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American Journal of Modern Physics

Kinematical Brownian Motion of the Harmonic Oscillator in Non-Commutative Space

Martin Tchoffo

Published Jan 1, 2014• 3 citations
Publisher

Abstract

In this work the Jacobi’s second equality in the form of stochastic equation and the Wiener path integral approach are used to evaluate the probability density of harmonic oscillator in non-commutative space. Using the factorization theorem and the Mastubara formalism, the thermodynamic parameters are determined. The structure of Fokker-Planck equation remained the same even in a commutative and non-commutative space. Moreover, the non-commutative parameter is depicted for increasing value of the entropy.

DOI
10.11648/j.ajmp.20140303.14

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In this work the Jacobi’s second equality in the form of stochastic equation and the Wiener path integral approach are used to evaluate the probability density of harmonic oscillator in non-commutative space. Using the factorization theorem and the Mastubara formalism, the thermodynamic parameters are determined. The structure of Fokker-Planck equation remained the same even in a commutative and non-commutative space. Moreover, the non-commutative parameter is depicted for increasing value of the entropy.
The theory of Brownian motion of a quantum oscillator is developed. The Brownian motion is described by a model Hamiltonian which is taken to be the one describing the interaction between this oscillator and a reservoir. Use is made of the master equation recently derived by the author, to obtain the equation of motion for the various reduced phase-space distribution functions that are obtained by mapping the density operator onto ccc-number functions. The equations of motion for the reduced phase-space distribution functions are found to be of the Fokker-Planck type. On transforming the Fokker-Planck equation to real variables, it is found to have the same form as the Fokker-Planck equation obtained by Wang and Uhlenbeck to describe the Brownian motion of a classical oscillator. The Fokker-Planck equation is solved for the conditional probability (Green's function) which is found to be in the form of a two-dimensional Gaussian distribution. This solution is then used to obtain various time-dependent quantum statistical properties of the oscillator. Next, the entropy for a quantum oscillator undergoing Brownian motion is calculated and we show that this system approaches equilibrium as $t\ensuremath{\rightarrow}\ensuremath{\infty}$. Finally we show that in the weak-coupling limit the Fokker-Planck equation reduces to the one obtained by making the usual rotating-wave approximation.
The system of oscillator interacting with vacuum is considered as a problem of random motion of quantum reactive harmonic oscillator (QRHO). It is formulated in terms of a wave functional regarded as complex probability process in the extended space. This wave functional obeys some stochastic differential equation (SDE). Based on the nonlinear Langevin type SDE of second order, introduced in the functional space R{W(t)}, the variables in original equation are separated. The general measure in the space R{W(t)} of the Fokker-Planck type is obtained and expression for total wave function (wave mixture) of random QRHO is constructed as functional expansion over the stochastic basis set. The pertinent transition matrix S_br is constructed. For Wiener type measure W(t) of functional space the exact representation for ''vacuum-vacuum'' transition probability is obtained. The thermodynamics of vacuum is described in detail for the asymptotic space R1_as. The exact values for Energy, shift and expansion of ground state of oscillator and its Entropy are calculated.
The problem of random motion of 1D quantum reactive harmonic oscillator (QRHO) is formulated in terms of a wave functional regarded as a complex probability process in an extended space. In the complex stochastic differential equation (SDE) for this process the variables are separated with the help of the Langevin-type model SDE introduced in the functional space. The complete positive Fokker-Plank measure of the functional space is obtained. The average wave function of roaming QRHO is obtained by means of functional integration over the process with the complete Fokker-Plank measure in the functional space. The local and averaged transition matrices of roaming QRHO were constructed. The thermodynamics of nonrelativistic vacuum is investigated in detail and expressions for the internal energy, Helmholtzian energy and entropy are obtained. The oscillator's ground state energy, its shift and broadening are calculated.
The system of oscillator interacting with vacuum is considered as a problem of random motion of quantum reactive harmonic oscillator (QRHO). It is formulated in terms of a wave functional regarded as complex probability process in the extended space. This wave functional obeys some stochastic differential equation (SDE). Based on the nonlinear Langevin type SDE of second order, introduced in the functional space R{W(t)}, the variables in original equation are separated. The general measure in the space R{W(t)} of the Fokker-Plank type is obtained and expression for total wave function (wave mixture) of random QRHO is constructed as functional expansion over the stochastic basis set. The pertinent transition matrix S_br is constructed. For Wiener type measure W(t) of functional space the exact representation for ''vacuum-vacuum'' transition probability is obtained. The thermodynamics of vacuum is described in detail for the asymptotic space R1_as. The exact values for Energy, shift and expansion of ground state of oscillator and its Entropy are calculated.