Centre-of-mass energy of hydrogenic ions in a magnetic field

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Abstract
With a canonical transformation, the Hamiltonian of a positively charged ion in a homogeneous magnetic field is transformed into the sum of three commuting operators and of a small term coupling the centre-of-mass and internal motions. The commuting operators are: (i) the corresponding Hamiltonian in the infinite-nuclear-mass approximation, (ii) a small additional Zeeman term, and (iii) a centre-of-mass operator. In the hydrogenic-ion case, the coupling term is treated as a perturbation and does not contribute to first order. Schematically, the perturbation parameter is the product of the reduced field B(2.35*105 T) and the electron-to-nuclear mass ratio. The commuting operators provide an exact first-order centre-of-mass correction to the energy, valid for any field strength. Higher order corrections are calculated approximately for strong fields. Accurate quantum excesses are also determined with a simple variational basis.