A Simplified Pöschl–Teller Potential: An Instructive Exercise for Introductory Quantum Mechanics
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Erin Brown [1] ; Lisandro Hernández de la Peña [1]
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[1] Kettering University
Kettering University
City of Flint, Estados Unidos
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- Localización: Journal of chemical education, ISSN 0021-9584, Vol. 95, Nº 11, 2018, págs. 1989-1995
- Idioma: inglés
- Texto completo no disponible (Saber más ...)
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Resumen
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.
