Waldyr Alves Rodrigues Jr.

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

It is explicitly given by

Plot of a sample Pöschl–Teller potential

$U(x)=-\frac{\lambda (\lambda +1)}{2}{\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}}^2(x)$

and the solutions of the time-independent Schrödinger equation

$-\frac{1}{2}{\psi }^{″}(x)+U(x)\psi (x)=E\psi (x)$

with this potential can be found by virtue of the substitution $u=\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\mathrm{(}\mathrm{x}\mathrm{)}$, which yields

${[{(}^{\prime }1-u^2){\psi }^{\prime }(u)]}^{\prime }+\lambda (\lambda +1)\psi (u)+\frac{2E}{1-u^2}\psi (u)=0$.

Thus the solutions $\psi (u)$ are just the Legendre functions $P_{\lambda }^{\mu }(u)$ with $E=\frac{-{\mu }^2}{2}$. 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.

See also

• Morse potential

External links

• Eigenstates for Pöschl-Teller Potentials

References list

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