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 $u={\mathrm {tanh(x)}}$, 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]

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

See also

• Morse potential

References list

1. "Edward Teller Biographical Memoir." by Stephen B. Libby and Andrew M. Sessler, 2009 (published in Edward Teller Centennial Symposium: modern physics and the scientific legacy of Edward Teller, World Scientific, 2010.
2. Pöschl, G.; Teller, E. (1933). "Bemerkungen zur Quantenmechanik des anharmonischen Oszillators". Zeitschrift für Physik. 83 (3–4): 143–151. Bibcode:1933ZPhy...83..143P. doi:10.1007/BF01331132.
3. Siegfried Flügge Practical Quantum Mechanics (Springer, 1998)
4. Lekner, John (2007). "Reflectionless eigenstates of the sech2 potential". American Journal of Physics. 875 (12): 1151–1157. Bibcode:2007AmJPh..75.1151L. doi:10.1119/1.2787015.

External links

• Eigenstates for Pöschl-Teller Potentials