China
Africa
Explore chapters and articles related to this topic
E
Published in Philip A. Laplante, Comprehensive Dictionary of Electrical Engineering, 2018
eigenfunction the name given to an eigenvector when the eigenvectors arise as solutions of particular types of integral equation. See eigensystem, eigenvector. eigenfunction expansion a method used to expand a given field in terms of eigenfunctions. It is particularly used in modal analysis of waveguide discontinuities. eigenstate a linear combination of quantum mechanical basis states that is constant in time. A quantum mechanical system starting in an eigenstate will remain unchanged in time except for an overall phase. The phase varies as the product of the eigenvalue and time. Quantum mechanical eigenstates are analogous to normal modes of coupled oscillator systems in classical mechanics. eigensystem a system where the output of a system is the input function multiplied by a constant. See eigenfunction, eigenvalue, eigenvector. eigenvalue the multiplicative scalar associated with an eigenfunction or an eigenvector. For example, if Ax = x, then is the eigenvalue
E
Published in Splinter Robert, Illustrated Encyclopedia of Applied and Engineering Physics, 2017
[atomic, computational, nuclear, quantum] A particular solution to a complex differential formulation that represent a characteristic function or value (i.e., eigenvalue) that can be used independently. The concept of eigenfunction originates from the use of “function space” or “Hilbert space.” The eigenfunction generally represents a particular function that is directly linked to the linear operator. Eigenfunctions are widely used in solving the Schrödinger equation. The eigenfunction is a solution to the operator (e.g., differential operator Fop) for every point where the operator is defined, which gives it the eigenfunction property. In the Schrödinger equation, for example, FopΨ=fΨ, the function Ψ represents the eigenfunction and f is the eigenvalue belonging to the eigenfunction and the operator applied to the function (also seeeigenvalue).
Eigenshells
Published in Sigrid Adriaenssens, Philippe Block, Diederik Veenendaal, Chris Williams, Shell Structures for Architecture, 2014
Panagiotis Michalatos, Sawako Kaijima
Eigenfunctions are basically the infinite dimensional equivalents of eigenvectors. If instead of a matrix one has a differential or integral operator K, then its eigenfunctions will be the functions f that satisfy the equation K(f)=λf for some scalar λ. Grossly simplifying for the sake of explaining, one can think of operators such as Δ as a matrix of infinite rows and columns. In the discrete case the rows and columns become finite.
Visualizing Laguerre polynomials as a complete orthonormal set for the inner product space ℙ n
Published in International Journal of Mathematical Education in Science and Technology, 2023
Named after the French mathematician Edmond Laguerre (1834–1886), the set forms an orthogonal set of polynomials with the associated integral inner product . In quantum mechanics, they appear in eigenfunctions satisfying the radial Schrödinger equation for the Hydrogen atom. This note illustrates the integral inner product in the vector space of polynomials of degree with real coefficients – the mapping defined via for – in a DGS/MATLAB-facilitated learning environment. Also explored are the defining properties (symmetry, linearity, positive definiteness) of the integral inner product along with other notions inherent in the inner product space, such as norm, distance, orthogonal projection, Cauchy–Schwarz Inequality, Triangle Inequality, Pythagorean Theorem, Parallelogram Law, orthogonality and orthonormality, orthonormal basis, and coordinates relative to an orthonormal basis. The article also demonstrates the diversity of ways through which Laguerre polynomials can be used as an orthonormal basis for the inner product space in a technology-assisted learning environment. More details on Laguerre Polynomials can be found in Arfken (1985), Hochstrasser (1972), Szegö (1975).
A Generalized Eigenvalue Formulation for Core-Design Applications
Published in Nuclear Science and Engineering, 2023
Nicolo’ Abrate, Sandra Dulla, Piero Ravetto, Paolo Saracco
The concepts of eigenvalue and eigenfunction of an operator are fundamental in a wide range of applications, from basic physics to statistics to engineering. Since their first appearance in the nineteenth century, they have rapidly become so popular that they are now considered a standard tool of applied mathematics. As reported in Ref. 1, there are at least four main reasons behind the success of eigenvalues and eigenfunctions: the possibility of using them as a basis to solve partial differential equations, when variables can be separatedtheir use in sensitivity analysis, for example, in connection with the physical phenomena of mechanical resonancestheir connection to stability and asymptotic analyses to determine the dominant response of a system to a perturbationtheir ability to provide the intimate behavior, i.e., the personality, of an operator by means of its spectrum.