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Perturbation Theory in the de Broglie-Bohm Interpretation of Quantum Mechanics

In the de Broglie–Bohm interpretation of quantum mechanics, the particle position and momentum are well defined, and the transition can be described as a continuous evolution of the quantum particle according to the time-dependent Schrödinger equation. There are no "quantum jumps." To study transitions in a two-level system, time-dependent perturbation theory must be used. These solutions are not exact solutions of the Schrödinger equation, but they are extremely accurate. For the particular case of a two-level system perturbed by a periodic external field (but without quantization of the transition-inducing field and ignoring radiation effects), an accurate solution can be derived (see Related Links).
Here a very special transition is studied, which is a particle in a trigonometric two-dimensional Pöschl–Teller potential, where the transition goes from ground state to a perturbed first excited superposition state . Time-dependent transition effect in the Bohm approach was first described in [1]. This system is characterized by the fact that the trajectory varies between periodic and ergodic motion controlled by the factor . In phase space during the quantum flow, the velocity exhibits two nodal points (vortices) and two saddle points, due to the singularities of the wavefunction, which leads to a chaotic motion in the configuration space. Chaos or ergodic motion emerges from the scattering process of the trajectory with the nodal points [2].

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Associated Legendre polynomials arise as the solution of the Schrödinger equation
,
with , , , , and so on. A degenerate, unnormalized wavefunction for the two-dimensional perturbed case, which leads to a transition from the state to the state , can be expressed by
,
where , are eigenfunctions, and are permuted eigenenergies of the corresponding stationary one-dimensional Schrödinger equation, with and . In the wavefunction , the perturbation term is given by , where the parameter is an arbitrary constant. The eigenfunctions , are defined by
,
where , are associated Legendre polynomials. are the quantum numbers with and . The wavefunction is taken from [3].
For , the velocity field obeys the time-independent part of the continuity equation with , where is the complex conjugate. For this special case (), the trajectory becomes strongly periodic, because of the sign changing for with of the velocity term.
In the program, if PlotPoints, AccuracyGoal, PrecisionGoal, and MaxSteps are increased (if enabled), the results will be more accurate.
References
[1] C. Dewdney and M. M. Lam, "What Happens during a Quantum Transition?," Information Dynamics (H. Atmanspacher and H. Scheingraber, eds.), New York: Plenum Press, 1991.
[2] C. Efthymiopoulos, C. Kalapotharakos, and G. Contopoulos, "Origin of Chaos near Critical Points of the Quantum Flow," Physical Review E, 79, 2009 036203. doi:10.1103/PhysRevE.79.036203, arXiv:0903.2655 [quant-ph].
[3] M. Trott, The Mathematica GuideBook for Symbolics, New York: Springer, 2006.
[4] G. Pöschl and E. Teller, "Bemerkungen zur Quantenmechanik des anharmonischen Oszillators," Zeitschrift für Physik, 83 (3–4), 1933 pp. 143–151, doi:10.1007/BF01331132.
[5] "Bohmian-Mechanics.net." (Jul 30, 2015) www.bohmian-mechanics.net/index.html.
[6] S. Goldstein. "Bohmian Mechanics." The Stanford Encyclopedia of Philosophy. (Jul 30, 2015) plato.stanford.edu/entries/qm-bohm.
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