Current density

In electromagnetism, current density is the electric current per unit area of cross section. It is defined as a vector whose magnitude is the electric current per cross-sectional area at a given point in space (i.e. it is a vector field). In SI units, the electric current density is measured in amperes per square metre.

Definition

Electric current density J is simply the electric current I (SI unit: A) per unit area A (SI unit: m²). Its magnitude is given by the limit:

For current density as a vector J, the surface integral over a surface S, followed by an integral over the time duration t₁ to t₂, gives the total amount of charge flowing through the surface in that time (t₂ − t₁):

The area required to calculate the flux is real or imaginary, flat or curved, either as a cross-sectional area or a surface. For example, for charge carriers passing through an electrical conductor, the area is the cross-section of the conductor, at the section considered.

The vector area is a combination of the magnitude of the area through which the mass passes through, A, and a unit vector normal to the area, . The relation is .

Probability current

In quantum mechanics, the probability current (sometimes called probability flux) is a mathematical quantity describing the flow of probability (i.e. probability per unit time per unit area). Intuitively, if one pictures the probability density as an inhomogeneous fluid, then the probability current is the rate of flow of this fluid. This is analogous to mass currents in hydrodynamics and electric currents in electromagnetism. It is a real vector, like electric current density. The notion of a probability current is useful in some of the formalism in quantum mechanics.

Definition (non-relativistic 3-current)

Free spin-0 particle

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

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,