A bidimensional quasi-adiabatic model for muon-catalyzed fusion in muonic hydrogen molecules

Non-relativistic solutions for three-body molecules within a Chern-Simons model

“…This is an expected fact in planar physics, in agreement with what happens in the case of ln(ρ) type interatomic interactions discussed in Ref. [23], where the lepton (electron or muon) is trapped in a potential well created by the nuclei. Now, we would like to stress that the ground state energy, in the case of molecules governed by the Chern-Simons potential, depends on the choice of λ factor, changing by a factor 10 going from the limit λ 0.2 × 10 −3 to the other λ 0.2 × 10 −5 .…”

“…Following the steps of references [21,22,23], the planar time-independent Schrödinger equation describing a three-body quantum system formed by two positive nuclei (in this work we will use just protons and deuterons) and one negative particle (electron or muon) orbiting them is: The Chern-Simons potential of mutual interactions, with…”

“…In order to numerically solve the Eq. ( 9), we first need to find the expression for W ± , following the steps of the references [23,21]. The functions W ± are given by the general expression…”

“…All the calculations were done with the CERN/ROOT package. The energies of different molecules, expressed in hartree, are shown in Table 3, and the values resulting from the potential of the type ln(ρ) in two dimensions and 1/ρ in three dimensions were recalculated from reference [23] with the same scales present in this paper. Molecule The ground state wave functions corresponding to the four different molecules considered here are shown in figure 2.…”

“…In Section 3, the two dimensional time-independent Schrödinger equation, which describes a three-body quantum system formed by two positive nuclei (in this work we will use just protons and deuterons) and one negative particle (electron or muon) is determined in an adiabatic approximation. Doing so, the usual W ± contributions to the effective molecular potential are calculated in Section 4, which allow us to obtain the interatomic potential of each molecule, following the steps of the references [21,22,23] and mainly that of Ref. [17].…”