Energy Ambiguity in Nonlinear Quantum Mechanics

Abstract

We observe that in nonlinear quantum mechanics, unlike in the linear theory, there exists, in general, a difference between the energy functional defined within the Lagrangian formulation as an appropriate conserved component of the canonical energy-momentum tensor and the energy functional defined as the expectation value of the corresponding nonlinear Hamiltonian operator. Some examples of such ambiguity are presented for a particularly simple model and some known modifications. However, we point out that there exist a class of nonlinear modifications of the Schr\"{o}dinger equation where this difference does not occur, which makes them more consistent in a manner similar to that of the linear Schr\"{o}dinger equation. It is found that necessary but not sufficient a condition for such modifications is the homogeneity of the modified Schr\"{o}dinger equation or its underlying Lagrangian density which is assumed to be ``bilinear'' in the wave function in some rather general sense. Yet, it is only for a particular form of this density that the ambiguity in question does not arise. A salient feature of this form is the presence of phase functionals. The present paper thus introduces a new class of modifications characterized by this desirable and rare property.