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Physica A Statistical Mechanics and its Applications

Electron-nuclear entanglement in hydrogen atom

Guo‐Qiang Zhu, Ke-Chun Du, You‐Quan Li

Published Aug 25, 2004• 7 citations
Publisher
DOI
10.1016/j.physa.2004.07.041

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The spectrum of atomic hydrogen has long been viewed as a Rosetta stone that bears the key to decode the writings of quantum mechanics in a vast variety of physical, chemical, and biological systems. Here, we show that, in addition to its role as a basic model of quantum mechanics, the hydrogen atom provides a fundamental building block of quantum information. Through its electron- and nuclear-spin degrees of freedom, the hydrogen atom is shown to lend a physically meaningful frame and a suitable Hilbert space for bipartite entanglement, the two-qubit concurrence and quantum coherence of which can be expressed in terms of the fundamental physical constants---the Planck and Boltzmann constants, electron and proton masses, the fine-structure constant, as well as the Bohr radius and the Bohr magneton. The intrinsic, natural entanglement that the hyperfine-structure (HFS) states of the H atom store at low temperatures rapidly decreases with a growth in temperature, vanishing above a ${\ensuremath{\tau}}_{c}\phantom{\rule{4pt}{0ex}}\ensuremath{\approx}5.35\phantom{\rule{4pt}{0ex}}\ensuremath{\mu}\mathrm{eV}$ threshold. An external magnetic field, however, can overcome this thermal loss of HFS entanglement. As one of the central findings of this paper, we show that an external magnetic field can induce and sustain an HFS entanglement, against all the odds of thermal effects, at temperatures well above the ${\ensuremath{\tau}}_{c}$ threshold, thus enabling magnetic-field-assisted entanglement engineering in low-temperature gases and solids.

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ABSTRACT: In the present work, we have computed the entanglement between the electronic and nuclear motions in two molecular model systems: the one-dimensional hydrogen molecular ion (H2+) and the Shin-Metiu model, considering the molecules as a bipartite systems: electron and nuclear motion. For that purpose, we have computed the Born-Oppenheimer and non-Born-Oppenheimer (Born-Huang) wave function in terms of the Fourier Grid Hamiltonian basis that expands both the electronic and nuclear wave functions. Also, according to the Schmidt decomposition theorem for bipartite systems, widely used in quantum-information theory, there is a much shorter but equivalent expansion in terms of the Schmidt bases for the electronic and nuclear sub-spaces. In these models of distinguishable coupled particles we have shown that the entanglements contents do not increase monotonically with the excitation energy. In the hydrogen molecular ion and in the ShinMetiu model, the entanglements contents for each Born-Oppenheimer electro-nuclear state is quantified through the von-Neumann and linear entropies and we have shown that entanglement serves as a witness of distinguishability of nuclear states related to different Born-Oppenheimer molecular energy curves or electronic excitation modes.
The spectrum of atomic hydrogen has long been viewed as a Rosetta stone that bears the key to decode the writings of quantum mechanics in a vast variety of physical, chemical, and biological systems. Here, we show that, in addition to its role as a basic model of quantum mechanics, the hydrogen atom provides a fundamental building block of quantum information. Through its electron- and nuclear-spin degrees of freedom, the hydrogen atom is shown to lend a physically meaningful frame and a suitable Hilbert space for bipartite entanglement, the two-qubit concurrence and quantum coherence of which can be expressed in terms of the fundamental physical constants---the Planck and Boltzmann constants, electron and proton masses, the fine-structure constant, as well as the Bohr radius and the Bohr magneton. The intrinsic, natural entanglement that the hyperfine-structure (HFS) states of the H atom store at low temperatures rapidly decreases with a growth in temperature, vanishing above a ${\ensuremath{\tau}}_{c}\phantom{\rule{4pt}{0ex}}\ensuremath{\approx}5.35\phantom{\rule{4pt}{0ex}}\ensuremath{\mu}\mathrm{eV}$ threshold. An external magnetic field, however, can overcome this thermal loss of HFS entanglement. As one of the central findings of this paper, we show that an external magnetic field can induce and sustain an HFS entanglement, against all the odds of thermal effects, at temperatures well above the ${\ensuremath{\tau}}_{c}$ threshold, thus enabling magnetic-field-assisted entanglement engineering in low-temperature gases and solids.
This paper discusses Bohr-Somerfield quantization theory and tries to explain how all earlier propositions about energy at atomic level failed to correspond to observed facts and further how this model solved the mathematical discrepancies leading to a near perfect model of energy exchange at the atomic level. The author has attempted here to show the historical connections, logical justifications as well as mathematical expressions for the energy of electron in hydrogen atom and prove how Bohr’s celebrated theory of atomic structure is an application of Planck’s theory of quanta to the Rutherford nuclear atom in an attempt, extraordinary fruitful, to define the nature of the orbits in which the electrons might revolve round the nucleus and to explain the origin of spectral lines of the elements. It further shows how Sommerfeld extended the model so as to include elliptical orbits. The introduction of the elliptic orbits has no new energy for the electron of the hydrogen atom. Bohr had also solved the dilemma arose due to the Rutherford’s atomic model and he admitting the failure of the classical theory, applied with remarkable success of the quantum theory to the Rutherford nuclear atom with revolving electron, which leads us to understand the significance of the Bohr’s atom model.  Keywords: Electromagnetic theory, Model, Electrical attraction, Energy, Atom.