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arXiv:math-ph/0502029 · Physical Review Letters

Proof that the Hydrogen-Antihydrogen Molecule Is Unstable

D. K. Gridnev, Carsten Greiner

Published Jun 10, 2005• 13 citations
arXivHTMLPDF

Abstract

In the framework of nonrelativistic quantum mechanics we derive a necessary condition for four Coulomb charges (m1(+), m2(-), m3(+), m4(-)), where all masses are assumed finite, to form the stable system. The obtained stability condition is physical and is expressed through the required minimal ratio of Jacobi masses. In particular, this provides the rigorous proof that hydrogen-antihydrogen and muonium-antimuonium molecules and hydrogen-positron-muon systems are unstable. It also proves that replacing hydrogen in the hydrogen-antihydrogen molecule with its heavier isotopes does not make the molecule stable. These are the first rigorous results on the instability of these systems.

DOI
10.1103/physrevlett.94.223402

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In the framework of nonrelativistic quantum mechanics we derive a necessary condition for four Coulomb charges (m1(+), m2(-), m3(+), m4(-)), where all masses are assumed finite, to form the stable system. The obtained stability condition is physical and is expressed through the required minimal ratio of Jacobi masses. In particular, this provides the rigorous proof that hydrogen-antihydrogen and muonium-antimuonium molecules and hydrogen-positron-muon systems are unstable. It also proves that replacing hydrogen in the hydrogen-antihydrogen molecule with its heavier isotopes does not make the molecule stable. These are the first rigorous results on the instability of these systems.
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We report a theoretical study of antihydrogen-hydrogen molecular resonance states consisting of a positron, an antiproton, an electron and a proton. The four particles strongly correlate and show different character from the hydrogen molecule. Because of the non-separability of the positron and electron motions from the antiproton and proton motions, the adiabatic approximation breaks down at the short antiproton-proton distance. Based on a non-adiabatic method, we directly solve the four-body problem and obtain the resonance energies and widths. In order to examine the roles of positron–antiproton and electron–proton correlations as well as positron–electron and antiproton–proton correlations, we introduce two different types of coordinate systems. One is suited for describing an antihydrogen–hydrogen configuration, and the other is a positronium–protonium configuration. The antihydrogen–hydrogen configuration contributes to the existence of the molecular resonance states, and the positronium–protonium configuration makes the resonance states unstable. Mixing of the two configurations results in an antihydrogen-hydrogen molecular resonance state, and the resonance state has an energy of −0.077 9(3) a.u. from the antihydrogen–hydrogen dissociation threshold with its lifetime 16 (2) fs.
The family of four-particle systems (M+{\mathrm{M}}^{+}M+m+{\mathrm{m}}^{+}m+${\mathrm{M}}^{\mathrm{\ensuremath{-}}}$${\mathrm{m}}^{\mathrm{\ensuremath{-}}}$) has been studied by means of Monte Carlo techniques. Nonadiabatic explicitly correlated wave functions for different values of the mass ratio M/m have been obtained using a variational Monte Carlo optimization method. These wave functions have been used in diffusion Monte Carlo simulations of (M+{\mathrm{M}}^{+}M+m+{\mathrm{m}}^{+}m+${\mathrm{M}}^{\mathrm{\ensuremath{-}}}$${\mathrm{m}}^{\mathrm{\ensuremath{-}}}$) to compute exact ground-state energies. Our results enlarge the stability range of the mass ratio for these and for similar less symmetric systems and address the problem of the stability of the hydrogen-antihydrogen system. For the special case of the dipositronium molecule (M=m) we report the ground-state energy, consistent with previous accurate calculations, and average values of various observables.
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We present recent works [15,16] on the thermodynamic limit of quantum Coulomb systems. We provide a general method which allows to show the existence of the limit for many different systems. Ordinary matter is composed of electrons (negatively charged) and nuclei (positively charged) interacting via Coulomb forces. The potential between two particles of charges z and z ′ located at x and x ′ in R 3 is zz ′ |x − x ′ |. There are two difficulties which occur when trying to describe systems composed of electrons and nuclei. Both have to do with the physical problem Talk given by M.L. at QMath10, 10th Quantum Mathematics International Conference, Moeciu, Romania, September 10–15, 2007. c○2008 by the authors. This work may be reproduced, in its entirety, for non-commercial purposes. 2 of stability of quantum systems. The first is due to the singularity of 1/|x | at 0: it is necessary to explain why a particle will not rush to a particle of the opposite charge. One of the first major triumphs of the theory of quantum mechanics is the explanation it gives of the stability of the hydrogen atom (and the complete description of its spectrum) and of other microscopic quantum Coulomb systems, via the uncertainty principle. Stability means that the total energy of the considered system cannot be arbitrarily negative. If there was no such lower bound to the energy it would be possible in principle to extract an infinite amount of energy. One often refers to this kind of stability as stability of the first kind [19,20]. If we denote by E(N) the ground state energy of the system under consideration, for N particles stability of the first kind can be written E(N)> −∞. (1) In proving (1) for Coulomb systems, a major role is played by the uncertainty principle which for nonrelativistic systems is mathematically expressed by the critical Sobolev embedding H 1 (R 3) ֒ → L 6 (R 3). The latter allows to prove Kato’s inequality
The quantum mechanics of two-electron systems is reviewed, starting with the ground state of the helium atom and heliumlike ions with central charge Z. The case of Z≥2 is rather straightforward. In contrast, for negative hydrogen ion with Z=1, the stability of H− cannot be achieved using a product of individual electron wavefunctions and requires explicit account of the anticorrelation among the two electrons. The wavefunction proposed by Chandrasekhar is revisited, where the permutation symmetry is first broken and then restored by a counterterm. More difficult problems can be studied using the same strategy such as the stability of hydrogenlike ions for any value of the proton-to-electron mass ratio M/m, the energy of the lowest spin-triplet state of helium and heliumlike ions, and the stability of the doubly excited hydrogen ion with unnatural parity. The positronium molecule, which was predicted years ago and discovered recently, can also be shown to be stable against spontaneous dissociation. Emphasis is placed on symmetry breaking, which can either spoil or improve the stability.