Steps toward a high precision evaluation of the hyperfine interactions in neutral muonic helium

The solution of the Schrödinger equation of the neutral muonic helium is sketched by an eigenfunction expansion method: the eigenfunctions of the two Coulombic centres problem (of chargesZ ₁=2,Z ₂=–1) are used to expand the wave function. Our Born expansion method is a generalization of the Born-Oppenheimer approximation to a system in which the two centres (He, ^–) do not have a stable equilibrium distance. The adiabatic approximation is solved, upper-lower bounds on the eigenvalue are given for a number of states. The hyperfine energy corrections are calculated in general terms and are given numerically for the ground state and for the first muonically and electronically excited states in the frames of the adiabatic approximation. Our best value fails to give the observed hyperfine splitting of the ground state by some 5 × 10^–4 (2 MHz).