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The density of states plays an important role in the kinetic theory of solids. The product of the density of states and the probability distribution function is the number of occupied states per unit volume at a given energy for a system in thermal equilibrium. This value is widely used to investigate various physical properties of matter.
In quantum mechanics, a density matrix (or density operator) is a matrix that describes an ensemble [1] of physical systems as quantum states (even if the ensemble contains only one system). It allows for the calculation of the probabilities of the outcomes of any measurements performed upon the systems of the ensemble using the Born rule .
In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can be interpreted as providing a relative likelihood that the value of the ...
A result is the Fermi–Dirac distribution of particles over these states where no two particles can occupy the same state, which has a considerable effect on the properties of the system. Fermi–Dirac statistics is most commonly applied to electrons, a type of fermion with spin 1/2.
We seek a probability distribution of states that maximizes the discrete Gibbs entropy = subject to two physical constraints: The probabilities of all states add to unity ( second axiom of probability ): ∑ i ρ i = 1. {\displaystyle \sum _{i}\rho _{i}=1.}
For the particle in a box, the probability density for finding the particle at a given position depends upon its state, and is given by (,) = { ((+)), < < +,, Thus, for any value of n greater than one, there are regions within the box for which P ( x ) = 0 {\displaystyle P(x)=0} , indicating that spatial nodes exist at which the particle ...
Electronic entropy is the entropy of a system attributable to electrons' probabilistic occupation of states. This entropy can take a number of forms. The first form can be termed a density of states based entropy. The Fermi–Dirac distribution implies that each eigenstate of a system, i, is occupied with a certain probability, p i. As the ...
The solid body shows the places where the electron's probability density is above a certain value (here 0.02 nm −3): this is calculated from the probability amplitude. The hue on the colored surface shows the complex phase of the wave function. In quantum mechanics, a probability amplitude is a complex number used for describing the behaviour ...