Exponential Convergent Nonlinear Estimation of the Quantum Density Operator for a Quantum Mechanical Statistical System Using Metric Projection Operators

2022 ◽  
Author(s):  
Mark J. Balas ◽  
Vinod P. Gehlot
Keyword(s):  
Density Operator ◽  
Nonlinear Estimation ◽  
Metric Projection ◽  
Quantum Mechanical ◽  
Projection Operators ◽  
Statistical System

scholarly journals ON SUPERLUMINAL FERMIONS WITHIN THE SECOND DERIVATIVE EQUATION

2012 ◽  
Vol 27 (14) ◽  
pp. 1250081 ◽  
Author(s):  
S. I. KRUGLOV
Keyword(s):  
Energy Momentum Tensor ◽  
Canonical Quantization ◽  
Vacuum Expectation ◽  
Momentum Tensor ◽  
Order Equation ◽  
Quantum Mechanical ◽  
Projection Operators ◽  
The Novel ◽  
First Order ◽  
Electromagnetic Interactions

We postulate the second-order derivative equation with four parameters for spin-1/2 fermions possessing two mass states. For some choice of parameters fermions propagate with the superluminal speed. Thus, the novel tachyonic equation is suggested. The relativistic 20-component first-order wave equation is formulated and projection operators extracting states with definite energy and spin projections are obtained. The Lagrangian formulation of the first-order equation is presented and the electric current and energy–momentum tensor are found. The minimal and nonminimal electromagnetic interactions of fermions are considered and Schrödinger's form of the equation and the quantum-mechanical Hamiltonian are obtained. The canonical quantization of the field in the first-order formalism is performed and we find the vacuum expectation of chronological pairing of operators.

scholarly journals Quantum fluctuations of elementary excitations in discrete media

2004 ◽  
Vol 2004 (1) ◽  
pp. 169-177
Author(s):  
Hermann Haken
Keyword(s):  
Quantum Noise ◽  
General Theorem ◽  
Polymer Chains ◽  
Quantum Mechanical ◽  
Projection Operators ◽  
Noise Sources ◽  
Elementary Excitations ◽  
Heisenberg Equations ◽  
Discrete Media

Elementary excitations (electrons, holes, polaritons, excitons, plasmons, spin waves, etc.) on discrete substrates (e.g., polymer chains, surfaces, and lattices) may move coherently as quantum waves (e.g., Bloch waves), but also incoherently (“hopping”) and may lose their phases due to their interaction with their substrate, for example, lattice vibrations. In the frame of Heisenberg equations for projection operators, these latter effects are often phenomenologically taken into account, which violates quantum mechanical consistency, however. To restore it, quantum mechanical fluctuating forces (noise sources) must be introduced, whose properties can be determined by a general theorem. With increasing miniaturization, in the nanotechnology of logical devices (including quantum computers) that use interacting elementary excitations, such fluctuations become important. This requires the determination of quantum noise sources in composite quantum systems. This is the main objective of my paper, dedicated to the memory of Ilya Prigogine.

The Quantum Mechanical Density Operator And Its Time Evolution: Quantum Dynamics Using The Quantum Liouville Equation

2006 ◽  
Author(s):  
Abraham Nitzan
Keyword(s):  
Phase Space ◽  
Probability Density ◽  
Time Evolution ◽  
Liouville Equation ◽  
Density Operator ◽  
Quantum Dynamics ◽  
Quantum Mechanical ◽  
Initial Condition ◽  
Initial State ◽  
Thermal Environments

The starting point of the classical description of motion is the Newton equations that yield a phase space trajectory (rN (t), pN (t)) for a given initial condition (rN (0), pN (0)). Alternatively one may describe classical motion in the framework of the Liouville equation (Section (1.2.2)) that describes the time evolution of the phase space probability density f (rN , pN ; t). For a closed system fully described in terms of a well specified initial condition, the two descriptions are completely equivalent. Probabilistic treatment becomes essential in reduced descriptions that focus on parts of an overall system, as was demonstrated in Sections 5.1–5.3 for equilibrium systems, and in Chapters 7 and 8 that focus on the time evolution of classical systems that interact with their thermal environments. This chapter deals with the analogous quantum mechanical problem. Within the limitations imposed by its nature as expressed, for example, by Heisenbergtype uncertainty principles, the Schrödinger equation is deterministic. Obviously it describes a deterministic evolution of the quantum mechanical wavefunction. The analog of the phase space probability density f (rN , pN ; t) is now the quantum mechanical density operator (often referred to as the “density matrix”), whose time evolution is determined by the quantum Liouville equation. Again, when the system is fully described in terms of a well specified initial wavefunction, the two descriptions are equivalent. The density operator formalism can, however, be carried over to situations where the initial state of the system is not well characterized and/or a reduced description of part of the overall system is desired. Such situations are considered later in this chapter.

Continuity of the metric-projection operators

1992 ◽  
Vol 44 (8) ◽  
pp. 1044-1047 ◽  
Author(s):  
S. V. Kuznetsov
Keyword(s):  
Metric Projection ◽  
Projection Operators

Quantum Theory of Spectral Modulation

1974 ◽  
Vol 52 (17) ◽  
pp. 1694-1702 ◽  
Author(s):  
Andrzej Zardecki ◽  
Claude Delisle
Keyword(s):  
Quantum Theory ◽  
Time Delay ◽  
Density Operator ◽  
Michelson Interferometer ◽  
Coherence Time ◽  
Path Difference ◽  
Intensity Modulation ◽  
Quantum Mechanical ◽  
Spectral Modulation ◽  
Thermal Light

Recent experiments of Delisle and Brochu, in which spectral modulation at a large path difference was observed, are analyzed in quantum mechanical terms. Starting with a stationary density operator for thermal light, a two beam superposition such as taking place in a Michelson interferometer, is investigated. It is shown that a simple interference law holds separately for each mode of radiation. Consequently, a coherent superposition of light should be observed for a time delay arbitrarily large compared to the coherence time. The latter determines merely the range of intensity modulation. Analysis of the Michelson interferometer accounting for the energy returned to the source is also presented.

ON THE QUANTUM MECHANICAL GENERALIZATION OF BLOCH'S TRANSPORT EQUATION

1963 ◽  
Vol 41 (3) ◽  
pp. 533-544 ◽  
Author(s):  
J. Hajdu
Keyword(s):  
Asymptotic Behavior ◽  
Steady State ◽  
Kinetic Equation ◽  
Magnetic Fields ◽  
Transport Equation ◽  
Density Operator ◽  
Quantum Mechanical ◽  
Simple Derivation ◽  
Asymptotic Equation ◽  
Weak Magnetic Fields

A simple derivation is given of a kinetic equation for a system of free electrons moving in uniform electric and magnetic fields, and interacting with fixed scattered. The kinetic equation describes the asymptotic behavior of the single-electron density operator if it approaches a steady value, or the asymptotic behavior of its average over oscillations if the density operator oscillates in time. This equation, which is effectively the quantum mechanical generalization of Bloch's transport equation, is identical with the one recently derived by Kosevich and Andreev using Bogolyubov's method. More general considerations show that this asymptotic equation is valid, and describes the approach to the steady state for weak magnetic fields when the relaxation time is much longer than the atomic time. For strong magnetic fields, the same statement holds if the density operator is averaged over its oscillations, whereas the unaveraged approach towards the steady state is governed by a somewhat different equation. The solutions of these two equations become identical in the most important limiting cases. The results obtained previously by different authors follow from the kinetic equation when further assumptions are introduced.

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