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arXiv:quant-ph/9709052 · American Journal of Physics

The hydrogen atom as an entangled electron–proton system

Paolo Tommasini, E.A.H. Timmermans, A. F. R. de Toledo Piza

Published Oct 1, 1998• 48 citations
arXivHTMLPDF

Abstract

We illustrate the description of correlated subsystems by studying the simple two-body hydrogen atom. We study the entanglement of the electron and proton coordinates in the exact analytical solution. This entanglement, which we quantify in the framework of the density matrix formalism, describes correlations in the electron–proton motion.

DOI
10.1119/1.18977

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We illustrate the description of correlated subsystems by studying the simple two-body hydrogen atom. We study the entanglement of the electron and proton coordinates in the exact analytical solution. This entanglement, which we quantify in the framework of the density matrix formalism, describes correlations in the electron–proton motion.
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.
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 center of mass coordinate and the relative coordinates. In the hydrogen atom the attractive force exerted by the proton on electron prevents it from escape. The potential energy V of the system is −e2/r where r is the distance between the electron and the proton. Suppose the nucleus is held fixed at the origin of coordinates. V (r) is independent of θ and φ, that is, insensitive to the direction of r and depends only on the scalar magnitude r. The potential energy remains invariant under rotation and is said to be spherically symmetric. Figure 12.1 shows V (r) as a function of r. In this chapter the hydrogen atom problem in threedimension is solved and momentum eigenfunction is obtained. The problem in D-dimension where D may be noninteger is analysed. The magnetic field produced by a hydrogen atom is worked out. Finally, the system in parabolic coordinates is discussed.
THE HIERARCHY OF QUANTUM SYSTEMS It is frequently noted that some quantum systems are composed of others – their subsystems: For example, a hydrogen atom is composed of a proton and an electron, and an α-particle is composed of two protons and two neutrons. It is less frequently noted that all such quantum systems may also be thought of as subsystems of more extensive systems: Thus, any ten α-particles constitute a quantum system, no matter how widely dispersed – and indeed this system has the important property that its quantum state is required to be symmetric under exchange of any pair of α-particles. In the context of nonrelativistic quantum mechanics, it is reasonable to assume that there is a fixed set of quantum systems, none of whose members has any nontrivial subsystems (i.e., none except itself and the null subsystem Ø , which is a subsystem of every quantum system); any system in this set will be called atomic (note that atoms are not atomic quantum systems in this sense – even a hydrogen atom is composed of an electron and a proton: An electron may be atomic in this sense, but the assumption that it is forms no part of quantum mechanics). All atomic systems together compose the universal quantum system ω , which may be finite or infinite, depending on the total number of atomic systems.
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 atom can be considered a two-particle system, an electron and a proton, with the Coulomb potential acting between them. This chapter shows how to solve the Schrodinger equation for the wave function of the two-particle system. It introduces the coordinates of the center of mass, which will allow us to split the Schrodinger equation into two independent equations, one for the center of mass motion and another for the relative motion of the electron and proton. The left- hand side of the Schrodinger equation depends solely on the spatial variables, whereas the right-hand side depends solely on time. The wave function can be written as a product of the spatial and time-dependent parts. It is very similar in form to the Schrodinger equation for the hydrogen atom except it involves two Laplacians. Since the Laplacians act on separate variables, the chapter helps to solve the Schrodinger equation using the method of separate variables.
We investigate the quantum dynamics of entanglement and fidelity in the hyperfine structure of hydrogen atoms under dephasing noise, modeled via the Lindblad master equation. The effective Hamiltonian captures the spin–spin interaction between the electron and proton, with dephasing incorporated through local Lindblad operators. Analytical solutions for the time-dependent density matrix are derived for various initial states, including separable, partially entangled, and maximally entangled configurations. Entanglement is quantified using the concurrence, while fidelity measures the similarity between the evolving state and the initial state. Numerical results demonstrate that entanglement exhibits oscillatory decay modulated by the dephasing rate, with anti-parallel spin states displaying greater robustness compared to parallel configurations, often leading to entanglement sudden death. Fidelity dynamics reveal similar damped oscillations, underscoring the interplay between coherent hyperfine evolution and environmental dephasing. These insights elucidate strategies for preserving quantum correlations in atomic systems, with implications for quantum information processing and metrology.