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The classical probability density is the probability density function that represents the likelihood of finding a particle in the vicinity of a certain
Classical_probability_density
Value for the flow of probability in quantum mechanics
current (i.e. the probability current density) is related to the probability density function via a continuity equation. The probability current is invariant
Probability_current
Mathematical tool in quantum physics
In quantum mechanics, a density matrix (or density operator) is a matrix used in calculating the probabilities of the outcomes of measurements performed
Density_matrix
Complex number whose squared absolute value is a probability
modulus of this quantity at a point in space represents a probability density at that point. Probability amplitudes provide a relationship between the quantum
Probability_amplitude
Fourier transform of the probability density function
admits a probability density function, then the characteristic function is the Fourier transform (with sign reversal) of the probability density function
Characteristic function (probability theory)
Characteristic_function_(probability_theory)
Formulation of classical mechanics in terms of Hilbert spaces
indeed probability density dynamics is recovered. Dynamics of the probability density (proof) In classical statistical mechanics, the probability density (with
Koopman–von Neumann classical mechanics
Koopman–von_Neumann_classical_mechanics
Branch of mathematics concerning probability
Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations
Probability_theory
Interpretation of probability
the previously dominant viewpoint, the classical interpretation. In the classical interpretation, probability was defined in terms of the principle of
Frequentist_probability
Approximation or recovery of classical mechanics in certain theories
deforms to statistical mechanics with deformation parameter 1/N. Classical probability density Ehrenfest theorem Madelung equations Fresnel integral Mathematical
Classical_limit
Wigner distribution function in physics as opposed to in signal processing
corrections to classical statistical mechanics. The goal was to link the wavefunction that appears in the Schrödinger equation to a probability distribution
Wigner quasiprobability distribution
Wigner_quasiprobability_distribution
Quantum mechanical model
state with little energy. As the energy increases, the probability density peaks at the classical "turning points", where the state's energy coincides with
Quantum_harmonic_oscillator
Mathematical entity to describe the probability of each possible measurement on a system
operator. The probability measure for a system in state ψ is given by:Pr(x∈B|ψ)=∫B⊂ℝ|ψ(x)|2dx, where |ψ(x)|2 is the probability density function for finding
Quantum_state
Description of physical properties at the atomic and subatomic scale
associates to each point in space a probability amplitude. Applying the Born rule to these amplitudes gives a probability density function for the position that
Quantum_mechanics
Description of a quantum-mechanical system
equation, led to a problem with probability density even though it was a relativistic wave equation. The probability density could be negative, which is physically
Schrödinger_equation
Computational quantum mechanical modelling method to investigate electronic structure
of additive terms. Classical density functional theory uses a similar formalism to calculate the properties of non-uniform classical fluids. Despite the
Density_functional_theory
Electric charge per unit length, area or volume
is for electric charge, not other densities like mass density, number density, probability density, and prevent conflict with the many other uses of λ,
Charge_density
Physics principle formulated by Niels Bohr
the replacement of classical mechanics with quantum mechanics. Quantum decoherence Classical limit Classical probability density Leggett–Garg inequality
Correspondence_principle
reformulation of the ideas of a classical Markov chain, replacing the classical definitions of probability with quantum probability. Very roughly, the theory
Quantum_Markov_chain
Distribution of an uncertain quantity
A prior probability distribution (often simply called the prior probability, prior distribution, or prior) of an uncertain quantity is its assumed probability
Prior_probability
Probability distribution
such as the normal, binomial, gamma, and Poisson distributions. The probability density function (pdf) of an exponential distribution is f(x;λ)={λe−λxx≥0
Exponential_distribution
Fundamental theorem in probability theory and statistics
almost surely (and therefore also converges in probability) to the expected value μ as n→∞. The classical central limit theorem describes the size and the
Central_limit_theorem
Probability distribution
In probability theory and statistics, Student's t distribution (or simply the t distribution) tν is a continuous probability distribution that generalizes
Student's_t-distribution
Calculation rule in quantum mechanics
also be employed to calculate probabilities (for measurements with discrete sets of outcomes) or probability densities (for continuous-valued measurements)
Born_rule
Quantum explanation of electromagnetic polarization
such as state vectors, probability amplitudes, unitary operators, and Hermitian operators, emerge naturally from the classical Maxwell's equations in
Photon_polarization
Ensemble of states at a constant temperature
density matrix is diagonal in this basis, with the diagonal entries each directly giving a probability. Example of canonical ensemble for a classical
Canonical_ensemble
Physics of many interacting particles
fluctuate about average values and are characterized by probability distributions. While classical thermodynamics is primarily concerned with thermodynamic
Statistical_mechanics
Interpretation of probability
Bayesian probability (/ˈbeɪziən/ BAY-zee-ən or /ˈbeɪʒən/ BAY-zhən) is an interpretation of the concept of probability, in which, instead of frequency or
Bayesian_probability
Discrete probability distribution
probability theory and statistics, the Poisson distribution (US: /pwɑːˈsɒn/ pwahss-ON, UK: /ˈpwɑːsɒn, ˈpwæsɒn/ PWA(H)SS-on) is a discrete probability
Poisson_distribution
Loss of quantum coherence
the reduced density matrix for the system: ρsys=Trenv(ρ)=|ψ⟩⟨ψ|⟨ϵ|ϵ⟩=|ψ⟩⟨ψ|. Now the probability of finding the system described by a density matrix ρ in
Quantum_decoherence
Average value of a random variable
distinct case of random variables dictated by (piecewise-)continuous probability density functions, as these arise in many natural contexts. All of these
Expected_value
Concept in statistics
necessarily yield probabilities of mutually exclusive states. Quasiprobability distributions also have regions of negative probability density, counterintuitively
Quasiprobability_distribution
Computational statistics technique
from a (target) probability density function f(x), which is proportional to f∝(x), using draws from a simpler (proposal) probability density g(x) as follows:
Rejection_sampling
its probability density is on the classical invariant manifolds near and all along that periodic orbit is systematically enhanced above the classical, statistically
Quantum_ergodicity
Mathematical rule for inverting probabilities
conditional probabilities, allowing the probability of a cause to be found given its effect. For example, with Bayes' theorem, the probability that a patient
Bayes'_theorem
Probability of shared birthdays
In probability theory, the birthday problem asks for the probability that, in a set of n randomly chosen people, at least two will share the same birthday
Birthday_problem
Random process independent of past history
In probability theory and statistics, a Markov chain or Markov process is a stochastic process describing a sequence of possible events in which the probability
Markov_chain
Mathematical description of quantum state
squared modulus of a wave function that depends upon position is the probability density of measuring a particle as being at a given place. The integral of
Wave_function
Metric in quantum mechanics
It is the quantum generalization of the Kolmogorov distance for classical probability distributions. The trace distance is defined as half of the trace
Trace_distance
Theory and paradigm of statistics
field of statistics based on the Bayesian interpretation of probability, where probability expresses a degree of belief in an event. The degree of belief
Bayesian_statistics
Specific quantum state of a quantum harmonic oscillator
just like a classical system, ⟨x̂(t)⟩=2ℏmωℜ[α(t)]=|α(0)|2ℏmωcos(σ−ωt) , ⟨p̂(t)⟩=2mℏωℑ[α(t)]=|α(0)|2mℏωsin(σ−ωt) . The probability density remains a Gaussian
Coherent_state
Specific probability distribution function, important in physics
write a probability distribution of speeds as the function f(v)=[m2πkBT]3/24πv2exp(−mv22kBT). This probability density function gives the probability, per
Maxwell–Boltzmann distribution
Maxwell–Boltzmann_distribution
Notions of probabilistic convergence, applied to estimation and asymptotic analysis
In probability theory, there exist several different notions of convergence of sequences of random variables, including convergence in probability, convergence
Convergence of random variables
Convergence_of_random_variables
Idealization of a large number of atomic-sized systems
other bases, the density matrix is not necessarily diagonal.) In classical mechanics, an ensemble is represented by a probability density function defined
Ensemble (mathematical physics)
Ensemble_(mathematical_physics)
Random change in the energy inside a volume
the above), whereas the classical thermal state is not (both the non-relativistic dynamics and the Gibbs probability density initial condition are not
Quantum_fluctuation
Probability theory term
probability distributions are treated on three levels: discrete probabilities, probability density functions, and measure theory. Conditioning leads to a non-random
Conditioning_(probability)
Function in thermodynamics and statistical physics
thermodynamic beta. Finally, the probability distribution ρi and entropy S are respectively ρi=1Ze−βEi,S=UT+kBlnZ. In classical mechanics, the position and
Partition function (statistical mechanics)
Partition_function_(statistical_mechanics)
Encoding an n-bit string in m qubits
classical bits into a quantum state of smaller dimension, such that any single bit of the original string can be retrieved with a certain probability
Quantum_random_access_code
Mathematical problem involving optimal stopping theory
stopping theory that is studied extensively in the fields of applied probability, statistics, and decision theory. It is also known as the marriage problem
Secretary_problem
Statistical distance measure
probability distributions on density matrices allows a definition of the quantum Jensen–Shannon divergence (QJSD). It is defined for a set of density
Jensen–Shannon_divergence
Generalization of gamma distribution to multiple dimensions
wireless channels. The Wishart distribution can be characterized by its probability density function as follows: Let X be a p × p symmetric matrix of random
Wishart_distribution
Solution method for linear differential equations
the probability density associated to the approximate wave function. The probability that the quantum particle will be found in the classically forbidden
WKB_approximation
Theorem of convex functions
respect to some probability distribution in the random variable X. If p(x) is the true probability density for X, and q(x) is another density, then applying
Jensen's_inequality
Types of numerical variables in mathematics
In statistical theory, the probability distributions of continuous variables can be expressed in terms of probability density functions. In continuous-time
Continuous or discrete variable
Continuous_or_discrete_variable
Interaction of a quantum system with a classical observer
measurement that can be defined, the probability distribution over the outcomes of that measurement can be computed from the density operator. The procedure for
Measurement in quantum mechanics
Measurement_in_quantum_mechanics
Measure of unpredictability of outcomes
defined as the logarithm of the number of outcomes with nonzero probability. As with the classical Shannon entropy and its quantum generalization, the von Neumann
Min-entropy
Family of distributions that generalize the multivariate normal distribution
and an ellipsoid, respectively, in iso-density plots. In statistics, the normal distribution is used in classical multivariate analysis, while elliptical
Elliptical_distribution
Rules in probabilistic logic
explicitly derived by Maurice Fréchet that govern the combination of probabilities about logical propositions or events logically linked together in conjunctions
Fréchet_inequalities
Computer hardware technology that uses quantum mechanics
vector behaves similarly to a (classical) probability vector, with one key difference: unlike probabilities, probability amplitudes are not necessarily
Quantum_computing
Physics phenomenon
state is separable if it is a probability distribution over uncorrelated states, or product states. By writing the density matrices as sums of pure ensembles
Quantum_entanglement
Term in quantum mechanics
density matrices. It expresses the probability that one state will pass a test to identify as the other. It is not a metric on the space of density matrices
Fidelity_of_quantum_states
Theorem in classical statistical mechanics
the probability density of finding another particle at a distance r from the given particle is equal to 4πr2ρg(r), where ρ = N/V is the mean density of
Equipartition_theorem
Partial differential equation
partial differential equation that describes the time evolution of the probability density function of the position or velocity of a particle under the influence
Fokker–Planck_equation
Probability distribution
probability distribution function and the probability density function of the 𝒩(0,1) standard normal distribution, then we have that the probability
Log-normal_distribution
Phenomenon in quantum systems
the eigenstates of a classically chaotic quantum system have enhanced probability density around the paths of unstable classical periodic orbits. The
Quantum_scar
Average uncertainty in variable's states
describe the state of the variable, considering the distribution of probabilities across all potential states. Given a discrete random variable X, which
Entropy_(information_theory)
American statistician (1929–2016)
years. During this time, he wrote what has become one of the classical texts in probability theory. In 1970, he accepted the chair of the statistics department
Emanuel_Parzen
Ensemble of states with an exactly specified total energy
a classical system consisting of one particle in a potential well. In classical mechanics, an ensemble is represented by a joint probability density function
Microcanonical_ensemble
Question in geometric probability
x≤l2sinθ. Now there are two cases. Integrating the joint probability density function gives the probability that the needle will cross a line: P=∫θ=0π2∫x=0l2sinθ4tπdxdθ=2ltπ
Buffon's_needle_problem
Key result in Hamiltonian mechanics and statistical mechanics
time-independent density is in statistical mechanics known as the classical a priori probability. Liouville's theorem applies to conservative systems, that is
Liouville's theorem (Hamiltonian)
Liouville's_theorem_(Hamiltonian)
Scientific hypothesis in mathematical physics
with few degrees of freedom but chaotic classical dynamics. It was proposed by Eugene Wigner in probability theory. The surmise was a result of Wigner's
Wigner_surmise
Set of equations describing the dynamics of a system of many interacting particles
particles. The equation for an s-particle distribution function (probability density function) in the BBGKY hierarchy includes the (s + 1)-particle distribution
BBGKY_hierarchy
one simple formula, the mean density of ℬ equals 1−exp(−λA) where Γ denotes the area of Cξ and A=E(Γ). The classical theory of stochastic geometry
Boolean model (probability theory)
Boolean_model_(probability_theory)
Stochastic process generalizing Brownian motion
Gaussian process. In particular, for t>0, Wt=Wt−W0∼N(0,t), that is, the probability density function of Wt is fWt(x)=12πte−x2/(2t). The covariance between Wt
Wiener_process
System comprising multiple qubits
system comprising multiple qubits. It is the quantum analogue of the classical hardware register. Quantum computers perform calculations by manipulating
Quantum_register
Twenty-first letter in the Greek alphabet
homomorphisms In probability theory, φ(x)=(2π)−12e−x22 is the probability density function of the standard normal distribution. In probability theory, φX(t)
Phi
Quantum mechanical phenomenon
finite probability of tunnelling through or reflecting from the surface barrier when their energies are close to the barrier energy. Classically, the electron
Quantum_tunnelling
Statistical function that defines the quantiles of a probability distribution
H(x)=−f′(x)f(x)=−ddxlnf(x) and f(x) is the probability density function. The forms of this equation, and its classical analysis by series and asymptotic solutions
Quantile_function
Entropy coding methods
different symbols – like into even and odd numbers, but with densities corresponding to the probability distribution of the symbols to encode. Then to add information
Asymmetric_numeral_systems
Statistical measure of how far values spread from their average
Var(X)=1n2∑i=1n∑j=1n12(xi−xj)2=1n2∑i∑j>i(xi−xj)2. If the random variable X has a probability density function f(x), and F(x) is the corresponding cumulative distribution
Variance
Measure of distinguishability between two quantum states
finite-dimensional. We first discuss the classical case. Suppose the probabilities of a finite sequence of events is given by the probability distribution P = {p1...pn}
Quantum_relative_entropy
Study of quantum systems changing with time
commutator of the Hamiltonian with the density matrix. This equation is the quantum mechanical analogue of the classical Liouville's theorem. For a closed
Quantum_dynamics
Physics experiment
Whole Time?". Wired. Couder, Y.; Fort, E. (2012). "Probabilities and trajectories in a classical wave–particle duality". Journal of Physics: Conference
Double-slit_experiment
Property of a thermodynamic system
the familiar classical definition of entropy for states with classical probabilities pi:S=−k𝖡∑ipilnpii.e. in such a basis the density matrix is diagonal
Entropy
Principle in Bayesian statistics
different forms of prior data. As a special case, a uniform prior probability density (Laplace's principle of indifference, sometimes called the principle
Principle_of_maximum_entropy
Mathematical approach to quantum optics
positive, and is not a bona-fide probability function. We wish to construct a function P(α) with the property that the density matrix ρ̂ is diagonal in the
Glauber–Sudarshan P representation
Glauber–Sudarshan_P_representation
Equation of statistical mechanics
integro-differential equation, and the unknown function in the equation is a probability density function in six-dimensional space of a particle position and momentum
Boltzmann_equation
Equations governing time evolution of physical systems
a kinetic scheme, and the process is Markovian (any jumping time probability density function for state i is an exponential, with a rate equal to the
Master_equation
Calculation of complex statistical distributions
Carlo methods create samples from a continuous random variable, with probability density proportional to a known function. These samples can be used to evaluate
Markov_chain_Monte_Carlo
Measure in quantum information theory
p(x)p(y). The quantum mechanical counterpart of classical probability distributions are modeled with density matrices. Consider a quantum system that can
Quantum_mutual_information
Mathematical statistics distance measure
where p and q denote the probability density functions of P and Q. More generally, if P and Q are probability measures on a measurable space 𝒳
Kullback–Leibler_divergence
Interpretation of quantum mechanics
allows probabilities to be assigned to various alternative histories of a system such that the probabilities for each history obey the rules of classical probability
Consistent_histories
Matrix-valued random variable
In probability theory and mathematical physics, a random matrix is a matrix-valued random variable—that is, a matrix in which some or all of its entries
Random_matrix
Probability distribution
the physicist Eugene Wigner, is the probability distribution defined on the domain [−R, R] whose probability density function f is a scaled semicircle,
Wigner semicircle distribution
Wigner_semicircle_distribution
Atom of the element hydrogen
cross-sections of the probability density that are color-coded (black represents zero density and white represents the highest density). The angular momentum
Hydrogen_atom
Theorem pertaining to the ontology of quantum mechanics
response functions, or conditional probability densities: A ( S , λ ) {\displaystyle A(S,\lambda )} is the probability that the measurement A {\displaystyle
Pusey–Barrett–Rudolph_theorem
Collection of random variables
In probability theory and related fields a stochastic (/stəˈkæstɪk/) or random process is a mathematical object usually defined as a family of random
Stochastic_process
Random motion of particles suspended in a fluid
with some probability density function φ(q) (i.e., φ(q) is the probability density for a jump of magnitude q, i.e., the probability density of the particle
Brownian_motion
List of statements that appear to contradict themselves
two of them have the same birthday. Borel's paradox: Conditional probability density functions are not invariant under coordinate transformations. Boy
List_of_paradoxes
Method for finding lost objects
hypothesis, construct a probability density function for the location of the object. Construct a function giving the probability of actually finding an
Bayesian_search_theory
Statistical distribution used in many-particle mechanics
kinetic energy: f(v)=(m2πkBT)3/24πv2e−mv22kBT where: f(v) is the probability density function of particle speeds, m is the mass of a particle, kB is the
Maxwell–Boltzmann_statistics
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