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

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

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:

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

See also

• Morse potential
• Trigonometric Rosen–Morse potential

External links

• Eigenstates for Pöschl-Teller Potentials