Pöschl–Teller potential

 In mathematical physics, a Pöschl–Teller potential, named after the physicists Herta Pöschl[1] (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.

DefinitionEdit

In its symmetric form is explicitly given by[2]

Symmetric Pöschl–Teller potential: {\displaystyle -{\frac {\lambda (\lambda +1)}{2}}\operatorname {sech} ^{2}(x)}. It shows the eigenvalues for μ=1, 2, 3, 4, 5, 6.
{\displaystyle V(x)=-{\frac {\lambda (\lambda +1)}{2}}\mathrm {sech} ^{2}(x)}

and the solutions of the time-independent Schrödinger equation

{\displaystyle -{\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 {\displaystyle P_{\lambda }^{\mu }(\tanh(x))} with E={\frac {-\mu ^{2}}{2}}, and {\displaystyle \lambda =1,2,3\cdots }, {\displaystyle \mu =1,2,\cdots ,\lambda -1,\lambda }. Moreover, eigenvalues and scattering data can be explicitly computed.[3] 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.[4]

The more general form of the potential is given by[2]

{\displaystyle V(x)=-{\frac {\lambda (\lambda +1)}{2}}\mathrm {sech} ^{2}(x)-{\frac {\nu (\nu +1)}{2}}\mathrm {csch} ^{2}(x).}

Rosen–Morse potentialEdit

A related potential is given by introducing an additional term:[5]

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