Probability density

A probability density is a nonnegative function whose integral over an event gives its probability.

A current in a probability continuity equation. For overdamped drift-diffusion, $\mathbf{J}=\mathbf{v}p-D\nabla p$. This transport definition does not require the quantum-mechanical wavefunction current formula.

The radial probability density $P(r)$ is defined so that $P(r)dr$ is the probability of finding a particle between radii $r$ and $r+dr$. The spherical volume element gives $P(r)=r^2\int |\psi (r,\Omega ){|}^{2}d\Omega$. For a spherically symmetric wavefunction, $P(r)=4\pi r^2|\psi (r){|}^2$; for $\psi =R(r)Y_l^m(\Omega )$ with normalized spherical harmonics, $P(r)=r^2|R(r){|}^2$. Its maximum need not coincide with the maximum of the density per unit volume: a hydrogenic $1s$ state has its largest spatial density at the origin but its most probable radius at $a_0/Z$.