Pöschl–Teller potential

Contents

Pöschl–Teller potential

Introduction

The Pöschl–Teller potential was introduced in 1933 by Heinz Pöschl and Eugène Teller, and it appears in studies connected to Erwin Schrödinger, Paul Dirac, Werner Heisenberg, Wolfgang Pauli, and Lev Landau as a solvable model for one-dimensional quantum systems; its roles have been discussed in contexts involving Max Born, Ludwig Boltzmann, Richard Feynman, Niels Bohr, and Enrico Fermi. The potential is prominent in comparisons with models treated by John von Neumann, Hermann Weyl, Felix Bloch, Eugene Wigner, and Hans Bethe and has been invoked in analyses related to Albert Einstein, Satyendra Nath Bose, Paul Ehrenfest, Otto Stern, and Walter Heitler.

The standard forms commonly studied are the trigonometric and hyperbolic variants described by parameters introduced by Léon Brillouin, Andrei Kolmogorov, Norbert Wiener, Israel Gelfand, and Harish-Chandra; one frequently used hyperbolic form is V(x)=−V0 sech^2(αx) or the related V(x)=V0 tanh^2(αx) with constants linked to work by Emmy Noether, Jacques Hadamard, Élie Cartan, Sophus Lie, and David Hilbert. The eigenvalue problems for these potentials are expressed using special functions associated with investigations by Erdős, Paul Erdős, Gábor Szegő, Salem, Harold Jeffreys, and George Birkhoff, and rely on Sturm–Liouville theory developed in studies involving Stanisław Ulam, John Littlewood, George Green, Bernhard Riemann, and Augustin Louis Cauchy.

Exact bound-state solutions were obtained using techniques related to the hypergeometric equation treated by Carl Friedrich Gauss, Bernhard Riemann, Gustav Kirchhoff, Sofia Kovalevskaya, and Évariste Galois; energies and normalized wavefunctions can be written in terms of associated Legendre and Jacobi polynomials explored by Adrien-Marie Legendre, Carl Gustav Jakob Jacobi, Niels Henrik Abel, Srinivasa Ramanujan, and Arthur Cayley. These solutions have been compared and contrasted in literature involving Eugene Wigner, Lev Landau, J. Robert Oppenheimer, Hans Bethe, and Wolfgang Pauli, and have served as pedagogical examples in courses taught at institutions such as University of Cambridge, Harvard University, Massachusetts Institute of Technology, Princeton University, and University of Göttingen.

The reflectionless nature of specific parameter choices was analyzed in relation to inverse scattering techniques developed by Murray Gell-Mann, Freeman Dyson, Mark Kac, John K. Skilling, and Vladimir Zakharov; these properties connect to soliton theory examined by Martin Kruskal, Norman Zabusky, Mikhail Novikov, Boris Kadomtsev, and Evgeny Korneev. Scattering phase shifts and transmission coefficients have been computed in contexts discussed by Richard Feynman, Julian Schwinger, Sin-Itiro Tomonaga, Lev Landau, and Sydney Coleman, and the reflectionless potentials form families related to results by Gardner, Green, Kruskal, Miura, and Lax.

Algebraic and supersymmetric approaches exploiting factorization methods connect the Pöschl–Teller potential to the supersymmetric quantum mechanics program developed by Edward Witten, Stanley Deser, Sheldon Glashow, Steven Weinberg, and Sergio Ferrara; partner potentials and shape invariance are discussed alongside works by L. Infeld, T. E. Hull, Abraham Shimony, Ernst Ising, and Yoichiro Nambu. Lie-algebraic techniques using representations related to Élie Cartan, Sophus Lie, Hermann Weyl, Harish-Chandra, and Issai Schur provide ladder-operator constructions and spectrum-generating algebras that mirror developments in research by Eugene Wigner, Paul Dirac, John von Neumann, Enrico Fermi, and George Uhlenbeck.

The Pöschl–Teller potential is applied in models of nanostructures discussed in reports from Bell Labs, IBM, Intel Corporation, Bell Telephone Laboratories, and Los Alamos National Laboratory, and appears in treatments of optical waveguides considered by Alexander Graham Bell, Guglielmo Marconi, Heinrich Hertz, Marconi Prize, and Charles Kao. It is used in studies of molecular vibrations and spectroscopy in research connected to Linus Pauling, Ahmed Zewail, Willis Lamb, Richard Smalley, and Fritz London and appears in semiclassical analyses referenced in works by Michael Berry, Marcel Grossmann, Roger Penrose, Stephen Hawking, and Kip Thorne. The potential also arises in field-theory reductions and integrable models investigated by Alexander Polyakov, Vladimir Fock, Ludwig Faddeev, Paul Dirac, and Nicolaas Bloembergen.