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Iranian Journal of Science

Marginal behavior of the quantum potential in Extended Phase Space

Samira Bahrami, S. Nasiri

Published Jul 6, 2011• 0 citations
Publisher

Abstract

Here, the concept of quantum potential that has been illustrated in Extended Phase Space (EPS) in previous works is explored for its marginal behavior. Unlike in configuration space, different representations of the quantum mechanics can be found in EPS when exploiting appropriate canonical transformations. These canonical transformations revealed that there exist several representations in which the quantum potential could be removed not only for a particle in constant, linear and quadratic potentials, but also for a free particle. In these representations one still has the form-invariance of the ordinary Hamilton-Jacobi equation. The present work is an attempt to show that the Squeezed Wigner representation is a possible representation in which, for large values of squeezing parameter, the quantum potential disappears from the dynamical equation for a particle in the general potential

DOI
10.22099/ijsts.2011.2122

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Here, the concept of quantum potential that has been illustrated in Extended Phase Space (EPS) in previous works is explored for its marginal behavior. Unlike in configuration space, different representations of the quantum mechanics can be found in EPS when exploiting appropriate canonical transformations. These canonical transformations revealed that there exist several representations in which the quantum potential could be removed not only for a particle in constant, linear and quadratic potentials, but also for a free particle. In these representations one still has the form-invariance of the ordinary Hamilton-Jacobi equation. The present work is an attempt to show that the Squeezed Wigner representation is a possible representation in which, for large values of squeezing parameter, the quantum potential disappears from the dynamical equation for a particle in the general potential
The behavior of the quantum potential is studied for a particle in a linear and a harmonic potential by means of an extended phase space technique.This is done by obtaining an expression for the quantum potential in momentum space representation followed by the generalization of this concept to extended phase space.It is shown that there exists an extended canonical transformation that removes the expression for the quantum potential in the dynamical equation.The situation, mathematically, is similar to disappearance of the centrifugal potential in going from the spherical to the Cartesian coordinates that changes the physical potential to an effective one.The representation where the quantum potential disappears and the modified Hamilton-Jacobi equation reduces to the familiar classical form, is one in which the dynamical equation turns out to be the Wigner equation.
In a previous work the concept of quantum potential is generalized into extended phase space (EPS) for a particle in linear and harmonic potentials.It was shown there that in contrast to the Schrödinger quantum mechanics by an appropriate extended canonical transformation one can obtain the Wigner representation of phase space quantum mechanics in which the quantum potential is removed from dynamical equation.In other words, one still has the form invariance of the ordinary Hamilton-Jacobi equation in this representation.The situation, mathematically, is similar to the disappearance of the centrifugal potential in going from the spherical to the Cartesian coordinates.Here we show that the Husimi representation is another possible representation where the quantum potential for the harmonic potential disappears and the modified Hamilton-Jacobi equation reduces to the familiar classical form.This happens when the parameter in the Husimi transformation assumes a specific value corresponding to Q-function.
Extended phase space (EPS) formulation of quantum statistical mechanics treats the ordinary phase space coordinates on the same footing and thereby permits the definite the canonical momenta conjugate to these coordinates . The extended lagrangian and extended hamiltonian are defined in EPS by the same procedure as one does for ordinary lagrangian and hamiltonian. The combination of ordinary phase space and their conjugate momenta exhibits the evolution of particles and their mirror images together. The resultant evolution equation in EPS for a damped harmonic oscillator, is such that the energy dissipated by the actual oscillator is absorbed in the same rate by the image oscillator leaving the whole system as a conservative system. We use the EPS formalism to obtain the dual hamiltonian of a damped harmonic oscillator, first proposed by Batemann, by a simple extended canonical transformations in the extended phase space. The extended canonical transformations are capable of converting the damped system of actual and image oscillators to an undamped one, and transform the evolution equation into a simple form. The resultant equation is solved and the eigenvalues and eigenfunctions for damped oscillator and its mirror image are obtained. The results are in agreement with those obtained by Bateman. At last, the uncertainty relation are examined for above system.
An ordinary unambiguous integral representation for the finite propagator of a quantum system is found by starting of a privileged skeletonization of the functional action in phase space, provided by the complete solution of the Hamilton-Jacobi equation. This representation allows to regard the propagator as the sum of the contributions coming from paths where the momenta generated by the complete solution of the Hamilton-Jacobi equation are conserved -as it does happen on the classical trajectory-, but are not restricted to having the classical values associated with the boundary conditions for the original coordinates.
The problem of expressing quantum mechanical expectation values as averages with respect to nonnegative density functions on phase space, by analogy with classical mechanics, is reexamined in the light of some earlier work on fuzzy phase spaces. It is shown that such phase space representations are possible if ordinary phase space is replaced by a so-called fuzzy phase space, on which the usual marginal distribution functions are redefined to conform to the fact that arbitrarily precise simultaneous measurements on position and momentum are not compatible with quantum mechanics. In the process a generalization of Wigner’s theorem on the nonexistence of phase space representations of quantum mechanics, which also satisfy the standard (classical) marginality conditions in position and momentum, is obtained. It is shown that a (continuous) representation of quantum mechanics exists on a given fuzzy phase space if an only if the corresponding confidence functions for position and momentum measurements satisfy the Heisenberg uncertainty relations.