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Quantum Guiding Equation Can Determine Classical Behavior of Earth for Large Quantum Numbers

Ali Soltanmanesh, Afshin Shafiee

Published Jan 15, 2018• 0 citations
OpenAlex

Abstract

For quantum systems, we expect to see classical behavior at the limit of large quantum numbers. Hence, we apply Bohmian approach to describe the Earth evolution around the Sun. We obtain possible trajectories of the Earth system with different initial conditions which converge to a definite stable orbit after a given time, known as the Kepler orbit. The trajectories have resulted from the guiding $p=\nabla S$ equation in Bohmian mechanics which relates the momentum of the system to the phase part of the wave function. Except for some special situations, Bohmian trajectories are not Newtonian in character. We show that the classical behavior of the Earth can be described as the consequence of the guiding equation at the limit of large quantum numbers.

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For quantum systems, we expect to see classical behavior at the limit of large quantum numbers. Hence, we apply Bohmian approach to describe the Earth evolution around the Sun. We obtain possible trajectories of the Earth system with different initial conditions which converge to a definite stable orbit after a given time, known as the Kepler orbit. The trajectories have resulted from the guiding p=∇Sp=\nabla Sp=∇S equation in Bohmian mechanics which relates the momentum of the system to the phase part of the wave function. Except for some special situations, Bohmian trajectories are not Newtonian in character. We show that the classical behavior of the Earth can be described as the consequence of the guiding equation at the limit of large quantum numbers.
Macroscopic objects, although quantum mechanical by nature, conform to Newtonian mechanics under normal observation. According to the quantum mechanical correspondence principle, quantum behavior is indistinguishable from classical behavior in the limit of very large quantum numbers. The purpose of this paper is to provide an example of the correspondence principle appropriate for an undergraduate quantum mechanics course by showing that Earth conforms to quantum mechanics while it orbits the Sun. It is argued that the planarity of Earth's orbit implies that the quantum numbers l and m l are equal and very large. It is additionally shown that for Earth in its orbit, the quantum number n is relatively indistinguishable from l and m l . Furthermore, it is demonstrated that the very large common value for the three quantum numbers implies a spatial probability distribution for Earth that is consistent with its actual distribution. In other words, the wavefunction of Earth, whose orbit will be assumed for simplicity to be circular, is consistent with a planar orbit with all angles within the plane equally probable and with a precisely defined Earth–Sun distance.
Bohmian mechanics is a causal interpretation of quantum mechanics in which particles describe trajectories guided by the wave function. The dynamics in the vicinity of nodes of the wave function, usually called vortices, is regular if they are at rest. However, vortices generically move during time evolution of the system. We show that this movement is the origin of chaotic behavior of quantum trajectories. As an example, our general result is illustrated numerically in the two-dimensional isotropic harmonic oscillator.
The method, described in the paper, incorporates ideas of the Bohmian formulation of quantum mechanics into a trajectory approach designed for systems comprised of light (quantum) and heavy (nearly classical) particles. The goal is to treat all degrees of freedom on the same footing but, for practical reasons, to include the quantum-mechanical effects into dynamics for just the light particles. The total wavefunction, taken as a product of slow and fast components representing the heavy and light particles respectively is represented in terms of trajectories. The fast component depends on the trajectory-guided configurations of the heavy particles. The initial conditions for the trajectories are defined by the full-dimensional initial wavefunction according to Bohm's prescription of associating the trajectory momentum with the gradient of the wavefunction phase. The time-dependent Schrödinger equation is solved for the quantum degrees of freedom using an ensemble of the quantum trajectories (or other time-propagation method) for each guiding trajectory of the classical subspace. Neglect of the quantum force acting on the heavy particles allows decoupling of the quantum calculations associated with different guiding trajectories. The approach allows reconstruction of the full-dimensional wavefunction and conserves the total wavefunction energy as illustrated for the vibrationally non-adiabatic model.
The quantum-classical correspondence is studied for a periodically driven quartic oscillator exhibiting integrable and chaotic dynamics, by studying the Bohmian trajectory of the corresponding "classical" Schrödinger equation. Phase plots and the Kolmogorov-Sinai entropy are computed and compared with the classical trajectory as well as the Bohmian trajectory obtained from the time dependent Schrödinger equation. Bohmian mechanics at the classical limit appears to mimick the behavior of a dissipative dynamical system.
Classical trajectories for the Coulomb potential are obtained from the large principle quantum-number limit of solutions to the nonrelativistic Schrödinger equation, by use of integral equations satisfied by the radial probability density function. These trajectories are found to be in excellent agreement with those computed directly from classical mechanics, in accordance with a statement of the Bohr Correspondence principle, except in a region very close to the center of force. PACS No.: 05.45.Mt
Bohmian mechanics is a causal interpretation of quantum mechanics in which particles describe trajectories guided by the wave function. The dynamics in the vicinity of nodes of the wave function, usually called vortices, is regular if they are at rest. However, vortices generically move during time evolution of the system. We show that this movement is the origin of chaotic behavior of quantum trajectories. As an example, our general result is illustrated numerically in the two-dimensional isotropic harmonic oscillator. © EDP Sciences.