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

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  • Author(s): Brown, Erin; de la Peña, Lisandro Hernández (ORCID de la Peña, Lisandro Hernández (ORCID 0000-0003-4646-3986)
  • Language:
    English
  • Source:
    Journal of Chemical Education. Nov 2018 95(11):1989-1995.
  • Publication Date:
    2018
  • Document Type:
    Journal Articles
    Reports - Evaluative
  • Additional Information
    • Availability:
      Division of Chemical Education, Inc. and ACS Publications Division of the American Chemical Society. 1155 Sixteenth Street NW, Washington, DC 20036. Tel: 800-227-5558; Tel: 202-872-4600; e-mail: [email protected]; Web site: http://pubs.acs.org/jchemeduc
    • Peer Reviewed:
      Y
    • Source:
      7
    • Education Level:
      Higher Education
      Postsecondary Education
    • Subject Terms:
      Quantum Mechanics; Mechanics (Physics); Science Instruction; College Science; Graduate Study; Undergraduate Study; Scientific Concepts; Equations (Mathematics)
    • Accession Number:
      10.1021/acs.jchemed.8b00029
    • ISSN:
      0021-9584
    • 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.
    • Abstract:
      As Provided
    • Publication Date:
      2018
    • Accession Number:
      EJ1196850