Pöschl–Teller potential

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In mathematical physics, a Pöschl–Teller potential, named after the physicists Herta Pöschl[1] (credited as G. Pöschl) and Edward Teller, is a special class of potentials for which the one-dimensional Schrödinger equation can be solved in terms of Special functions.

Definition[edit]

In its symmetric form is explicitly given by[2]

Symmetric Pöschl–Teller potential: . It shows the eigenvalues for μ=1, 2, 3, 4, 5, 6.

and the solutions of the time-independent Schrödinger equation

with this potential can be found by virtue of the substitution , which yields

.

Thus the solutions are just the Legendre functions with , and , . Moreover, eigenvalues and scattering data can be explicitly computed.[3] In the special case of integer , the potential is reflectionless and such potentials also arise as the N-soliton solutions of the Korteweg-de Vries equation.[4]

The more general form of the potential is given by[2]

Rosen-Morse Potential[edit]

A related potential is given by an additional term.

[5]

See also[edit]

References list[edit]

  1. ^ "Edward Teller Biographical Memoir." by Stephen B. Libby and Andrew M. Sessler, 2009 (published in Edward Teller Centennial Symposium: modern physics and the scientific legacy of Edward Teller, World Scientific, 2010.
  2. ^ a b Pöschl, G.; Teller, E. (1933). "Bemerkungen zur Quantenmechanik des anharmonischen Oszillators". Zeitschrift für Physik. 83 (3–4): 143–151. Bibcode:1933ZPhy...83..143P. doi:10.1007/BF01331132. 
  3. ^ Siegfried Flügge Practical Quantum Mechanics (Springer, 1998)
  4. ^ Lekner, John (2007). "Reflectionless eigenstates of the sech2 potential". American Journal of Physics. 875 (12): 1151–1157. Bibcode:2007AmJPh..75.1151L. doi:10.1119/1.2787015. 
  5. ^ Barut, A. O.; Inomata, A.; Wilson, R. (1987). "Algebraic treatment of second Poschl-Teller, Morse-Rosen and Eckart equations". Journal of Physics A: Mathematical and General. 20 (13): 4083. Bibcode:1987JPhA...20.4083B. doi:10.1088/0305-4470/20/13/017. ISSN 0305-4470. 

External links[edit]