Pöschl–Teller Potential

In mathematical physics, a Pöschl–Teller potential, named after the physicists Herta Pöschl (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

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

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