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Without the free field E 0 (t) in this equation the operator x(t) would be exponentially dampened, and commutators like [z(t),p z (t)] would approach zero for t ≫ 1 / τω 2 0 . With the vacuum field included, however, the commutator is iħ at all times, as required by unitarity, and as we have just shown. A similar result is easily ...
A free particle with mass in non-relativistic quantum mechanics is described by the free Schrödinger equation: (,) = (,) where ψ is the wavefunction of the particle at position r and time t . The solution for a particle with momentum p or wave vector k , at angular frequency ω or energy E , is given by a complex plane wave :
In physics, mean free path is the average distance over which a moving particle (such as an atom, a molecule, or a photon) travels before substantially changing its direction or energy (or, in a specific context, other properties), typically as a result of one or more successive collisions with other particles.
Beyond this distance, the diffusioosmotic velocity does not vary with distance from the surface. The driving force for diffusioosmosis is thermodynamic, i.e., it acts to reduce the free energy if the system, and so the direction of flow is away from surface regions of low surface free energy, and towards regions of high surface free energy.
The diffusion equation is a parabolic partial differential equation.In physics, it describes the macroscopic behavior of many micro-particles in Brownian motion, resulting from the random movements and collisions of the particles (see Fick's laws of diffusion).
The three-body problem is a special case of the n-body problem, which describes how n objects move under one of the physical forces, such as gravity. These problems have a global analytical solution in the form of a convergent power series, as was proven by Karl F. Sundman for n = 3 and by Qiudong Wang for n > 3 (see n-body problem for details
Free energy relationships establish the extent at which bond formation and breakage happen in the transition state of a reaction, and in combination with kinetic isotope experiments a reaction mechanism can be determined. Free energy relationships are often used to calculate equilibrium constants since they are experimentally difficult to ...
In the gas phase, is often defined as the diffusional mean free path, by assuming that a simple approximate relation is exact: =, where is the root mean square speed of the gas molecules: =, where is the mass of the diffusing species. This approximate equation becomes exact when used to define the diffusional mean free path.