Composite Coherent States Approximation for One-Dimensional Multi-Phased Wave Functions

AbstractThe coherent states approximation for one-dimensional multi-phased wave functions is considered in this paper. The wave functions are assumed to oscillate on a characteristic wave length 0(ε) with ε ≪ 1. A parameter recovery algorithm is first developed for single-phased wave function based on a moment asymptotic analysis. This algorithm is then extended to multi-phased wave functions. If cross points or caustics exist, the coherent states approximation algorithm based on the parameter recovery will fail in some local regions. In this case, we resort to the windowed Fourier transform technique, and propose a composite coherent states approximation method. Numerical experiments show that the number of coherent states derived by the proposed method is much less than that by the direct windowed Fourier transform technique.

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Amplitudes of characteristic waves and non-reflecting outflow conditions for high-fidelity numerical simulations

Proceedings of the Institution of Mechanical Engineers Part G Journal of Aerospace Engineering ◽

10.1177/09544100211017380 ◽

2021 ◽

pp. 095441002110173

Author(s):

Yongle Du ◽

Jinsheng Cai

Keyword(s):

Boundary Conditions ◽

Numerical Experiments ◽

Jacobian Matrix ◽

Wave Number ◽

Multidimensional Space ◽

High Fidelity ◽

Characteristic Wave ◽

One Dimensional ◽

Characteristic Theory ◽

Different Levels

Non-reflecting boundary conditions are crucial for aeroacoustic simulations to prevent non-physical reflections at boundaries from contaminating the interior acoustic solutions. Various non-reflecting outflow conditions, mainly based on the characteristic theory, have been developed. By using a locally one-dimensional inviscid approximation, the characteristic wave amplitudes are identified as the projection of flow solutions into the eigenspace of the flux Jacobian matrix. When the weighted transverse terms are included, the definitions are essentially changed and produce different levels of non-physical reflections. This study aims to resolve the ambiguity in defining the amplitudes of characteristic waves. A new analysis in the frequency and wave number domain is proposed to identify the characteristic wave amplitudes in multidimensional space, and a second order in terms of the lateral wave number non-reflecting outflow condition that is more physically reasonable is derived. Numerical experiments are carried out to evaluate the performances of these non-reflecting conditions.

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Quantum Boxes, Stringed Instruments

10.1093/oso/9780198808275.003.0008 ◽

2017 ◽

Author(s):

Frank S. Levin

Keyword(s):

Energy Level ◽

Schrödinger Equation ◽

Quantum System ◽

Schrodinger Equation ◽

Probability Distributions ◽

Wave Functions ◽

Energy Level Diagram ◽

One Dimensional ◽

Stringed Instruments ◽

Symmetry Properties

Chapter 7 illustrates the results obtained by applying the Schrödinger equation to a simple pedagogical quantum system, the particle in a one-dimensional box. The wave functions are seen to be sine waves; their wavelengths are evaluated and used to calculate the quantized energies via the de Broglie relation. An energy-level diagram of some of the energies is constructed; on it are illustrations of the corresponding wave functions and probability distributions. The wave functions are seen to be either symmetric or antisymmetric about the midpoint of the line representing the box, thereby providing a lead-in to the later exploration of certain symmetry properties of multi-electron atoms. It is next pointed out that the Schrödinger equation for this system is identical to Newton’s equation describing the vibrations of a stretched musical string. The different meaning of the two solutions is discussed, as is the concept and structure of linear superpositions of them.

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