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{\lambda(\lambda+1)}{2}\mathrm{sech}^2(x) $$ and the solutions of the time-independent Schrödinger equation

-\frac{1}{2}\psi''(x)+ V(x)\psi(x)=E\psi(x) $$ with this potential can be found by virtue of the substitution $$u=\mathrm{tanh(x)}$$, which yields

\left[(1-u^2)\psi'(u)\right]'+\lambda(\lambda+1)\psi(u)+\frac{2E}{1-u^2}\psi(u)=0 $$. Thus the solutions $$\psi(u)$$ are just the Legendre functions $$P_\lambda^\mu(\tanh(x))$$ with $$E=-\frac{\mu^2}{2}$$, and $$\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{\lambda(\lambda+1)}{2}\mathrm{sech}^2(x) - \frac{\nu(\nu+1)}{2}\mathrm{csch}^2(x). $$

Rosen–Morse potential
A related potential is given by introducing an additional term:



V(x) =-\frac{\lambda(\lambda+1)}{2}\mathrm{sech}^2(x) - g \tanh x. $$