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In crystallography, direct methods are a family of methods for estimating the phases of the Fourier transform of the scattering density from the corresponding magnitudes. . The methods generally exploit constraints or statistical correlations between the phases of different Fourier components that result from the fact that the scattering density must be a positive real nu
Consider the average number of particles with particle properties denoted by a particle state vector (x,r) (where x corresponds to particle properties like size, density, etc. also known as internal coordinates and, r corresponds to spatial position or external coordinates) dispersed in a continuous phase defined by a phase vector Y(r,t) (which again is a function of all such vectors which ...
A quantum master equation is a generalization of the idea of a master equation.Rather than just a system of differential equations for a set of probabilities (which only constitutes the diagonal elements of a density matrix), quantum master equations are differential equations for the entire density matrix, including off-diagonal elements.
A density matrix with only diagonal elements can be modeled as a classical random process, therefore such an "ordinary" master equation is considered classical. Off-diagonal elements represent quantum coherence which is a physical characteristic that is intrinsically quantum mechanical.
where ϕ(r, t) is the density of the diffusing material at location r and time t and D(ϕ, r) is the collective diffusion coefficient for density ϕ at location r; and ∇ represents the vector differential operator del. If the diffusion coefficient depends on the density then the equation is nonlinear, otherwise it is linear.
Density functional theory (DFT) is a computational quantum mechanical modelling method used in physics, chemistry and materials science to investigate the electronic structure (or nuclear structure) (principally the ground state) of many-body systems, in particular atoms, molecules, and the condensed phases.
In physics and quantum chemistry, specifically density functional theory, the Kohn–Sham equation is the non-interacting Schrödinger equation (more clearly, Schrödinger-like equation) of a fictitious system (the "Kohn–Sham system") of non-interacting particles (typically electrons) that generate the same density as any given system of interacting particles.
where ρ(r a,r b) represents the joint electronic density, or the probability density of finding electron a at r a and electron b at r b. Within this notation, ρ(r a,r b) dr a dr b represents the probability of finding the two electrons in their respective volume elements dr a and dr b.