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Journal of Chemical Education

A Simplified Pöschl–Teller Potential: An Instructive Exercise for Introductory Quantum Mechanics

E. L. Brown, Lisandro Hernández de la Peña

Published Aug 29, 2018• 15 citations
Publisher

Abstract

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.

DOI
10.1021/acs.jchemed.8b00029

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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.
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The computation of allowed energy levels for a particle bounded to a finite square well potential is ubiquitous in modern physics and introductory quantum mechanics textbooks. For stationary bound states, the matching conditions for the wave functions lead to a pair of transcendental equations whose roots correspond to the energy eigenvalues. However, the graphical solutions available do not make clear the dependence of the energies on the well potential parameters. In this note, I present a simpler graphical solution involving only one dimensionless parameter that determines a straight-line crossing identical sinusoidal curves. I then reduce this solution to a single cosine curve, and from a three-point interpolation, I derive an approximate formula for all energy levels valid for any square quantum well and that demands only a pocket calculator.
The Dirac-modified Poschl-Teller problem with position dependent mass is investigated within the framework of the asymptotic iteration method in N dimensions. Any -state solutions are obtained by using the exponential approximation for the centrifugal term. The energy eigenvalues and two-component spinor corresponding eigenfunctions are determined for different  screening parameters explicitly. The corresponding eigenfunctions obtained in the form of hypergeometric functions are normalized and plotted for different quantum states. Effects on modified Poschl-Teller potential, bound state energy eigenvalues and normalized corresponding eigenfunctions of the  screening parameters are investigated and its results are discussed.    Key words: Dirac equation, position-dependent mass (PDM), Poschl-Teller (PT) potential, asymptotic iteration method, bound state.