International Journal of Geometric Methods in Modern Physics

QUANTIZATION OF SECOND-ORDER CONSTRAINED LAGRANGIAN SYSTEMS USING THE WKB APPROXIMATION

Eqab M. Rabei, EYAD H. HASSAN, H. B. Ghassib, Sami I. Muslih

Published Jun 1, 2005• 12 citations

Abstract

A general theory is given for quantizing both constrained and unconstrained systems with second-order Lagrangian, using the WKB approximation. In constrained systems, the constraints become conditions on the wave function to be satisfied in the semiclassical limit. This is illustrated with two examples.

DOI: 10.1142/s0219887805000661

Related papers

Quantization of constrained systems using the WKB approximation

Eqab M. Rabei, Khaled I. Nawafleh, H. B. Ghassib

A general theory for solving the Hamilton-Jacobi partial differential equations for constrained Hamiltonian systems is proposed. The quantization of constrained systems is then applied using the WKB approximation. The constraints become conditions on the wave function to be satisfied in the semiclassical limit.

Physical Review AAug 5, 200238 citations

Quantization of Reparametrized Systems Using the WKB Method

Khaled I. Nawafleh, Eqab M. Rabei, H. B. Ghassib

The quantization of reparametrized systems is discussed using the WKB approximation. The Hamilton-Jacobi function, the equations of motion and the wave function---which the conditions constrain in the semiclassical limit---are determined. The success of this approach is then demonstrated for two applications. The first is an illustrative example of nonrelativistic dynamics. The second is the well-known motion of a relativistic particle in an external electromagnetic field.

TURKISH JOURNAL OF PHYSICSJun 2, 20054 citations

Semiclassical Mechanics of Constrained Systems

O. Yu. Shvedov

Semiclassical mechanics of systems with first-class constraints is developed. Starting from the quantum theory, one investigates such objects as semiclassical states and observables, semiclassical inner product, semiclassical gauge transformations and evolution. Quantum mechanical semiclassical substitutions (not only the WKB-ansatz) can be viewed as "composed semiclassical states" being infinite superpositions of wave packets with minimal uncertainties of coordinates and momenta ("elementary semiclassical states"). Each elementary semiclassical state is specified by a set (X,f) of classical variables X (phase, coordinates, momenta) and quantum function f ("shape of the wave packet" or "quantum state in the background X"). A notion of an elemantary semiclassical state can be generalized to the constrained systems, provided that one uses the refined algebraic quantization approach based on modifying the inner product rather than on imposing the constrained conditions on physical states. The inner product of physical states is evaluated. It is obtained that classical part of X the semiclassical state should belong to the constrained surface; otherwise, the semiclassical state (X,f) will have zero norm for all f. Even under classical constraint conditions, the semiclassical inner product is degenerate. One should factorize then the space of semiclassical states. Semiclassical gauge transformations and evolution of semiclassical states are studied. The correspondence with semiclassical Dirac approach is discussed.

arXiv:hep-th/0111265Nov 28, 20015 citations

QUANTIZATION OF DISSIPATIVE SYSTEMS WITH SECOND-ORDER LAGRANGIAN

Ola A. Jarab’ah, Eyad Hasan Hasan, Khaled I. Nawafleh

This paper examined dissipative systems with second order Lagrangian in the framework of the Hamilton-Jacobi method. The principal function is determined using the method of separation of variables in the same manner as for regular time-independent Lagrangian. The equations of motions are obtained from the action function. By finding the action function, the wave function can be constructed and the quantization has been carried out using the WKB approximation. One example is given to illustrate our formalism.

European Scientific Journal ESJMar 31, 20142 citations

Bohr–Sommerfeld, WKB, and modified semiclassical quantization rules

D. J. W. Geldart, David Kiang

The WKB approximation for energy eigenvalues can be advantageously introduced to students when the Bohr–Sommerfeld quantization rule is discussed. A simple heuristic derivation of these quantization rules is given. Exceptional cases occur when conditions necessary for the validity of semiclassical approximations are violated, but appropriate modifications can still provide useful semiclassical approximations even in such cases. Various situations are illustrated by examples.

American Journal of PhysicsFeb 1, 198611 citations

A proposal of quantization of systems without standard lagrangians

Dehai Bao, Zhongyuan Zhu

For many classical systems which disobey Helmholtz conditions and thus possess no standard Lagrangian formulation on velocity phase space TQ , ordinary quantization procedures cannot be directly applied. By shifting to an enlarged space TTQ and viewing the systems as typically constrained ones, we propose a quantization method which seems to be practicable as illustrated by the examples.

Acta Physica Sinica (Overseas Edition)Jun 1, 19962 citations