Microarticle Equivalent period for a stationary quantum system

A generalization of the Kepler’s third law has been proposed for classical and quantum N-body systems in a Newtonian gravitation field. This implies the definition of the equivalent of a period for a stationary quantum system. In this paper, it is shown that a significant quantum definition for the equivalent of a period is possible and coincides with the quantities defined phenomenologically for the generalization of the Kepler’s third law. The Kepler’s third law has certainly a great historical significance, but this relation between the period and the size of the orbit, or the period and the energy of the orbit, applies only for classical two-body systems. Nevertheless, a generalization for classical and quantum Nbody systems in gravitational interaction has been proposed recently [1–3]. A problem is the necessity to define the equivalent of a period for a stationary quantum system. In [3], a formula has been proposed for the two-body system with a combination of some mean values of observables, and another one for the N-body systems with semiclassical considerations. It seems thus desirable to have a clear and unique definition for a quantum period. The idea is to build from a classical system a formula which can be computed for the equivalent quantum system. Let us consider a particle of mass mmoving nonrelativistically along a bounded trajectory C (this is also valid for a relative two-body motion with a reduced mass m). In one dimension, the motion is periodic, while in three dimensions, the periodicity is only guaranteed for a harmonic oscillator or a Coulomb system. If the motion has a period , the action I is computed by = = + p r r I d m dt 1 2 · 1 2 . C t t 2 0 0 (1) Defining the classical mean value of a quantity A by the integral