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In mathematical physics, a '''Pöschl&ndash;Teller potential''', named after the physicists Herta Pöschl<ref>[https://e-reports-ext.llnl.gov/pdf/376159.pdf "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.]</ref> (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.
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==Definition==
 
It is explicitly given by<ref>{{cite doi|10.1007/BF01331132}}</ref> [[File:Poeschl teller potential.png|frame|Plot of a sample Pöschl&ndash;Teller potential]]
 
: <math>
U(x) =-\frac{\lambda(\lambda+1)}{2}\mathrm{sech}^2(x)
</math>
and the solutions of the time-independent Schrödinger equation
: <math>
-\frac{1}{2}\psi''(x)+ U(x)\psi(x)=E\psi(x)
</math>
with this potential can be found by virtue of the substitution <math>u=\mathrm{tanh(x)}</math>, which yields
: <math>
\left[(1-u^2)\psi'(u)\right]'+\lambda(\lambda+1)\psi(u)+\frac{2E}{1-u^2}\psi(u)=0
</math>.
Thus the solutions <math>\psi(u)</math> are just the [[Legendre functions]] <math>P_\lambda^\mu(u)</math> with <math>E=\frac{-\mu^2}{2}</math>. Moreover, eigenvalues and scattering data can be explicitly computed.<ref>[[Siegfried Flügge]] ''Practical Quantum Mechanics'' (Springer, 1998)</ref> In the special case of integer <math>\lambda</math>, the potential is reflectionless and such potentials also arise as the N-soliton solutions of the [[Korteweg-de Vries equation]].
 
==See also==
* [[Morse potential]]
 
==External links==
* [http://demonstrations.wolfram.com/EigenstatesForPoeschlTellerPotentials/ Eigenstates for Pöschl-Teller Potentials]
 
==References list==
{{Reflist}}
 
{{DEFAULTSORT:Poschl-Teller Potential}}
[[Category:Quantum mechanics]]
[[Category:Mathematical physics]]
[[Category:Edward Teller]]
 
 
{{physics-stub}}
{{applied-math-stub}}

Revision as of 07:29, 16 February 2014

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