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ACTA PHOTONICA SINICA

Variant Measures and Visualized Statistical Distributions

Jeffrey Zheng, 郑昊航 ZHENG Chris

Published Jan 1, 2011• 18 citations
Publisher

Abstract

Using measure model and statistical approches to simulate quantum interaction on binary level systems,multiple interactive conditions are simulated via double path interference measures.From quantum interaction,Einstein,Mach-Zehnder,Stern-Gerlach and CHSH measuring parameters are investigated.Using multiple variable logic functions and variant principle,N bit vectors of 0-1 input/output pairs form variant quaternion to establish variant double path simulation model.Using probability,symmetry/anti-symmetry,sychronous/asychronous conditions,simulation system generates two groups of eight histograms to express their statistical distributions in relevant conditions.

DOI
10.3788/gzxb20114009.1397

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Using measure model and statistical approches to simulate quantum interaction on binary level systems,multiple interactive conditions are simulated via double path interference measures.From quantum interaction,Einstein,Mach-Zehnder,Stern-Gerlach and CHSH measuring parameters are investigated.Using multiple variable logic functions and variant principle,N bit vectors of 0-1 input/output pairs form variant quaternion to establish variant double path simulation model.Using probability,symmetry/anti-symmetry,sychronous/asychronous conditions,simulation system generates two groups of eight histograms to express their statistical distributions in relevant conditions.
Quantum measures play a key role in quantum interactions. In this paper, a variant simulation system – a variant double path model – is proposed that uses multiple variable logic functions and variant principle schemes applied to input/output relationships as a variant quaternion. Under this mechanism, a two-state system of quantum interactions can be simulated by the variant double path model. Visual distributions can be expressed by their probability, synchronous/asynchronous, and symmetry/anti-symmetry statistical properties. The simulation system can output eight histograms with different symmetric properties. Sample results are illustrated by two groups of 16 histograms for their intrinsic distributions. Critical evidence from Afshar's experiments is discussed. This particle model is fully capable of simulating double path experiments in systematic methodologies.
Double slit experiments play a key role in Quantum Theory in distinct particle and wave interactions according to Feynman. In this paper, double path models together with variant logic principles are applied to establish a simulation system enabling the exhaustive testing of given targets. Using Einstein quanta interaction, different quaternion measures are investigated. Under conditions of Symmetry / Anti-symmetry and Synchronous / Asynchronous interaction, eight groups of statistical results are generated and presented as eight histograms showing the distributions. From this set of simulation results, it can be seen that not only is the synchronous condition the key factor in generating quantum wave interference patterns but also that the asynchronous condition is the key factor in classical particle distributions. Sample results are illustrated and explanations are discussed.
Abstract Double-slit experiment plays a key role in Quantum Theory to distinct particle and wave interactions according to Feynman’s claims. In this chapter, double path model and variant logic principle are applied to establish a simulation system for exhaustive testing targets. Using Einstein quanta interaction, different measure quaternion structures are investigated. Under Symmetry/Anti-symmetry and Synchronous/Asynchronous interaction conditions, eight groups of statistical results are generated as eight histograms to show their distributions. From this set of simulation results, it can be recognized that the synchronous condition is the key fact to generate quantum wave interference patterns and, in addition, the asynchronous condition is the key fact to make classic particle distributions. Sample results are illustrated and explanations are discussed.
Two quantum interactive experiments are shown in strong wave-like interference properties,ie Young double slit experiment and very lower intensive and very long time explosin experiment for quanta.This paper is based on a classical statistic scheme: Bayes statistics on Conditional Probability to propose conditional probability measurements and relevant statistcal models to visualized complex properties of their interactions.Variant measurement simulation model is established and corresponding measring equations and perameters are formulated.Two typical cases are selected and each time has corresponding to four groups of results in 16 histograms.Under this construction,it is feasible to understand the intrinsic wave-like properties of quantum interactions.Under the conditional probability model,intrinsic wave-like statistical distributions are observed on both normal conditions and interferenced conditions in their spatial statistical distributions respectively.
The purpose of this article is to deeply study the evolution process and measurement behavior of wave function in quantum mechanics by computer simulation. Based on Schrodinger equation and quantum measurement postulate, a set of numerical simulation framework is constructed, and typical models such as one-dimensional infinite potential well, simple harmonic oscillator, double-slit interference and two-level system are simulated by Crank-Nicolson method. Probability sampling mechanism is introduced in the simulation process to reproduce the wave function collapse caused by measurement, and the key results in the evolution process are shown. It includes the change of maximum probability density at different time steps (0.78 → 0.63 → 0.78), the exchange of kinetic energy and potential energy under the condition of energy conservation, the distribution characteristics of measurement positions in the double-slit interference experiment and the statistical frequency of the measurement results of the two-level system (0.503:0.497). The research shows that the designed simulation method can effectively reproduce the dynamic evolution law of quantum system and the statistical characteristics of measurement, and provide a visual auxiliary tool for understanding quantum phenomena.
From a quantum statistical viewpoint, four typical quantum states are Fock, Sub-Poissonian, Poissonian and SuperPoissonian states. Quantum interactions are focus among Fock and Poissonian states. Using quantum statistics, model and simulation, this paper proposes two models: matrix and variant transformations: 1. MT Matrix Transformation – eigenvalue states; 2. VT Variant Transformation – invariant states to analyze three random sequences: 1) random; 2) conditional random in a constant; 3) periodic pattern. Four procedures are proposed. Fast Fourier Transformation FFT is applied as one of MT schemes and two invariant scheme of VT schemes are applied, three random sequences are in M segments and each segment has a length m to generate a measuring sequence. Shifting operations are applied on each random sequence to create m+1 spectrum distributions. For FFT, a pair of eigenvalues are selected as the output. Two types of 1D and 2D variant maps are generated to illustrate multiple parameter selections to generate a series of results. Since sequences 1) and 3) are related simple, more cases are focus on sequences 2). Better than FFT, VT distinguishes various Fock, Sub-Poissonian, Poissonian states in random analysis to distinguish three random sequences as three levels of statistical ensembles: Micro-canonical, Canonical, and Grand-Canonical ensembles. Applying two transformations, quantum statistics, model and simulation of modern quantum theory and applications can be explored.