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Concept# Mathematical formulation of quantum mechanics

Summary

The mathematical formulations of quantum mechanics are those mathematical formalisms that permit a rigorous description of quantum mechanics. This mathematical formalism uses mainly a part of functional analysis, especially Hilbert spaces, which are a kind of linear space. Such are distinguished from mathematical formalisms for physics theories developed prior to the early 1900s by the use of abstract mathematical structures, such as infinite-dimensional Hilbert spaces (L2 space mainly), and operators on these spaces. In brief, values of physical observables such as energy and momentum were no longer considered as values of functions on phase space, but as eigenvalues; more precisely as spectral values of linear operators in Hilbert space.
These formulations of quantum mechanics continue to be used today. At the heart of the description are ideas of quantum state and quantum observables, which are radically different from those used in previous models of physical reality. While the m

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Vincent Maronnier, Marco Picasso, Jacques Rappaz

A numerical model is presented for the simulation of complex fluid flows with free surfaces. The unknowns are the velocity and pressure fields in the liquid region, together with a function defining the volume fraction of liquid. Although the mathematical formulation of the model is similar to the volume of fluid (VOF) method, the numerical schemes used to solve the problem are different. A splitting method is used for the time discretization. At each time step, two advection problems and a generalized Stokes problem are to be solved. Two different grids are used for the space discretization. The two advection problems are solved on a fixed, structured grid made out of small rectangular cells, using a forward characteristic method. The generalized Stokes problem is solved using a finite element method on a fixed, unstructured mesh. Numerical results are presented for several test cases: the filling of an S-shaped channel, the filling of a disk with core, the broken dam in a confined domain. (C) 1999 Academic Press.

1999Vincent Maronnier, Marco Picasso, Jacques Rappaz

A numerical model is presented for the simulation of complex fluid flows with free surfaces in three space dimensions. The model described in Maronnier et al. (J. Comput. Phys. 1999; 155(2):439) is extended to three dimensional situations. The mathematical formulation of the model is similar to that of the volume of fluid (VOF) method, but the numerical procedures are different. A splitting method is used for the time discretization. At each time step, two advection problems-one for the predicted velocity field and the other for the volume fraction of liquid-are to be solved. Then, a generalized Stokes problem is solved and the velocity field is corrected. Two different grids are used for the space discretization. The two advection problems are solved on a fixed, structured grid made out of small cubic cells, using a forward characteristic method. The generalized Stokes problem is solved using continuous, piecewise linear stabilized finite elements on a fixed, unstructured mesh of tetrahedrons. The three-dimensional implementation is discussed. Efficient postprocessing algorithms enhance the quality of the numerical solution. A hierarchical data structure reduces memory requirements. Numerical results are presented for complex geometries arising in mold filling. Copyright (C) 2003 John Wiley Sons, Ltd.

2003Andrei Ardelean, Edoardo Charbon, Kazuhiro Morimoto, Ming-Lo Wu

Advances in high-speed imaging techniques have opened new possibilities for capturing ultrafast phenomena such as light propagation in air or through media. Capturing light in flight in three-dimensional xyt space has been reported based on various types of imaging systems, whereas reconstruction of light-in-flight information in the fourth dimension z has been a challenge. We demonstrate the four-dimensional light-in-flight imaging based on the observation of a superluminal motion captured by a new time-gated megapixel single-photon avalanche diode camera. A high-resolution light-in-flight video is generated without laser scanning, camera translation, interpolation, or dark noise subtraction. An unsupervised machine-learning technique is applied to analyze the measured spatiotemporal data set. A theoretical formula is introduced to perform least-square regression for numerically solving a nonlinear inverse problem, and extra-dimensional information is recovered without prior knowledge. The algorithm relies on the mathematical formulation equivalent to the superluminal motion in astrophysics, which is scaled by a factor of a quadrillionth. The reconstructed light-in-flight trajectory shows good agreement with the actual geometry of the light path. Applicability of the reconstruction approach to more complex scenes with multiple overlapped light trajectories is verified based on a data set generated by Monte Carlo simulations. Our approach could potentially provide novel functionalities to high-speed imaging applications such as non-line-of-sight imaging and time-resolved optical tomography.