Continuous Transitions Between Quantum and Classical Motions (1608.07963v2)

Abstract: Using a nonlinear Schr\"{o}dinger equation for the wave function of all systems, continuous transitions between quantum and classical motions are demonstrated for (i) the double-slit set up, (ii) the 2D harmonic oscillator and (iii) the hydrogen-like atom, all of which are of empirical interest.

• The paper proposes a theoretical framework using a modified nonlinear Schrödinger equation to model continuous transitions between quantum and classical motion.
• Simulations of the double-slit, harmonic oscillator, and hydrogen atom validate the framework by showing continuous transitions driven by environmental coupling.
• The research offers an alternative explanation for classical emergence, potentially useful for understanding mesoscopic systems and informing quantum technologies.

Continuous Transitions Between Quantum and Classical Motions: A Detailed Exploration

This paper by Partha Ghose and Klaus von Bloh addresses the enigmatic transition between quantum mechanics and classical mechanics using a modified version of the Schrödinger equation. The authors propose a theoretical framework that enables continuous transitions between quantum and classical motions, with a focus on empirical examples such as the double-slit experiment, the two-dimensional harmonic oscillator, and the hydrogen-like atom.

Conceptual Framework and Methodology

The primary objective of this research is to bridge the theoretical gap between classical and quantum mechanics, a longstanding issue in physics. The paper revisits and builds upon previous work by Ghose, utilizing the causal and ontological interpretation of quantum mechanics to illustrate these transitions. A key insight from this approach is that quantum characteristics of a system can be quenched by strong coupling with the environment, aligning with classical behavior, while reduced coupling leads to quantum behavior.

The authors employ a nonlinear Schrödinger equation formulated to encompass both classical and quantum systems. This equation incorporates a coupling term, $\lambda(t)Q$, where $Q$ is the quantum potential and $\lambda(t)$ is a time-dependent function modulating the strength of the environment's influence. The paper explores this concept through simulations of specific systems widely recognized in physics: the double-slit setup, a two-dimensional harmonic oscillator, and a hydrogen-like atom.

Numerical Simulations and Results

1. Double-Slit Experiment: The paper simulates the famous double-slit interference patterns by varying the coupling to the environment. As the coupling decreases, the interference patterns emerge from classical particle trajectories, effectively demonstrating the transition from classical to quantum states. The authors provide numerical results, showing how trajectories are visibly shifted as the system moves between quantum and classical regimes.
2. 2D Harmonic Oscillator: A superposition of harmonic oscillator states is examined to show transitions in a two-dimensional space. The results illustrate how classical elliptical orbits develop from Bohmian trajectories as the coupling increases. This example showcases the mechanics of entangled states and their classical signatures when influenced by an environment.
3. Hydrogen-Like Atom: For the hydrogen-like atom, the transition from quantum to classical orbits is demonstrated. The trajectories of electrons in Bohmian mechanics smoothly transition into classical Keplerian orbits, which align with conventional Newtonian physics as the system's coupling constant changes.

These simulations underscore the effectiveness of the nonlinear Schrödinger framework in explaining mesoscopic systems, where both quantum and classical traits coexist.

Theoretical Implications and Future Directions

This work significantly contributes to understanding how classical behavior emerges from quantum dynamics, an area often addressed through decoherence theories. By proposing a nonlinear dynamic model that inherently includes coupling to the environment, the authors offer a compelling alternative that might circumvent some limitations of decoherence, such as basis dependence.

Furthermore, the practical implications of this research could extend to areas where classical and quantum systems overlap, like quantum computing and nanoscale physics, where understanding and controlling transitions could be crucial.

The authors openly suggest that their methodology could be adapted to paper other mesoscopic systems, thus inviting further research and validation through experimental and theoretical advancements. Future research could explore different environmental couplings or modify the model parameters to simulate various physical systems.

Conclusion

The paper by Ghose and von Bloh presents an innovative approach to addressing the intricate relationship between quantum mechanics and classical physics. By demonstrating how a modified Schrödinger equation can simulate smooth transitions via mesoscopic states, this research deepens our conceptual and practical understanding of physics at the blurry boundary between quantum and classical domains. The robustness of their framework, validated through detailed examples, suggests a fertile ground for future exploration in both theoretical and applied physics.