# Probability Current Density | Quantum Mechanics

Quantum Mechanics
Basic Quantum Mechanics

Probability Current Density

As the time changes from t=0, the probability of finding the particle in some region of space may increase.

If the probability increases in some region, then it should decrease in some other region such that the total probability of finding the particle in the entire space should be equal to one.

We can assume this as a flow of probability from one region to another region, like a fluid or current. Therefore, the probability flow satisfies the equation of continuity i.e.

$\overrightarrow{\nabla}.\overrightarrow{J}+\frac{\partial \rho}{\partial t}=0$

where $\rho$ = probability density $=\psi*\psi$

and

$\overrightarrow{J}=$ probability current density $=-\frac{i\hbar}{2m}\left[\psi^{*}\overrightarrow{\nabla}\psi-\psi\overrightarrow{\nabla}\psi^{*}\right]$

The magnitude of probability current density represents the flux of the particle i.e. number of particles passing through per unit area per unit time and the direction of probability current density is along the direction of the flow of the particles.