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 Schrödinger equation where this difference does not occur, which makes them more consistent in a manner similar to that of the linear Schrödinger equation. It is found that necessary but not sufficient a condition for such modifications is the homogeneity of the modified Schrödinger 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.

Slightly extended, new references added, Latex, 15 pages

References in corpus (2)

• Nonlinear von Neumann-type equations
• Gisin Nonlocality of the Doebner-Goldin 2-Particle Equation

Cited by in corpus (7)

• Informal Resource Letter - Nonlinear quantum mechanics on arXiv up to August 2004
• Quantum-gravity in a dynamical system perspective
• Non-separability without Non-separability in Nonlinear Quantum Mechanics
• On Solutions to the Nonlinear Phase Modification of the Schroedinger Equation
• Extension of the Staruszkiewicz Modification of the Schroedinger Equation
• Higher Order Modification of the Schroedinger Equation
• Nonlinear Phase Modification of the Schroedinger Equation