Eigenstates for Pöschl-Teller Potentials

It has been long known that the Schrödinger equation for a class of potentials of the form , usually referred to as Pöschl–Teller potentials, is exactly solvable. The eigenvalue problem
(in units with has physically significant solutions for , for both bound and continuum states. For , we find the solution , , which follows simply from the derivative relation . More generally, the Schrödinger equation has the bound state solutions
, , , ,
where the are associated Legendre polynomials.
The Schrödinger equation has, in addition, continuum positive-energy eigenstates with . The trivial case gives a free particle . The first two nontrivial solutions are and . These represent waves traveling left to right. A remarkable property of Pöschl-Teller potentials is that they are "reflectionless", meaning that waves are 100% transmitted through the barrier with no reflected waves.



  • [Snapshot]
  • [Snapshot]
  • [Snapshot]


G. Pöschl and E. Teller, "Bemerkungen zur Quantenmechanik des Anharmonischen Oszillators", Z. Phys., 83(3,4), 1933 pp. 143–151.
A recent discussion of reflectionless scattering is given by J. Lekner, "Reflectionless Eigenstates of the Potential," American Journal of Physics, 75(12), 2007 pp. 1151–1157.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2018 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+