A path-integral solution for unstable systems

Michel Poirier

The path-integral formulation of quantum mechanics has promoted the study of classical periodic trajectories. In this spirit, the three-body Coulomb problem with a nucleus of infinite mass is considered in the special case of a rigid rotation. Using only Newton's fundamental law for a classical atomic system, one can show that both electrons must be equidistant from the nucleus, and one obtains the solutions previously derived by Klar [Phys. Rev. Lett. 57, 66 (1986); Z. Phys. D 3, 353 (1986); J. Opt. Soc. Am. B 4, 788 (1987)]. Nevertheless, if one allows the charge at rest to be arbitrary, one finds that for Z less than unity unsymmetrical solutions may exist. The stability of each of these configurations is considered. In each case an exponentially increasing solution is present, but the unstable character is less marked in the unsymmetrical configuration. Possible consequences for atomic systems are discussed.

Yoshio Ohnuki

De‐Hone Lin

A method for calculating the relativistic path integral solution via sum over perturbation series is given. As an application the exact path integral solution of the relativistic Aharonov–Bohm–Coulomb system is obtained by the method. Different from the earlier treatment based on the space–time transformation and infinite multiple-valued trasformation of Kustaanheimo–Stiefel in order to perform path integral, the method developed in this contribution involves only the explicit form of a simple Green’s function and an explicit path integral is avoided.

Mustafa Keskín, Mehmet Arı, Paul H. E. Meijer

Horacio Grinberg, Julio Marañón