Pöschl–Teller potential

In its symmetric form is explicitly given by{{Cite journal | last1 = Pöschl | first1 = G. | last2 = Teller | first2 = E. | doi = 10.1007/BF01331132 | title = Bemerkungen zur Quantenmechanik des anharmonischen Oszillators | journal = Zeitschrift für Physik | volume = 83 | issue = 3–4 | pages = 143–151 | year = 1933 |bibcode = 1933ZPhy...83..143P | s2cid = 124830271 }} Image:Poschl-Teller_potential.svg

: $$

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 $u=\backslash mathrm\{tanh(x)\}$, 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 $\backslash psi(u)$ are just the Legendre functions $P\_\backslash lambda^\backslash mu(\backslash tanh(x))$ with $E=-\backslash frac\{\backslash mu^2\}\{2\}$, and $\backslash lambda=1,\; 2,\; 3\backslash cdots$, $\backslash mu=1,\; 2,\; \backslash cdots,\; \backslash lambda-1,\; \backslash lambda$. Moreover, eigenvalues and scattering data can be explicitly computed.Siegfried Flügge Practical Quantum Mechanics (Springer, 1998) In the special case of integer $\backslash lambda$, the potential is reflectionless and such potentials also arise as the N-soliton solutions of the Korteweg–De Vries equation.{{Cite journal | last1 = Lekner | first1 = John | doi = 10.1119/1.2787015 | title = Reflectionless eigenstates of the sech2 potential | journal = American Journal of Physics | volume = 875 | issue = 12 | pages = 1151–1157 | year = 2007|bibcode = 2007AmJPh..75.1151L }}

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) .

A related potential is given by introducing an additional term:{{Cite journal|last1=Barut|first1=A. O.|last2=Inomata|first2=A.|last3=Wilson|first3=R.|date=1987|title=Algebraic treatment of second Poschl-Teller, Morse-Rosen and Eckart equations|url=http://stacks.iop.org/0305-4470/20/i=13/a=017|journal=Journal of Physics A: Mathematical and General|language=en|volume=20|issue=13|pages=4083|doi=10.1088/0305-4470/20/13/017|issn=0305-4470|bibcode=1987JPhA...20.4083B}}

:$$

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

• Morse potential
• Trigonometric Rosen–Morse potential

{{Reflist}}

• [http://demonstrations.wolfram.com/EigenstatesForPoeschlTellerPotentials/ Eigenstates for Pöschl-Teller Potentials]

{{DEFAULTSORT:Poschl-Teller Potential}}

Category:Quantum mechanical potentials

Category:Mathematical physics

Category:Edward Teller

Category:Quantum models

{{quantum-stub}}

{{math-physics-stub}}