Probability current

In quantum mechanics, the probability current (sometimes called probability flux) is a mathematical quantity describing the flow of probability in terms of probability per unit time per unit area. Specifically, if one describes the probability density as a heterogeneous fluid, then the probability current is the rate of flow of this fluid. This is equivalent to mass currents in hydrodynamics and electric currents in electromagnetism. It is a real vector, like electric current density. The concept of a probability current is a useful formalism in quantum mechanics.

In non-relativistic quantum mechanics, the probability current j of the wave function Ψ in one dimension is defined as

where Ψ* denotes the complex conjugate of the wave function. In three dimensions, this generalizes to

where ħ is the reduced Planck constant, m is the particle's mass, Ψ is the wavefunction, and ∇ denotes the del or gradient operator.

This can be simplified in terms of the momentum operator,

to obtain

These definitions use the position basis (i.e. for a wavefunction in position space, momentum space is possible).

The above definition should be modified for a system in an external electromagnetic field. In SI units, a charged particle of mass m and electric charge q includes a term due to the interaction with the electromagnetic field;

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