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

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

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

and the solutions of the time-independent Schrödinger equation

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

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

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

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

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

Rosen–Morse potential

A related potential is given by introducing an additional term:

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

See also

• Morse potential
• Trigonometric Rosen–Morse potential

External links

• Eigenstates for Pöschl-Teller Potentials