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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.
The fact that the density is positive definite and convected according to this continuity equation implies that one may integrate the density over a certain domain and set the total to 1, and this condition will be maintained by the conservation law. A proper relativistic theory with a probability density current must also share this feature.
The electric current is = / =, it follows that the current density vector is the vector normal (i.e. parallel to v) and of magnitude / = j = ρ v . {\displaystyle \mathbf {j} =\rho \mathbf {v} .} The surface integral of j over a surface S , followed by an integral over the time duration t 1 to t 2 , gives the total amount of charge flowing ...
In RQM, while ψ(r, t) is a wavefunction, the probability interpretation is not the same as in non-relativistic QM. Some RWEs do not predict a probability density ρ or probability current j (really meaning probability current density) because they are not positive-definite functions of space and time. The Dirac equation does: [25]
Furthermore, the square of the amplitude (determining the probability density) is inversely proportional to p(x), reflecting the length of time the classical particle spends near x. The system behavior in a small neighborhood of the turning point does not have a simple classical explanation, but can be modeled using an Airy function .
The Born rule is a postulate of quantum mechanics that gives the probability that a measurement of a quantum system will yield a given result. In one commonly used application, it states that the probability density for finding a particle at a given position is proportional to the square of the amplitude of the system's wavefunction at that position.
The probability density is = | |, this equation is exactly the continuity equation, appearing in many situations in physics where we need to describe the local conservation of quantities. The best example is in classical electrodynamics, where j corresponds to current density corresponding to electric charge, and the density is the charge-density.
The probability distribution of the sum of two or more independent random variables is the convolution of their individual distributions. The term is motivated by the fact that the probability mass function or probability density function of a sum of independent random variables is the convolution of their corresponding probability mass functions or probability density functions respectively.