# Quantum Motion of Two Particles in a 3D Trigonometric Pöschl-Teller Potential

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Exact solutions of the nonrelativistic wave equations contain all the necessary information for the quantum system and have important applications in particle physics. This Demonstration discusses a solution of the Schrödinger equation in three-dimensional configuration space with the trigonometric Pöschl–Teller potential in the Bohm approach.

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Contributed by: Klaus von Bloh (March 2015)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

Associated Legendre polynomials arise as the solution of the Schrödinger equation:

,

with the integers , , , , and so on. A degenerate, unnormalized wavefunction with time period for for the three-dimensional case can be expressed by:

,

where , , are eigenfunctions, and are permuted eigenenergies of the corresponding stationary one-dimensional Schrödinger equation with . The eigenfunctions are defined by

,

where , , are associated Legendre polynomials. The parameter is a constant phase shift , and are the quantum numbers with and . The wavefunction is taken from [2].

For this Demonstration, the wavefuction is defined by:

.

Due to the permuted wavefunction structure and depending on the constant phase shift , most of the nodal points are positioned only in the , plane.

The velocity field is calculated from the gradient of the phase from the total wavefunction in the eikonal form (often called polar form) . The time-dependent phase function from the total wavefunction is:

, with

.

The corresponding velocity field becomes time independent (autonomous) because of the gradient of the phase function.

The velocity in the direction becomes zero if , which is fulfilled for . The velocities in the other directions become zero for . The length of the curve depends on the constant phase shift .

In the program, if PlotPoints, AccuracyGoal, PrecisionGoal, and MaxIterations are increased, the results will be more accurate.

References

[1] G. Pöschl, E. Teller, "Bemerkungen zur Quantenmechanik des anharmonischen Oszillators," *Zeitschrift für Physik*,* *83 (3–4), 1933 pp. 143–151. doi:10.1007/BF01331132.

[2] M. Trott, *The Mathematica GuideBook for Symbolics*, New York: Springer-Verlag, 2006.

[3] "Bohmian-Mechanics.net." (Mar 16, 2015) www.bohmian-mechanics.net/index.html.

[4] S. Goldstein. "Bohmian Mechanics." *The Stanford Encyclopedia of Philosophy*. (Mar 16, 2015)plato.stanford.edu/entries/qm-bohm.

[5] K. von Bloh. *The Quantum Motion of Eight Particles in a 3D Trigonometric Pöschl–Teller Potential*. [Video]. (Mar 16, 2015) www.youtube.com/watch?v=ejJF3cSWa4Y.

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