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).[more]
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 . 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 .[less]
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 .
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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