elementary introduction is given to the subject of Supersymmetry in Quantum Mechanics. We demonstrate with explicit examples that given a solvable problem in quantum mechanics with n bound states, one can construct new exactly solvable n Hamiltonians having n-1,n-2,...,0 bound states. The relationship between the eigenvalues, eigenfunctions and scattering matrix of the supersymmetric partner potentials is derived and a class of reflectionless potentials are explicitly constructed. We extend the operator method of solving the one-dimensional harmonic oscillator problem to a class of potentials called shape invariant potentials. Further, we show that given any potential with at least one bound state, one can very easily construct one continuous parameter family of potentials having same eigenvalues and s-matrix. The supersymmetry inspired WKB approximation (SWKB) is also discussed and it is shown that unlike the usual WKB, the lowest order SWKB approximation is exact for the shape invariant potentials. Finally, we also construct new exactly solvable periodic potentials by using the machinery of supersymmetric quantum mechanics.

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