Classical probability density distributions with uncertainty relations for ground states of simple non-relativistic quantum-mechanical systems

In this chapter, we show how the newest developments in quantum mechanics of the late 1920s were very quickly compared with general relativity, with attempts made to demonstrate their mutual coherence. This involved focusing on the basic mathematical structures that formed the first concrete representations of quantum mechanical systems. The aim was structural harmonisation, rather than quantization. Likewise, we will find that conceptual debates, especially having to do with measurement and the uncertainty relations, as well as new cosmological discoveries (based on applications of general relativity) were also quickly compared, often with surprising results such as explanations of discreteness and predictions of particle production in curved spaces. We see two primary motivations pushing this research forward: coherence (into which the more formal approaches also fit) and utility (that is attempting to gain a better grip on the quantum theory).

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Controllability of quantum mechanical systems by root space decomposition of su(N)

Journal of Mathematical Physics ◽

10.1063/1.1467611 ◽

2002 ◽

Vol 43 (5) ◽

pp. 2051 ◽

Cited By ~ 121

Author(s):

Claudio Altafini

Keyword(s):

Mechanical Systems ◽

Quantum Mechanical ◽

Root Space ◽

Space Decomposition ◽

Quantum Mechanical Systems

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Assembly variation analysis of complicated products based on rigid–flexible hybrid vector loop

Proceedings of the Institution of Mechanical Engineers Part B Journal of Engineering Manufacture ◽

10.1177/0954405418816851 ◽

2018 ◽

Vol 233 (10) ◽

pp. 2099-2114 ◽

Cited By ~ 1

Author(s):

Zhenyu Liu ◽

Shien Zhou ◽

Chan Qiu ◽

Jianrong Tan

Keyword(s):

Probability Density ◽

Loop Model ◽

Feature Points ◽

Variation Analysis ◽

Manufacturing Error ◽

Density Distributions ◽

Mechanical Products ◽

Flexible Parts ◽

Probability Density Distributions ◽

Product Assembly

The performance of mechanical products is closely related to their key feature errors. It is essential to predict the final assembly variation by assembly variation analysis to ensure product performance. Rigid–flexible hybrid construction is a common type of mechanical product. Existing methods of variation analysis in which rigid and flexible parts are calculated separately are difficult to meet the requirements of these complicated mechanical products. Another methodology is a result of linear superposition with rigid and flexible errors, which cannot reveal the quantitative relationship between product assembly variation and part manufacturing error. Therefore, a kind of complicated products’ assembly variation analysis method based on rigid–flexible vector loop is proposed in this article. First, shapes of part surfaces and sidelines are estimated according to different tolerance types. Probability density distributions of discrete feature points on the surface are calculated based on the tolerance field size with statistical methods. Second, flexible parts surface is discretized into a set of multi-segment vectors to build vector-loop model. Each vector can be orthogonally decomposed into the components representing position information and error size. Combining the multi-segment vector set of flexible part with traditional rigid part vector, a uniform vector-loop model is constructed to represent rigid and flexible complicated products. Probability density distributions of discrete feature points on part surface are regarded as inputs to calculate assembly variation values of products’ key features. Compared with the existing methods, this method applies to the assembly variation prediction of complicated products that consist of both rigid and flexible parts. Impact of each rigid and flexible part’s manufacturing error on product assembly variation can be determined, and it provides the foundation of parts tolerance optimization design. Finally, an assembly example of phased array antenna verifies effectiveness of the proposed method in this article.

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A new Description of Probability Density Distributions of Polar Mesospheric Clouds (PMC)

10.5194/acp-2018-642 ◽

2018 ◽

Author(s):

Uwe Berger ◽

Gerd Baumgarten ◽

Jens Fiedler ◽

Franz-Josef Lübken

Keyword(s):

Probability Density ◽

Mass Density ◽

Particle Radius ◽

Number Density ◽

Data Set ◽

Statistical Probability ◽

Polar Mesospheric Clouds ◽

Density Distributions ◽

Mesospheric Clouds ◽

Probability Density Distributions

Abstract. In this paper we present a new description about statistical probability density distributions (pdfs) of Polar Mesospheric Clouds (PMC) and noctilucent clouds (NLC). The analysis is based on observations of maximum backscatter, ice mass density, ice particle radius, and number density of ice particles measured by the ALOMAR RMR-lidar for all NLC seasons from 2002 to 2016. From this data set we derive a new class of pdfs that describe the statistics of PMC/NLC events which is different from previously statistical methods using the approach of an exponential distribution commonly named g-distribution. The new analysis describes successfully the probability statistic of ALOMAR lidar data. It turns out that the former g-function description is a special case of our new approach. In general the new statistical function can be applied to many kinds of different PMC parameters, e.g. maximum backscatter, integrated backscatter, ice mass density, ice water content, ice particle radius, ice particle number density or albedo measured by satellites. As a main advantage the new method allows to connect different observational PMC distributions of lidar, and satellite data, and also to compare with distributions from ice model studies. In particular, the statistical distributions of different ice parameters can be compared with each other on the basis of a common assessment that facilitate, for example, trend analysis of PMC/NLC.

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