A Simplified Pöschl–Teller Potential: An Instructive Exercise for Introductory Quantum Mechanics,Journal of Chemical Education

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A Simplified Pöschl–Teller Potential: An Instructive Exercise for Introductory Quantum Mechanics
Journal of Chemical Education ( IF 2.5 ) Pub Date : 2018-08-29 00:00:00 , DOI: 10.1021/acs.jchemed.8b00029
Erin Brown ₁ , Lisandro Hernández de la Peña ₁

Affiliation

The quantum mechanics of a simplified Pöschl–Teller potential in one dimension is analyzed at an elementary level. The number of bound states of this potential is determined by a parameter l characterizing the depth of the potential energy. The resulting collection of potentials with varying depth, as well as their ground state energies, is found by requiring the ground state wave function to be the hyperbolic secant function raised to power l. A step-by-step operator-based strategy for the construction of the eigenfunctions corresponding to all other bound states and their eigenenergies is discussed. We also show how all these results can be alternatively obtained by means of the Legendre differential equation, familiar from the rigid-rotor and hydrogen atom problems. The connection between all these problems is discussed while highlighting the use of this model for teaching introductory quantum mechanics.

中文翻译：

简化的Pöschl-Teller电位：量子力学入门课程

在基本层面上分析了一维简化的Pöschl–Teller势的量子力学。该电势的束缚状态数由表征电势能深度的参数l确定。通过要求基态波函数是升为幂l的双曲正割函数，可以发现深度变化的电势及其基态能量的集合。。讨论了基于逐步的基于算子的策略，用于构造与所有其他束缚态及其本征能量相对应的本征函数。我们还展示了如何通过勒格德勒微分方程（对于刚性转子和氢原子问题很熟悉）可以替代地获得所有这些结果。讨论了所有这些问题之间的联系，同时着重介绍了该模型在教授入门量子力学中的使用。

更新日期：2018-08-29

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