Pöschl–Teller potential: Difference between revisions
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\left[(1-u^2)\psi'(u)\right]'+\lambda(\lambda+1)\psi(u)+\frac{2E}{1-u^2}\psi(u)=0 |
\left[(1-u^2)\psi'(u)\right]'+\lambda(\lambda+1)\psi(u)+\frac{2E}{1-u^2}\psi(u)=0 |
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</math>. |
</math>. |
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Thus the solutions <math>\psi(u)</math> are just the [[Legendre functions]] <math>P_\lambda^\mu(\tanh(x))</math> with <math>E=\frac{ |
Thus the solutions <math>\psi(u)</math> are just the [[Legendre functions]] <math>P_\lambda^\mu(\tanh(x))</math> with <math>E=-\frac{\mu^2}{2}</math>, and <math>\lambda=1, 2, 3\cdots</math>, <math>\mu=1, 2, \cdots, \lambda-1, \lambda</math>. Moreover, eigenvalues and scattering data can be explicitly computed.<ref>[[Siegfried Flügge]] ''Practical Quantum Mechanics'' (Springer, 1998)</ref> In the special case of integer <math>\lambda</math>, the potential is reflectionless and such potentials also arise as the N-soliton solutions of the [[Korteweg–De Vries equation]].<ref>{{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 }}</ref> |
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The more general form of the potential is given by<ref name="Poschl-Teller"></ref> |
The more general form of the potential is given by<ref name="Poschl-Teller"></ref> |
Latest revision as of 05:48, 26 April 2024
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[edit]
In its symmetric form is explicitly given by[2]
![](http://upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Poschl-Teller_potential.svg/400px-Poschl-Teller_potential.svg.png)
and the solutions of the time-independent Schrödinger equation
with this potential can be found by virtue of the substitution , which yields
- .
Thus the solutions are just the Legendre functions with , and , . Moreover, eigenvalues and scattering data can be explicitly computed.[3] In the special case of integer , 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]
Rosen–Morse potential[edit]
A related potential is given by introducing an additional term:[5]
See also[edit]
References list[edit]
- ^ ""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" (PDF). Archived from the original (PDF) on 2017-01-18. Retrieved 2011-11-29.
- ^ a b 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. S2CID 124830271.
- ^ Siegfried Flügge Practical Quantum Mechanics (Springer, 1998)
- ^ 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.
- ^ 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. 20 (13): 4083. Bibcode:1987JPhA...20.4083B. doi:10.1088/0305-4470/20/13/017. ISSN 0305-4470.
External links[edit]