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.

Definition

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 ${\displaystyle u=\mathrm {tanh(x)} }$, which yields

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

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

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

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

See also

• Morse potential
• Trigonometric Rosen–Morse potential