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arXiv:1302.1628 · Journal of Physics Conference Series

Hydrogen atom as a quantum-classical hybrid system

Fei Zhan, Biao Wu

Published Jun 10, 2013• 0 citations
arXivHTMLPDF

Abstract

Hydrogen atom is studied as a quantum-classical hybrid system, where the proton is treated as a classical object while the electron is regarded as a quantum object. We use a well known mean-field approach to describe this hybrid hydrogen atom; the resulting dynamics for the electron and the proton is compared to their full quantum dynamics. The electron dynamics in the hybrid description is found to be only marginally different from its full quantum counterpart. The situation is very different for the proton: in the hybrid description, the proton behaves like a free particle; in the fully quantum description, the wave packet center of the proton orbits around the center of mass. Furthermore, we find that the failure to describe the proton dynamics properly can be regarded as a manifestation of the fact that there is no conservation of momentum in the mean-field hybrid approach. We expect that such a failure is a common feature for all existing approaches for quantum-classical hybrid systems of Born-Oppenheimer type.

DOI
10.1088/1742-6596/442/1/012006

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Hydrogen atom is studied as a quantum-classical hybrid system, where the proton is treated as a classical object while the electron is regarded as a quantum object. We use a well known mean-field approach to describe this hybrid hydrogen atom; the resulting dynamics for the electron and the proton is compared to their full quantum dynamics. The electron dynamics in the hybrid description is found to be only marginally different from its full quantum counterpart. The situation is very different for the proton: in the hybrid description, the proton behaves like a free particle; in the fully quantum description, the wave packet center of the proton orbits around the center of mass. Furthermore, we find that the failure to describe the proton dynamics properly can be regarded as a manifestation of the fact that there is no conservation of momentum in the mean-field hybrid approach. We expect that such a failure is a common feature for all existing approaches for quantum-classical hybrid systems of Born-Oppenheimer type.
The hydrogen spectrum is explored as a sensitive probe of a massive object being in a superposition of two distinct locations. Here I solve the simpler but instructive semiclassical problem of the proton hopping stochastically between two sites but forbidden to be anywhere else, analogous to the superposition over two locations. Although the model is semiclassical, it captures symmetry properties and the structure of the energy spectrum that one would expect in a fully quantum treatment. The hydrogen atom is a well-understood system whose spectroscopy reaches exceptional precision, and its semiclassical description displays symmetry with respect to two fixed locations. I develop a classical model for an electron and a proton, with the proton stochastically hopping between two fixed sites; and describe its classical dynamics mathematically via a one-particle effective Hamiltonian. For this Hamiltonian I identify a set of invariance properties that are characteristic of heteronuclear diatomic molecules and are not as extensive as the SO(4) symmetries present in the traditional models of the hydrogen atom. I connect the classical description of the proton-hopping model with the two-fixed-centre problem with time-dependent charges. I find that there is a lower limit to the hopping distance of the proton below which state-of-the-art measurements cannot resolve the spectral signature predicted; and also an upper limit above which approximation methods are not applicable. I gain meaningful insight into the mechanisms that shape the quantum dynamics of proton tunnelling in hydrogen by analyzing the classical counterpart, leveraging the shared symmetries between quantum and classical analogous systems. My results predict an observable spectral shift in the ground-state energy when the proton is delocalized over two sites separated by a distance large enough to be considered macroscopically distinct. With this research, I support the idea that proton tunnelling inside the hydrogen atom is worth exploring as a probe for a delocalized massive object.
Recently a new formulation of quantum mechanics has been suggested which is based on the concept of signed particles, that is, classical objects provided with a position, a momentum and a sign simultaneously. In this article, we comment on the plausibility of simulating atomic systems beyond the Born–Oppenheimer approximation by means of the signed particle formulation of quantum mechanics. First, to show the new perspective offered by this new formalism, we provide an example studying quantum tunnelling through a simple Gaussian barrier in terms of the signed particle formulation. Then, we perform a direct simulation of the hydrogen atom as a full quantum two-body system, showing that the formalism can be a very promising tool for first-principle-only quantum chemistry.
The unique properties of the hydrogen atom, especially those that result from the dynamic symmetry of the 1/r potential, are presented from both classical and quantal points of view. Using these properties as a starting point, the response of a classical hydrogen atom, an electron executing a Keplerian ellipse about a proton, to a weak electric field is described. Because the field is weak, the elliptical orbit may be treated as a dynamical element itself, variations in its shape and orientation depicting the classical Stark effect. The behavior of this orbit is then correlated with the quantum mechanical description of the first-order Stark effect in hydrogen, a subject that is included in most introductory courses in quantum mechanics.
Modern quantum theory introduces quantum structures (decompositions into subsystems) as a new discourse that is not fully comparable with the classical-physics counterpart.To this end, socalled Entanglement Relativity appears as a corollary of the universally valid quantum mechanics that can provide for a deeper and more elaborate description of the composite quantum systems.In this paper we employ this new concept to describe the hydrogen atom.We offer a consistent picture of the hydrogen atom as an open quantum system that naturally answers the following important questions: 1) how do the so called "quantum jumps" in atomic excitation and de-excitation occur?and 2) why does the classically and seemingly artificial "center-of-mass + relative degrees of freedom" structure appear as the primarily operable form in most of the experimental reality of atoms?
The hydrogen atom is the simplest of all atoms consisting of proton of charge e and the electron of charge -e. It is a two particle system. A two-body problem can always be reduced to a one-body problem by choosing the centre of mass coordinate and the relative coordinates. The existence of degenerate energy eigenvalues implies that linear combination of the corresponding eigenfunctions are solutions of the wave equation with the same energy. The isotopes of atomic hydrogen, deuterium and tritium also provide examples of hydrogenic systems. Since the mass of deuterium is md˜2mp and that of tritium is mt˜3mp, the reduced masses of them will be slightly different from that of hydrogen. In addition to deuterium and tritium, there exists other hydrogenic systems in which the role of proton is played by another positive particle.
Even after extensive research in Quantum Mechanics, we are still unable to visualize instant-to-instant motion of an electron in hydrogen atom. That is because in QM treatment, potential energy term has been mistakenly assumed to be time-independent instead of depending on the instant-to-instant varying position of the orbiting electron [1]. This has led to wrong and weird solutions for the electron motion in hydrogen atom. Before the advent of wave mechanics, Sommerfeld model of elliptical electron orbits was able to explain most features of hydrogen spectra, except for the features associated with electron spin and magnetic moment interactions. However, the Sommerfeld elliptical orbits were of kinematic origin and could not provide visualization of instant-to-instant dynamic motion of the orbiting electron. Contrary to the QM perspective, we find that central core of the electron behaves as a classical particle while its electrostatic field behaves as a wave phenomenon. As such an electron under Coulomb force moves strictly in accordance with Newtonian laws of motion. In this paper, we develop dynamic electron orbits in hydrogen atom by using energy and angular momentum conservation principle in central force field. We have shown that during photon emission, angular momentum of the orbiting electron is changed by ħ due to recoil action. This may be the origin of various quantization rules. During emission of a photon, elliptical orbit transitions are also computed and plotted. Orbit transition time is of the order of 10-16 seconds. We have extended this methodology to compute electron orbits in hydrogen molecular bond and computed the H2 bond energy.