Pöschl–Teller potential explained

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]

V(x) =- (λ ( λ+1 ))/(2) sech ^{2(x)} and the solutions of the time-independent Schrödinger equation- (1)/(2) \psi''(x) + V(x)\psi(x) = E\psi(x) with this potential can be found by virtue of the substitution u = tanh(x) , which yields\left[( 1- u ^{2 )\psi'(u)\right]' +λ ( λ+1 )\psi(u) + (2 E)/(1- u ^{2})} \psi(u) =0 .Thus the solutions \psi(u) are just the Legendre functions P ^{\mu(\tanh(x))} _{λ} with E =- (\mu ^{2})/(2) , and λ=1 , 2 , 3 … , \mu =1 , 2 , … , λ-1 , λ . Moreover, eigenvalues and scattering data can be explicitly computed.^[3] In the special case of integer λ , 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

V(x) =- (λ ( λ+1 ))/(2) sech ^{2(x)} - (\nu(\nu +1 ))/(2) csch ^{2(x)} .

Rosen–Morse potential

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

V(x) =- (λ ( λ+1 ))/(2) sech ^{2(x)} - g \tanh x.

See also

• Morse potential
• Trigonometric Rosen–Morse potential

External links

• Eigenstates for Pöschl-Teller Potentials

Notes and References

1. Web site: "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. . 2011-11-29 . https://web.archive.org/web/20170118171614/https://e-reports-ext.llnl.gov/pdf/376159.pdf . 2017-01-18 . dead .
2. Pöschl . G. . Teller . E. . 10.1007/BF01331132 . Bemerkungen zur Quantenmechanik des anharmonischen Oszillators . Zeitschrift für Physik . 83 . 3–4 . 143–151 . 1933 . 1933ZPhy...83..143P . 124830271 .
3. [Siegfried Flügge]
4. Lekner . John . 10.1119/1.2787015 . Reflectionless eigenstates of the sech2 potential . American Journal of Physics . 875 . 12 . 1151–1157 . 2007. 2007AmJPh..75.1151L .
5. Barut. A. O.. Inomata. A.. Wilson. R.. 1987. Algebraic treatment of second Poschl-Teller, Morse-Rosen and Eckart equations. Journal of Physics A: Mathematical and General. en. 20. 13. 4083. 10.1088/0305-4470/20/13/017. 0305-4470. 1987JPhA...20.4083B.