Chaos and Husimi Distribution Function in Quantum Mechanics

1985 ◽  
Vol 55 (7) ◽  
pp. 645-648 ◽  
Author(s):  
Kin'ya Takahashi ◽  
Nobuhiko Saitô
Keyword(s):  
Distribution Function ◽  
Quantum Mechanics ◽  
Husimi Distribution

scholarly journals A relativistic formulation of the de la Peña-Cetto stochastic quantum mechanics

2018 ◽  
Vol 64 (2) ◽  
pp. 158
Author(s):  
Alejandro Cabo Montes de Oca ◽  
A. González Lezcano
Keyword(s):  
Distribution Function ◽  
Quantum Mechanics ◽  
Joint Distribution ◽  
Lorentz Invariance ◽  
Complex Function ◽  
Space Time ◽  
Zeroth Order ◽  
Momentum Distribution Function ◽  
Iterative Equations ◽  
Stochastic Forces

A covariant generalization of a non-relativistic stochastic quantum mechanics introduced by de la Peña and Cetto is formulated. The analysis is done in space-time and avoids the use of a non-covariant time evolution parameter in order to search for Lorentz invariance. The covariant form of the set of iterative equations for the joint coordinate and momentum distribution function Q(x; p) is derived and expanded in power series of the coupling of the particle with the stochastic forces. Then, particular solutions of the zeroth order in the charge of the iterative equations for Q(x; p) are considered. For them, it follows that the space-time probability density ρ(x) and the function S(x) which gradient defines the mean value of the momentum at the space time point x, define a complex function ψ(x) which exactly satisfies the Klein-Gordon (KG) equation. These results for the zeroth order solution reproduce the ones formerly and independently derived in the literature. It is alsoargued that when the KG solution is either of positive or negative energy, the total number of particles conserves in the random motion. Other solutions for the joint distribution function in lowest order, satisfying the positive condition are also presented here. The are consistent with the assumed lack of stochastic forces implied by the zeroth order equations. It is also argued that such joint distributions, after considering the action of the stochastic forces, might furnish an explanation of the quantum mechanical properties, as associated to ensembles of particles in which the vacuum makes such particles behave in a similar way as Couder’s droplets moving over oscillating liquid surfaces. Some remarks on the solutions of the positive joint distribution problem proposed in the Olavos’s analysis are also presented.

scholarly journals Liouville's theorem and quantum mechanics - time quantization and reality

2020 ◽  
Vol 9 ◽  
pp. 395
Author(s):  
C. Syros ◽  
G. S. Ioannidis ◽  
G. Raptis
Keyword(s):  
Distribution Function ◽  
Quantum Mechanics ◽  
Quantum Theory ◽  
Phase Space ◽  
Statistical Mechanics ◽  
Liouville Equation ◽  
Time Variable ◽  
Planck Constant ◽  
Quantum Numbers ◽  
Liouville’S Theorem

The chrono-topology, as introduced axiomatically in a different context, is also supported by Liouville's theorem of statistical mechanics. It is shown that, if time is quantized, the distribution function (d.f.) becomes real. An elementary solution, g, of the classical Liouville equation has been found in phase-space and time, which can be used to construct any differentiable d.f, F(g), satisfying the same Liouville equation. The conditions imposed on F(g) are reality and additivity. The reality requirement, {Im F(g)=0) quantizes: (i) F(g) and makes it time-independent, (ii). The time variable, (iii) The energy. As a verification of chronotopology, the Planck constant h has been calculated on the basis of the time quantization. The d.f. F(g) becomes, after the time quantization, a real generalized Maxwell-Boltzmann d.f, F(g) = exp[g(p, g; l1,l2,..,lN)], depending on Ν quantum numbers. These facts are significant for quantum theory, because they uncover an intrinsic relationship between Liouville's theorem and quantum mechanics.

Onsagerian Quantum Mechanics

2016 ◽  
pp. 3353-3373
Author(s):  
G. Vincze ◽  
A. Szasz
Keyword(s):  
Distribution Function ◽  
Quantum Mechanics ◽  
Energy Dissipation ◽  
Probability Distribution ◽  
Probability Distribution Function ◽  
Hamiltonian Formalism ◽  
Quantum Mechanical ◽  
Basic Principles ◽  
Characteristic Values

We describe the basic quantum-mechanical categories and properties of the thermodynamical basis of Onsager’s theorem. 3 basic principles are used: 1. energy dissipation; 2. Hamiltonian formalism; 3. Onsager’s linearity. We obtain the 2 characteristic values of the observables, their main-value and the deviation, the first and second momentums of the probability distribution function, which we also derived also from the same principles. 

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