Probability current

In quantum mechanics, the probability current (sometimes called probability flux) is a concept describing the flow of probability density. In particular, if one pictures the probability density as an inhomogeneous fluid, then the probability current is the rate of flow of this fluid (the density times the velocity).

Definition

In non-relativistic quantum mechanics, the probability current $\vec j$ of the wave function Ψ is defined as

$\vec j = \frac{\hbar}{2mi}\left(\Psi^* \vec \nabla \Psi - \Psi \vec \nabla \Psi^*\right) = \frac\hbar m \mbox{Im}(\Psi^*\vec\nabla\Psi)=\mbox{Re}(\Psi^* \frac{\hbar}{im} \vec\nabla \Psi)$

in the position basis and satisfies the quantum mechanical continuity equation

$\frac{\partial \rho}{\partial t} + \vec \nabla \cdot \vec j = 0$

with the probability density $\rho\,$ defined as

$\rho = |\Psi|^2 \,$.

If one were to integrate both sides of the continuity equation with respect to volume, so that

$\int_V \left( \frac{\partial |\Psi|^2}{\partial t} \right) dV + \int_V \left( \vec \nabla \cdot \vec j \right) dV = 0$

then the divergence theorem implies the continuity equation is equivalent to the integral equation

$\frac{\partial}{\partial t} \int_V |\Psi|^2 dV + \int_S \vec j \cdot \vec {dA} = 0$

where the V is any volume and S is the boundary of V. This is the conservation law for probability in quantum mechanics.

In particular, if $\Psi\,$ is a wavefunction describing a single particle, the integral in the first term of the preceding equation (without the time derivative) is the probability of obtaining a value within V when the position of the particle is measured. The second term is then the rate at which probability is flowing out of the volume V. Altogether the equation states that the time derivative of the change of the probability of the particle being measured in V is equal to the rate at which probability flows into V.

Examples

Plane wave

The probability current associated with the (three dimensional) plane wave

$\Psi = A e^{i\vec k \cdot \vec r} e^{-i \omega t}$

is

$\vec j = \frac{\hbar}{2mi} |A|^2 \left( e^{-i\vec k \cdot \vec r} \vec \nabla e^{i\vec k \cdot \vec r} - e^{i\vec k \cdot \vec r} \vec \nabla e^{-i\vec k \cdot \vec r} \right) = |A|^2 \frac{\hbar \vec k}{m}.$

This is just the square of the amplitude of the wave times the particle's velocity,

$\vec v = \frac{\vec p}{m} = \frac{\hbar \vec k}{m}$.

Note that the probability current is nonzero despite the fact that plane waves are stationary states and hence

$\frac{d|\Psi|^2}{dt} = 0\,$

everywhere. This demonstrates that a particle may be in motion even if its spatial probability density has no explicit time dependence.

Particle in a box

The energy eigenstates of a particle in a box of one spatial dimension and of length L are, for 0 < x < L,

$\Psi_n = \sqrt{\frac{2}{L}} \sin \left( \frac{n\pi}{L} x \right)$

and zero elsewhere. The associated probability currents are

$j_n = \frac{i\hbar}{2m}\left( \Psi_n^* \frac{\partial \Psi_n}{\partial x} - \Psi_n \frac{\partial \Psi_n^*}{\partial x} \right) = 0$

since $\Psi_n = \Psi_n^*.$

Definition in an external field

The standard definition should be modified for a system in an external electromagnetic field. E. g. for a charged particle the Hamiltonian is:

$H = \frac{1}{2m}\left[\vec{p} - \frac{q}{c} \vec{A}(\vec{x},t)\right]^2 + q \cdot \phi(\vec{x},t)$

where $\phi(\vec{x},t)$ is a scalar potential, $\vec{A}(\vec{x},t)$ is vector potential of electromagnetic field, $\vec{p} = \frac{\hbar}{i} \vec{\nabla}$ is momentum operator, q is the charge of the particle. In the way very similar to the derivation of probability flux current without a field, one obtains:

$\vec j = \frac{\hbar}{2mi}\left(\Psi^* \vec \nabla \Psi - \Psi \vec \nabla \Psi^*\right) - \frac{q}{mc} \vec{A}(\vec{x},t) |\Psi|^2$