On classical and quantum mechanical problem of two material points in three-dimensional Lobachevsky space

Abstract

The classical and quantum problems of motion of two particles in a three-dimensional Lobachevsky space is formulated with respect to center of mass with arbitrary position. The Hamilton-Jacobi and Schrödinger equations of the problem are formulated and their solutions are found. It is shown that the reduced mass of the system depends on the relative distance. The classical and quantum problems of a rigid rotator in three-dimensional sphere and Lobachevsky space are formulated and solved. The dependences of the oscillation periods of the rotator on the ratio of the masses of the particles forming it are studied for a fixed total mass in the cases of spaces of constant curvature.