The algebra of polynomials in operators that represent generalized coordinate and momentum and depend on the Planck constant is defined. The Planck constant is treated as the parameter taking values between zero and some nonvanishing $h_0$ . For the second of these two extreme values, introduced operatorial algebra becomes equivalent to the algebra of observables of quantum mechanical system defined in the standard manner by operators in the Hilbert space. For the vanishing Planck constant, the generalized algebra gives the operator formulation of classical mechanics since it is equivalent to the algebra of variables of classical mechanical system defined, as usually, by functions over the phase space. In this way, the semiclassical limit of kinematical part of quantum mechanics is established through the generalized operatorial framework.The algebra of polynomials in operators that represent generalized coordinate and momentum and depend on the Planck constant is defined. The Planck constant is treated as the parameter taking values between zero and some nonvanishing $h_0$. For the second of these two extreme values, introduced operatorial algebra becomes equivalent to the algebra of observables of quantum mechanical system defined in the standard manner by operators in the Hilbert space. For the vanishing Planck constant, the generalized algebra gives the operator formulation of classical mechanics since it is equivalent to the algebra of variables of classical mechanical system defined, as usually, by functions over the phase space. In this way, the semiclassical limit of kinematical part of quantum mechanics is established through the generalized operatorial framework.

arXiv:quant-ph/0103044Mar 9, 20010 citations

Semiclassical Approximation to Quantum Mechanics

WORLD SCIENTIFIC eBooksJan 1, 201247 citations

An Operator Formulation of Classical Mechanics and Semiclassical Limit

Slobodan Prvanović, Z. Marić, Belgrade +1 more

The generalized h-dependent operator algebra is defined ( $0\leq h \leq h_o$ ). For h= h_o it becomes equivalent to the quantum mechanical algebra of observables and for h=0 it is equivalent to the classical one. We show this by proposing how the main features of both mechanics can be defined in operator form.The generalized h-dependent operator algebra is defined ($0\leq h \leq h_o$). For h= h_o it becomes equivalent to the quantum mechanical algebra of observables and for h=0 it is equivalent to the classical one. We show this by proposing how the main features of both mechanics can be defined in operator form.

arXiv:quant-ph/9911033Nov 8, 19991 citation

The semiclassical limit of quantum mechanics: A qualitative theory via stochastic mechanics

Giovanni Jona-Lasinio, Fabio Martinelli, Elisabetta Scoppola

Physics ReportsNov 1, 198141 citations

Symmetric Classical Mechanics

Nuno Costa Dias, João Nuno Prata

We investigate the possibility that the semiclassical limit of quantum mechanics might be correctly described by a classical dynamical theory, other than standard classical mechanics. Using a set of classicality criteria proposed in a related paper, we show that the time evolution of a set of quantum initial data satisfying these criteria is fully consistent with the predictions of a new theory of classical dynamics. The dynamical structure of the new theory is given by the Moyal bracket. This is a Lie bracket that was first derived as the dynamical structure of the Moyal-Weyl-Wigner formulation of quantum mechanics. We present a new derivation of the Moyal bracket, this time in the context of the semiclassical limit of quantum mechanics and thus prove that both classical and quantum dynamics can be formulated in terms of the same canonical structure.We investigate the possibility that the semiclassical limit of quantum mechanics might be correctly described by a classical dynamical theory, other than standard classical mechanics. Using a set of classicality criteria proposed in a related paper, we show that the time evolution of a set of quantum initial data satisfying these criteria is fully consistent with the predictions of a new theory of classical dynamics. The dynamical structure of the new theory is given by the Moyal bracket. This is a Lie bracket that was first derived as the dynamical structure of the Moyal-Weyl-Wigner formulation of quantum mechanics. We present a new derivation of the Moyal bracket, this time in the context of the semiclassical limit of quantum mechanics and thus prove that both classical and quantum dynamics can be formulated in terms of the same canonical structure.

arXiv:quant-ph/0003005Mar 1, 20000 citations

h_0

. For the second of these two extreme values, introduced operatorial algebra becomes equivalent to the algebra of observables of quantum mechanical system defined in the standard manner by operators in the Hilbert space. For the vanishing Planck constant, the generalized algebra gives the operator formulation of classical mechanics since it is equivalent to the algebra of variables of classical mechanical system defined, as usually, by functions over the phase space. In this way, the semiclassical limit of kinematical part of quantum mechanics is established through the generalized operatorial framework.

The generalized h-dependent operator algebra is defined (

0\leq h \leq h_o

). For h= h_o it becomes equivalent to the quantum mechanical algebra of observables and for h=0 it is equivalent to the classical one. We show this by proposing how the main features of both mechanics can be defined in operator form.