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: $-\frac{\lambda(\lambda +1)}{2} \operatorname{sech}^2(x)$. 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.^[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]

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

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. 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. doi:10.1119/1.2787015.

External links

• Eigenstates for Pöschl-Teller Potentials