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

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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].


Contributed by:Klaus von Bloh (July 2015)
Open content licensed under CC BY-NC-SA



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.

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[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] "" (Jul 30, 2015)

[6] S. Goldstein. "Bohmian Mechanics." The Stanford Encyclopedia of Philosophy. (Jul 30, 2015)

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