2019
DOI: 10.2478/amcs-2019-0032
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The Phase–Space Approach to time Evolution of Quantum States in Confined Systems: the Spectral Split–Operator Method

Abstract: Using the phase space approach, we consider the quantum dynamics of a wave packet in an isolated confined system with three different potential energy profiles. We solve the Moyal equation of motion for the Wigner function with the highly efficient spectral split-operator method. The main aim of this study is to compare the accuracy of the employed algorithm through analysis of the total energy expectation value, in terms of deviation from its exact value. This comparison is performed for the second and fourth… Show more

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Cited by 5 publications
(3 citation statements)
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“…is the time increment. Calculation-efficient form of the time evolution operator can be derived by applying the symmetric Strang splitting formula 89 91 Using the partial Fourier transforms in the first or second variable defined as follows, one can obtain a formula for a single step of the time evolution of the WDF in the form where the auxiliary function is defined as the central difference of the potential energies, namely Since Eq. ( 4 ) is defined for , in order to perform numerical computation the phase space is limited to the box of size with periodic boundary conditions imposed by the numerical method.…”
Section: Methodsmentioning
confidence: 99%
“…is the time increment. Calculation-efficient form of the time evolution operator can be derived by applying the symmetric Strang splitting formula 89 91 Using the partial Fourier transforms in the first or second variable defined as follows, one can obtain a formula for a single step of the time evolution of the WDF in the form where the auxiliary function is defined as the central difference of the potential energies, namely Since Eq. ( 4 ) is defined for , in order to perform numerical computation the phase space is limited to the box of size with periodic boundary conditions imposed by the numerical method.…”
Section: Methodsmentioning
confidence: 99%
“…Although this basic splitting formula is sufficient for our needs, let us note that the high-order variants of the splitting formula are more precise, but also unavoidably more complicated when applied to the Moyal equation, as has been discussed in Ref. 68 . Going back to the symmetric Strang splitting formula ( 18 ), it can be noted that each operator is unitarily equivalent to some multiplication operator owing to the adequate Fourier transform, specifically and where the symbol denotes the ordinary Fourier transform for the x variable with dual variable .…”
Section: Theoretical Frameworkmentioning
confidence: 99%
“…Artificial neural networks and other tools of machine learning still evolve, and play a more and more important role in data processing. Today, the whole world is on the verge of a quantum revolution where data are going to be encoded as quantum states and processed with the use of laws of quantum mechanics (Nielsen and Chuang, 2010;Kołaczek et al, 2019) and machine learning methods are also developed for quantum computational systems. Researchers deal with the different kinds of quantum machine learning (Biamonte et al, 2017;Schuld et al, 2014;, e.g., quantum neural networks (Narayanan and Menneer, 2000;Zoufal et al, 2019), quantum kNN * Corresponding author methods (Wiebe et al, 2015), quantum self-organized maps (Weigang, 1998), or the quantum k-means method (Veenman and Reinders, 2005).…”
Section: Introductionmentioning
confidence: 99%