dorsal/arxiv
View SchemaEnergy Ambiguity in Nonlinear Quantum Mechanics
| Authors | Waldemar Puszkarz |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/9802001 |
| URL | https://arxiv.org/abs/quant-ph/9802001 |
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.
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"abstract": "We observe that in nonlinear quantum mechanics, unlike in the linear theory,\nthere exists, in general, a difference between the energy functional defined\nwithin the Lagrangian formulation as an appropriate conserved component of the\ncanonical energy-momentum tensor and the energy functional defined as the\nexpectation value of the corresponding nonlinear Hamiltonian operator. Some\nexamples of such ambiguity are presented for a particularly simple model and\nsome known modifications. However, we point out that there exist a class of\nnonlinear modifications of the Schr\\\"{o}dinger equation where this difference\ndoes not occur, which makes them more consistent in a manner similar to that of\nthe linear Schr\\\"{o}dinger equation. It is found that necessary but not\nsufficient a condition for such modifications is the homogeneity of the\nmodified Schr\\\"{o}dinger equation or its underlying Lagrangian density which is\nassumed to be ``bilinear\u0027\u0027 in the wave function in some rather general sense.\nYet, it is only for a particular form of this density that the ambiguity in\nquestion does not arise. A salient feature of this form is the presence of\nphase functionals. The present paper thus introduces a new class of\nmodifications characterized by this desirable and rare property.",
"arxiv_id": "quant-ph/9802001",
"authors": [
"Waldemar Puszkarz"
],
"categories": [
"quant-ph"
],
"title": "Energy Ambiguity in Nonlinear Quantum Mechanics",
"url": "https://arxiv.org/abs/quant-ph/9802001"
},
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