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Bohmian Quantum Trajectories for Coherent States of the Pöschl-Teller Potential

The trigonometric Pöschl–Teller potential (proposed for the first time in 1933 [1]) approximates the diatomic molecular potential energy. In the causal interpretation, superposed states, constant phase shifts, and non-factorizability are necessary for quantum motion to occur. Under certain circumstances, the motion might be chaotic (see Related Links). This Demonstration studies a two-dimensional version of the trigonometric Pöschl–Teller potential, which is in a special superposition state. If the perturbation term becomes zero (), the wave density (given by the square of the Schrödinger wavefunction) is not time dependent, but the corresponding autonomous differential equation system for the velocity field leads to a periodic motion in the configuration space. With the perturbation term, the Bohmian trajectory forms periodic, quasi-periodic, weak ergodic, or chaotic curves while interacting with the nodal points. Further investigations are necessary to capture the full dynamics of this system. The graphic shows the squared wavefunction, the trajectory (white), the nodal points (blue), the contour lines of the potential (gray), and the velocity field (red).

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Associated Legendre polynomials arise as the solution of the Schrödinger equation:
,
with , , , , and so on. A degenerate, unnormalized wavefunction with time period for for the two-dimensional case can be expressed by:
,
where , are eigenfunctions, and are permuted eigenenergies of the corresponding stationary one-dimensional Schrödinger equation with . In the wavefunction , the perturbation term is given by . The eigenfunctions are defined by
,
where , are associated Legendre polynomials. The parameter is an arbitrary constant, is a constant phase shift (here ), and are the quantum numbers with and . The wavefunction is taken from [2].
For this Demonstration the three cases are distinguished by:
,
,
.
The velocity field is calculated from the gradient of the phase from the total wavefunction in the eikonal form (often called polar form) . For the three cases, the phase functions are:
,
,
.
For , the velocity field becomes time independent (autonomous).
In the program, if PlotPoints, AccuracyGoal, PrecisionGoal, and MaxSteps 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, 2006.
[3] "Bohmian-Mechanics.net." (Sep 26, 2014) www.bohmian-mechanics.net/index.html.
[4] S. Goldstein. "Bohmian Mechanics." The Stanford Encyclopedia of Philosophy. (Sep 26, 2014) plato.stanford.edu/entries/qm-bohm.
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