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Introduction to Quantum Mechanics
Published in Caroline Desgranges, Jerome Delhommelle, A Mole of Chemistry, 2020
Caroline Desgranges, Jerome Delhommelle
However, nothing seems to explain the entire spectrum of a blackbody. Planck (1858–1947) is the first to solve this puzzle by realizing that the energy of light cannot be understood through classical mechanics. Indeed, he assumes that the amount of energy (E) that can be absorbed, or emitted, by a blackbody can only be of the form E = nhν where n = 0, 1, 2… meaning that E can only take the following values: 0, hν, 2hν, 3hν… Using these values for E, Planck converts the continuous integrals used by Raleigh and Jeans to discrete sums over an infinite number of terms. Making that simple change gives Planck a new formula for the spectrum of the blackbody radiation. And this equation gets it right! It exactly describes the blackbody spectrum, both at low and high frequencies! Planck’s idea is revolutionary! Planck’s equation is now known as Planck’s quantization rule, and h is named Planck’s constant (h = 6.626 × 10–34 Js). In short, Planck assumes that electromagnetic radiation can only be emitted or absorbed in discrete packets, called quanta of energy (hν)! This is the birth of quantum physics! Let us add that the concept of blackbody has been proved to exist. Indeed, the cosmic microwave background (CMB) radiation, emitted 380,000 years after the Big Bang, shows a spectrum that matches perfectly the blackbody radiation at a temperature of 2.73K. This is discovered by Mather and Smoot, who receive the Nobel Prize in Physics in 2006.
Bohmian Quantum Gravity and Cosmology
Published in Xavier Oriols, Jordi Mompart, Applied Bohmian Mechanics, 2019
Nelson Pinto-Neto, Ward Struyve
The power spectrum determines the temperature fluctuations of the cosmic microwave background. Let us consider this in a bit more detail. Let T (n) denote the temperature of the cosmic microwave background in the direction n, with T¯ its average over the sky. The temperature anisotropy δT (n)/T¯, where δT (n) = T (n) − T¯, can be expanded in terms of spherical harmonics () δT(n)T¯=∑l=2∞∑m=−lm=lalmYlm(n).
Energy Today
Published in Anco S. Blazev, Global Energy Market Trends, 2021
There is also radiation from the universe as a whole, which is called cosmic microwave background radiation. Luckily, each of the cosmic radiation sources is far beyond the safety of Earth’s atmosphere. Unfortunately, in order to be of use to people, the cosmic energy must be captured and controlled, something we are simply unable to do at this point.
Distribution-guided heuristic search for nonlinear parameter estimation with an application in semiconductor manufacturing
Published in IISE Transactions, 2020
Hyungjin Kim, Chuljin Park, Yoonshik Kang
The rational background of Assumption 2 can be described in two stages: (i) the asymptotic distribution of estimates from nonlinear least squares estimation and (ii) the empirical distribution of the measures of interest. Wu (1981) presented the asymptotic distribution of estimates obtained by nonlinear least squares following the normal assumption under some conditions. Based on the literature, estimated parameters repeatedly obtained by the nonlinear least squares approach may result in the approximated normal distribution. In addition, the measure of interest we considered usually represents physical quantities (e.g., line length of a geometric structure or electrical conductivity). In the motivating example of Section 2.1, a vector of CD values can be assumed to follow a multivariate normal distribution (May and Spanos, 2006). Other than the OCD example, one can also find some cases in large-scale inverse problems, such as the cosmic microwave background and reservoir analysis (Biegler et al., 2011) and subsurface electrical conductivity (Siqueira et al., 2015), whose parameters are satisfied with Assumption 2. In this article, the distribution of parameter vector is referred to as the hypothetical underlying distribution if the mean vector and the covariance matrix of the distribution are supposable (e.g., pre-known or setting to default values) under Assumption 2.