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USSR Computational Mathematics and Mathematical Physics

The two-centre problem of quantum mechanics

Л. И. Пономарев, Т. П. Пузынина

Published Jan 1, 1968• 6 citations
Publisher
DOI
10.1016/0041-5553(68)90095-5

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In describing the motion of atoms and clusters, we face with choosing quantum mechanics or classical mechanics under different conditions. In principle, there exist two criteria for this choice, but they do contradict in some cases though they are in agreement for other cases. Actually, this problem is closely related with the effective centre-of-mass method, the underlying application of quantum mechanics. It is shown that quantum mechanics must be selected for particle's motion when the de Broglie wave length of the mass centre is larger than the particle size, and in such case the effective centre-of-mass can be used in Quantum Mechanics. In order to test this conclusion, an easy-manufactured experiment is suggested.
A one-particle non-relativistic quantum mechanical solvable model in two-dimensional space is given. The Hamiltonian is the sum of kinetic and interaction parts. Interactions are separable and can be centred at n arbitrary points of the plane. Conditions for the existence and for the number of bound states in finite linear chains are formulated in terms of the parameters of the interactions and intercentre distances. Scattering problems are also considered. Finally, when the interactions are centred in a single centre, it is shown that the model remains solvable in the presence of a uniform magnetic field of arbitrary intensity.