skip to main content
OSTI.GOV title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: Quantum mechanics of a constrained particle on an ellipsoid: Bein formalism and Geometric momentum

Abstract

In this work we apply the Dirac method in order to obtain the classical relations for a particle on an ellipsoid. We also determine the quantum mechanical form of these relations by using Dirac quantization. Then by considering the canonical commutation relations between the position and momentum operators in terms of curved coordinates, we try to propose the suitable representations for momentum operator that satisfy the obtained commutators between position and momentum in Euclidean space. We see that our representations for momentum operators are the same as geometric one.

Authors:
;
Publication Date:
OSTI Identifier:
22617370
Resource Type:
Journal Article
Resource Relation:
Journal Name: Annals of Physics; Journal Volume: 372; Other Information: Copyright (c) 2016 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; COMMUTATION RELATIONS; COMMUTATORS; EUCLIDEAN SPACE; GEOMETRY; QUANTIZATION; QUANTUM MECHANICS

Citation Formats

Panahi, H., E-mail: t-panahi@guilan.ac.ir, and Jahangiri, L., E-mail: laleh.jahangiry@yahoo.com. Quantum mechanics of a constrained particle on an ellipsoid: Bein formalism and Geometric momentum. United States: N. p., 2016. Web. doi:10.1016/J.AOP.2016.04.013.
Panahi, H., E-mail: t-panahi@guilan.ac.ir, & Jahangiri, L., E-mail: laleh.jahangiry@yahoo.com. Quantum mechanics of a constrained particle on an ellipsoid: Bein formalism and Geometric momentum. United States. doi:10.1016/J.AOP.2016.04.013.
Panahi, H., E-mail: t-panahi@guilan.ac.ir, and Jahangiri, L., E-mail: laleh.jahangiry@yahoo.com. 2016. "Quantum mechanics of a constrained particle on an ellipsoid: Bein formalism and Geometric momentum". United States. doi:10.1016/J.AOP.2016.04.013.
@article{osti_22617370,
title = {Quantum mechanics of a constrained particle on an ellipsoid: Bein formalism and Geometric momentum},
author = {Panahi, H., E-mail: t-panahi@guilan.ac.ir and Jahangiri, L., E-mail: laleh.jahangiry@yahoo.com},
abstractNote = {In this work we apply the Dirac method in order to obtain the classical relations for a particle on an ellipsoid. We also determine the quantum mechanical form of these relations by using Dirac quantization. Then by considering the canonical commutation relations between the position and momentum operators in terms of curved coordinates, we try to propose the suitable representations for momentum operator that satisfy the obtained commutators between position and momentum in Euclidean space. We see that our representations for momentum operators are the same as geometric one.},
doi = {10.1016/J.AOP.2016.04.013},
journal = {Annals of Physics},
number = ,
volume = 372,
place = {United States},
year = 2016,
month = 9
}
  • The canonical quantization is a procedure for quantizing a classical theory while preserving the formal algebraic structure among observables in the classical theory to the extent possible. For a system without constraint, we have the so-called fundamental commutation relations (CRs) among positions and momenta, whose algebraic relations are the same as those given by the Poisson brackets in classical mechanics. For the constrained motion on a curved hypersurface, we need more fundamental CRs otherwise neither momentum nor kinetic energy can be properly quantized, and we propose an enlarged canonical quantization scheme with introduction of the second category of fundamental CRsmore » between Hamiltonian and positions, and those between Hamiltonian and momenta, whereas the original ones are categorized into the first. As an N − 1 (N ⩾ 2) dimensional hypersurface is embedded in an N dimensional Euclidean space, we obtain the proper momentum that depends on the mean curvature. For the spherical surface, a long-standing problem in the form of the geometric potential is resolved in a lucid and unambiguous manner, which turns out to be identical to that given by the so-called confining potential technique. In addition, a new dynamical group SO(N, 1) symmetry for the motion on the sphere is demonstrated.« less
  • The authors develop, in the framework of two-particle relativistic quantum mechanics, the formalism needed to describe massless bound state systems and their internal dynamics. It turns out that the dynamics here is two-dimensional, besides the contribution of the spin degrees of freedom, provided by the two space-like transverse components of the relative coordinate four-vector, decomposed in an appropriate light cone basis. This is in contrast with the massive bound state case, where the dynamics is three-dimensional. They also construct the scalar product of the theory. They apply this formalism to several types of composite systems.
  • General structural features of dynamical theories can be exhibited in relations between classical and quantum mechanics; the essential structure is a Lie algebra of basic dynamical-variable fanctions providing transformationgroup infinitesimal generators. It is shown that the particular representation chosen is not important for the dynamical structure analysis by considering possibilities of transcribing classical and quantum mechanics each into the natural representation of the other. This leads to an outlining of the formal theory of generalized dynamics by constructing a class of Lie algebras including those of classical and quantum mechanics as special cases. (L.N.N.)
  • We analyze two approaches to the quantum-classical Liouville (QCL) formalism that differ in the order of two operations: Wigner transformation and projection onto adiabatic electronic states. The analysis is carried out on a two-dimensional linear vibronic model where geometric phase (GP) effects arising from a conical intersection profoundly affect nuclear dynamics. We find that the Wigner-then-Adiabatic (WA) QCL approach captures GP effects, whereas the Adiabatic-then-Wigner (AW) QCL approach does not. Moreover, the Wigner transform in AW-QCL leads to an ill-defined Fourier transform of double-valued functions. The double-valued character of these functions stems from the nontrivial GP of adiabatic electronic statesmore » in the presence of a conical intersection. In contrast, WA-QCL avoids this issue by starting with the Wigner transform of single-valued quantities of the full problem. As a consequence, GP effects in WA-QCL can be associated with a dynamical term in the corresponding equation of motion. Since the WA-QCL approach uses solely the adiabatic potentials and non-adiabatic derivative couplings as an input, our results indicate that WA-QCL can capture GP effects in two-state crossing problems using first-principles electronic structure calculations without prior diabatization or introduction of explicit phase factors.« less