Radial probability density ($P(r)=r^2\int |\psi (r,\Omega ){|}^{2}d\Omega$)

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 $a_0/Z$.