Quantum Mechanics in Hilbert Space

Prugovečki, Eduard; Pan, Yuh Kang

doi:10.1119/1.1987530

Cited by 21 publications

(38 citation statements)

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“…Formal theory 4 of quantum scattering has been studied in depth from the mathematical standpoint and the mathematical properties 5,6 of quantum scattering operators, such as the Hilbert space for the operators, are well understood at present. On the other hand, it cannot be said the same of the classical scattering theory based on the classical Liouville operator, and even the mathematical nature of the function space for the classical Liouville operator is not well clarified at present, almost all of mathematical operations involving the classical Liouville operators being performed by simple analogy to the quantum counterparts; see, for example, Refs.…”

Section: The Eigenvalue Problem For the Classical Liouville Operatormentioning

confidence: 99%

“…They appear in the statistical mechanical formulas of transport coefficients in kinetic theory of matter and present an important problem to resolve in the final stage of implementing the dense fluid kinetic theory to make it connect with experiments. Although practical methods of computing the classical collision operators are essential to making the kinetic theory a useful molecular theory of matter, they have not been given much attention beyond the formal theory level in contrast to their quantum mechanical analogs in the literature, [4][5][6] and computation of them and related expressions appearing in the kinetic theory of dense fluids still poses a theoretical and computational challenge in nonequilibrium statistical mechanics. In this paper, we would like to take up the subject for study and make an attempt to mitigate the situation.…”

Section: Introductionmentioning

confidence: 99%

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Eigenfunctions for Liouville Operators, Classical Collision Operators, and Collision Bracket Integrals in Kinetic Theory Made Amenable to Computer Simulations

Eu¹

2012

Bulletin of the Korean Chemical Society

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In the kinetic theory of dense fluids the many-particle collision bracket integral is given in terms of a classical collision operator defined in the phase space. To find an algorithm to compute the collision bracket integrals, we revisit the eigenvalue problem of the Liouville operator and re-examine the method previously reported [Chem. Phys. 1977, 20, 93]. Then we apply the notion and concept of the eigenfunctions of the Liouville operator and knowledge acquired in the study of the eigenfunctions to cast collision bracket integrals into more convenient and suitable forms for numerical simulations. One of the alternative forms is given in the form of time correlation function. This form, on a further manipulation, assumes a form reminiscent of the ChapmanEnskog collision bracket integrals, but for dense gases and liquids as well as solids. In the dilute gas limit it would give rise precisely to the Chapman-Enskog collision bracket integrals for two-particle collision. The alternative forms obtained are more readily amenable to numerical simulation methods than the collision bracket integrals expressed in terms of a classical collision operator, which requires solution of classical Lippmann-Schwinger integral equations. This way, the aforementioned kinetic theory of dense fluids is made fully accessible by numerical computation/simulation methods, and the transport coefficients thereof are made computationally as accessible as those in the linear response theory.

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Section: The Eigenvalue Problem For the Classical Liouville Operatormentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Eigenfunctions for Liouville Operators, Classical Collision Operators, and Collision Bracket Integrals in Kinetic Theory Made Amenable to Computer Simulations

Eu¹

2012

Bulletin of the Korean Chemical Society

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“…Inner product spaces have an unconsciously de ned norm based on the inner product of the space itself that does not the parallelogram equality [17,18]:…”

Section: Theorem 4 Taking Into Account the Initial Conditions Associmentioning

confidence: 99%

Analysis of time-fractional hunter-saxton equation: a model of neumatic liquid crystal

2016

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Abstract:In this work, a theoretical study of di usion of neumatic liquid crystals was done using the concept of fractional order derivative. This version of fractional derivative is very easy to handle and obey to almost all the properties satis ed by the conventional Newtonian concept of derivative. The mathematical equation underpinning this physical phenomenon was solved analytically via the so-called homotopy decomposition method. In order to show the accuracy of this iteration method, we constructed a Hilbert space in which we proved its stability for the time-fractional Hunder-Saxton equation.

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“…In an attempt to embrace also nonMarkovian situations we have combined well-known stochastic methods jointly with common tools of functional analysis employed in quantum mechanics [13,14] to state the following result: Any Lindblad-type evolution with selfadjoint Lindblad operators, whether Markovian or nonMarkovian, can be understood as an averaged random unitary evolution. This result, whose fully-fledged mathematical proof is provided elsewhere [15], reveals the fact that the Lindblad structure of ME for the case of selfadjoint Lindblad operators is beyond the Markov approximation and can also be recovered in nonMarkovian situations.…”

Section: Introductionmentioning

confidence: 99%

Damped quantum interference using stochastic calculus

Salgado¹,

Sánchez-Gómez²

2003

Journal of Modern Optics

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It is shown how the phase-damping master equation, either in Markovian and nonMarkovian regimes, can be obtained as an averaged random unitary evolution. This, apart from offering a common mathematical setup for both regimes, enables us to solve this equation in a straightforward manner just by solving the Schrödinger equation and taking the stochastic expectation value of its solutions after an adequate modification. Using the linear entropy as a figure of merit (basically the loss of quantum coherence) the distinction of four kinds of environments is suggested.

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Quantum Mechanics in Hilbert Space

Cited by 21 publications

References 0 publications

Eigenfunctions for Liouville Operators, Classical Collision Operators, and Collision Bracket Integrals in Kinetic Theory Made Amenable to Computer Simulations

Eigenfunctions for Liouville Operators, Classical Collision Operators, and Collision Bracket Integrals in Kinetic Theory Made Amenable to Computer Simulations

Analysis of time-fractional hunter-saxton equation: a model of neumatic liquid crystal

Damped quantum interference using stochastic calculus

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