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Journal of Physics B Atomic Molecular and Optical Physics

The Berry phase and the hydrogen molecular ion in an external magnetic field

W. N. Cottingham, N Hassan

Published Jan 28, 1990• 9 citations
Publisher

Abstract

The authors consider the rotational spectrum of the hydrogen molecular ion in a magnetic field. It is shown that the Berry phase factor for this system acts like an additional magnetic field which effectively reduces the external field by about 5%.
DOI
10.1088/0953-4075/23/2/014

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The Berry phase for a variety of systems comprising of two angular momenta is discussed. These include the electron and proton in the ground state of the hydrogen atom (taking into account the hyperfine interaction), the positronium atom, the spin and orbital angular momenta of a single electron, muon capture by the deuteron in an external magnetic field, etc. Though the time scales involved, the underlying intrinsic Hamiltonians are quite different, as also the possible experimental probes, the geometric nature of the results for the Berry phase due to a time varying externally imposed magnetic field is found to be quite robust. Some indications are also put forward as to the possible interesting studies with time varying electric fields. The objective of this work is an attempt to broaden the scope of such studies in both the experimental and theoretical directions.
The experimental observation of effects due to Berry's phase in quantum systems is certainly one of the most impressive demonstrations of the correctness of the superposition principle in Quantum Mechanics. Since Berry's original paper in 1984, the spin 1/2 coupled with rotating external magnetic field has been one of the most studied model where those phases appear. We also consider this soluble model. A detailed analysis of the coupled differential equations and comparison with exact results teach us why the usual procedure (of neglecting non-diagonal terms) is mathematically sound.
We consider the hydrogen molecular ion H2+H^+_2H2+​ in the presence of a strong homogeneous magnetic field. In this regime, the effective Hamiltonian is almost one dimensional with a potential energy which looks like a sum of two Dirac delta functions. This model is solvable, but not close enough to our exact Hamiltonian for relevant strenght of the magnnetic field. However we show that the correct values of the equilibrium distance as well as the binding energy of the ground state of the ion, can be obtained when incorporating perturbative corrections up to second order. Finally, we show that $ He_2^{3+}$ exists for sufficiently large magnetic fields.
The experimental observation of effects due to Berry’s phase in quantum systems is certainly one of the most impressive demonstrations of the correctness of the superposition principle in quantum mechanics. Since Berry’s original paper in 1984, the spin 12 coupled with a rotating external magnetic field has been one of the most studied models where those phases appear. We also consider a special case of this soluble model. A detailed analysis of the coupled differential equations and comparison with exact results teach us why the usual procedure (of neglecting nondiagonal terms) is mathematically sound.