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Probability current information


In quantum mechanics, the probability current (sometimes called probability flux) is a mathematical quantity describing the flow of probability. Specifically, if one thinks of probability as a heterogeneous fluid, then the probability current is the rate of flow of this fluid. It is a real vector that changes with space and time. Probability currents are analogous to mass currents in hydrodynamics and electric currents in electromagnetism. As in those fields, the probability current (i.e. the probability current density) is related to the probability density function via a continuity equation. The probability current is invariant under gauge transformation.

The concept of probability current is also used outside of quantum mechanics, when dealing with probability density functions that change over time, for instance in Brownian motion and the Fokker–Planck equation.[1]

  1. ^ Paul, Wolfgang; Baschnagel, Jörg (1999). Stochastic Processes : From Physics to Finance. Berlin: Springer. p. 84. ISBN 3-540-66560-9.

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Continuity equation

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V}(t)=\int _{V}\Psi ^{*}\Psi dV=\int _{V}|\Psi |^{2}dV.} The probability current (probability flux) is j ( r , t ) = ℏ 2 m i [ Ψ ∗ ( ∇ Ψ ) − Ψ ( ∇ Ψ ∗ )...

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The posterior probability is a type of conditional probability that results from updating the prior probability with information summarized by the likelihood...

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In mathematics, a probability measure is a real-valued function defined on a set of events in a σ-algebra that satisfies measure properties such as countable...

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Transmission coefficient

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coefficient is defined in terms of the incident and transmitted probability current density J according to: T = J → t r a n s ⋅ n ^ J → i n c ⋅ n ^ ...

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In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number...

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The word probability has been used in a variety of ways since it was first applied to the mathematical study of games of chance. Does probability measure...

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Win probability is a statistical tool which suggests a sports team's chances of winning at any given point in a game, based on the performance of historical...

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Frequentist probability or frequentism is an interpretation of probability; it defines an event's probability as the limit of its relative frequency in...

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Markov chain

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stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event...

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Wave packet

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region of space. The momentum profile A(k) remains invariant. The probability current is j = ρ v = 1 2 i ( ψ ∗ ∇ ψ − ψ ∇ ψ ∗ ) = ρ ( k 0 + 4 t ( x − k...

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Free probability

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quantum information theory and other theories were established. Free probability is currently undergoing active research. Typically the random variables lie...

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In probability theory and related fields, a stochastic (/stəˈkæstɪk/) or random process is a mathematical object usually defined as a sequence of random...

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Step potential

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coefficient is defined as the ratio of the transmitted probability current to the incoming probability current. However, the quantities directly involved in this...

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Flux

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\right).} This is sometimes referred to as the probability current or current density, or probability flux density. As a mathematical concept, flux is...

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Dirac equation

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=\phi ^{*}\phi } and this density is convected according to the probability current vector J = − i ℏ 2 m ( ϕ ∗ ∇ ϕ − ϕ ∇ ϕ ∗ ) {\displaystyle J=-{\frac...

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Glossary of elementary quantum mechanics

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transform. Probability amplitude A probability amplitude is of the form ⟨ α | ψ ⟩ {\displaystyle \langle \alpha |\psi \rangle } . Probability current Having...

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