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The tentative rate equation determined by the method of initial rates is therefore normally verified by comparing the concentrations measured over a longer time (several half-lives) with the integrated form of the rate equation; this assumes that the reaction goes to completion. For example, the integrated rate law for a first-order reaction is
that can be solved once we know the solution of the leading order equation. But we know that in this case we can use the adiabatic approximation. When () does not depend on time one gets the Wigner-Kirkwood series that is often used in statistical mechanics. Indeed, in this case we introduce the unitary transformation
Therefore, as q increases from 0 to 1, the solution U(x; q) of the zeroth-order deformation equation varies (or deforms) from the chosen initial guess u 0 (x) to the solution u(x) of the considered nonlinear equation. Expanding U(x; q) in a Taylor series about q = 0, we have the homotopy-Maclaurin series
The following solutions for the zeroth and first order problem in ... Example: Mathieu equation. This is an example of parametric resonance.
In the zeroth-order example above, the quantity "a few" was given, but in the first-order example, the number "4" is given. A first-order approximation of a function (that is, mathematically determining a formula to fit multiple data points) will be a linear approximation, straight line with a slope: a polynomial of degree 1. For example:
In the distorted-wave Born approximation (DWBA), the incident waves are solutions | to a part of the problem = + that is treated by some other method, either analytical or numerical. The interaction of interest V {\displaystyle V} is treated as a perturbation V 2 {\displaystyle V^{2}} to some system V 1 {\displaystyle V^{1}} that can be solved ...
In quantum physics, Fermi's golden rule is a formula that describes the transition rate (the probability of a transition per unit time) from one energy eigenstate of a quantum system to a group of energy eigenstates in a continuum, as a result of a weak perturbation.
Zero order reaction. Zero-order process (statistics), a sequence of random variables, each independent of the previous ones; Zero order process (chemistry), a chemical reaction in which the rate of change of concentration is independent of the concentrations; Zeroth-order approximation, an approximation of a function by a constant