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Register a new userCoherent States and Modified de Broglie-Bohm Complex Quantum Trajectories
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Moncy Vilavinal John
Publication date
2011
fields
Physics
and research's language is
English
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This paper examines the nature of classical correspondence in the case of coherent states at the level of quantum trajectories. We first show that for a harmonic oscillator, the coherent state complex quantum trajectories and the complex classical trajectories are identical to each other. This congruence in the complex plane, not restricted to high quantum numbers alone, illustrates that the harmonic oscillator in a coherent state executes classical motion. The quantum trajectories are those conceived in a modified de Broglie-Bohm scheme and we note that identical classical and quantum trajectories for coherent states are obtained only in the present approach. The study is extended to Gazeau-Klauder and SUSY quantum mechanics-based coherent states of a particle in an infinite potential well and that in a symmetric Poschl-Teller (PT) potential by solving for the trajectories numerically. For the coherent state of the infinite potential well, almost identical classical and quantum trajectories are obtained whereas for the PT potential, though classical trajectories are not regained, a periodic motion results as t --> infty.
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A usual assumption in the so-called {it de Broglie - Bohm} approach to quantum dynamics is that the quantum trajectories subject to typical `guiding wavefunctions turn to be quite irregular, i.e. {it chaotic} (in the dynamical systems sense). In the present paper, we consider mainly cases in which the quantum trajectories are {it ordered}, i.e. they have zero Lyapunov characteristic numbers. We use perturbative methods to establish the existence of such trajectories from a theoretical point of view, while we analyze their properties via numerical experiments. Using a 2D harmonic oscillator system, we first establish conditions under which a trajectory can be shown to avoid close encounters with a moving nodal point, thus avoiding the source of chaos in this system. We then consider series expansions for trajectories both in the interior and the exterior of the domain covered by nodal lines, probing the domain of convergence as well as how successful the series are in comparison with numerical computations or regular trajectories. We then examine a H{e}non - Heiles system possessing regular trajectories, thus generalizing previous results. Finally, we explore a key issue of physical interest in the context of the de Broglie - Bohm formalism, namely the influence of order in the so-called {it quantum relaxation} effect. We show that the existence of regular trajectories poses restrictions to the quantum relaxation process, and we give examples in which the relaxation is suppressed even when we consider initial ensembles of only chaotic trajectories, provided, however, that the system as a whole is characterized by a certain degree of order.
We use an alternative interpretation of quantum mechanics, based on the Bohmian trajectory approach, and show that the quantum effects can be included in the classical equation of motion via a conformal transformation on the background metric. We apply this method to the Robertson-Walker metric to derive a modified version of Friedmanns equations for a Universe consisting of scalar, spin-zero, massive particles. These modified equations include additional terms that result from the non-local nature of matter and appear as an acceleration in the expansion of the Universe. We see that the same effect may also be present in the case of an inhomogeneous expansion.
Complex quantum trajectories, which were first obtained from a modified de Broglie-Bohm quantum mechanics, demonstrate that Borns probability axiom in quantum mechanics originates from dynamics itself. We show that a normalisable probability density can be defined for the entire complex plane, though there may be regions where the probability is not locally conserved. Examining this for some simple examples such as the harmonic oscillator, we also find why there is no appreciable complex extended motion in the classical regime.
It is shown that a normalisable probability density can be defined for the entire complex plane in the modified de Broglie-Bohm quantum mechanics, which gives complex quantum trajectories. This work is in continuation of a previous one that defined a conserved probability for most of the regions in the complex space in terms of a trajectory integral, indicating a dynamical origin of quantum probability. There it was also shown that the quantum trajectories obtained are the same characteristic curves that propagate information about the conserved probability density. Though the probability density we now adopt for those regions left out in the previous work is not conserved locally, the net source of probability for such regions is seen to be zero in the example considered, allowing to make the total probability conserved. The new combined probability density agrees with the Borns probability everywhere on the real line, as required. A major fall out of the present scheme is that it explains why in the classical limit the imaginary parts of trajectories are not observed even indirectly and particles are confined close to the real line.
De Broglie - Bohm (dBB) theory is a deterministic theory, built for reproducing almost all Quantum Mechanics (QM) predictions, where position plays the role of a hidden variable. It was recently shown that different coincidence patterns are predicted by QM and dBB when a double slit experiment is realised under specific conditions and, therefore, an experiment can test the two theories. In this letter we present the first realisation of such a double slit experiment by using correlated photons produced in type I Parametric Down Conversion. Our results confirm QM contradicting dBB predictions.
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