Extension of the Staruszkiewicz Modification of the Schroedinger Equation

Abstract

We present an extension of Staruszkiewicz's modification of the Schr\"{o}dinger equation which preserves its main and unique feature: in the natural system of units the modification terms do not contain any dimensional constants. The extension, similarly as the original, is formulated in a three-dimensional space and derives from a Galilean invariant Lagrangian. It is pointed out that this model of nonlinearity violates the separability of compound systems in the fundamentalist approach to this issue. In its general form, this modification does not admit stationary states for all potentials for which such states exist in linear quantum mechanics. This is, however, possible for a suitable choice of its free parameters. It is only in the original Staruszkiewicz modification that the energy of these states remains unchanged, which marks the uniqueness of this variant of the modification.