Pöschl–Teller Potential
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In mathematical physics, a Pöschl–Teller potential, named after the physicists Herta Pöschl (credited as G. Pöschl) and
Edward Teller Edward Teller (; January 15, 1908 – September 9, 2003) was a Hungarian and American Theoretical physics, theoretical physicist and chemical engineer who is known colloquially as "the father of the hydrogen bomb" and one of the creators of ...
, is a special class of potentials for which the one-dimensional
Schrödinger equation The Schrödinger equation is a partial differential equation that governs the wave function of a non-relativistic quantum-mechanical system. Its discovery was a significant landmark in the development of quantum mechanics. It is named after E ...
can be solved in terms of
special functions Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications. The term is defined by ...
.


Definition

In its symmetric form is explicitly given by : V(x) =-\frac\mathrm^2(x) and the solutions of the time-independent Schrödinger equation : -\frac\psi''(x)+ V(x)\psi(x)=E\psi(x) with this potential can be found by virtue of the substitution u=\mathrm, which yields : \left 1-u^2)\psi'(u)\right+\lambda(\lambda+1)\psi(u)+\frac\psi(u)=0 . Thus the solutions \psi(u) are just the Legendre functions P_\lambda^\mu(\tanh(x)) with E=-\frac, and \lambda=1, 2, 3\cdots, \mu=1, 2, \cdots, \lambda-1, \lambda. Moreover, eigenvalues and scattering data can be explicitly computed. In the special case of integer \lambda, the potential is reflectionless and such potentials also arise as the N-soliton solutions of the Korteweg–De Vries equation. The more general form of the potential is given by : V(x) =-\frac\mathrm^2(x) - \frac\mathrm^2(x) .


Rosen–Morse potential

A related potential is given by introducing an additional term: : V(x) =-\frac\mathrm^2(x) - g \tanh x.


See also

* Morse potential * Trigonometric Rosen–Morse potential


References list


External links


Eigenstates for Pöschl-Teller Potentials
Quantum mechanical potentials Mathematical physics Edward Teller Quantum models {{math-physics-stub