= Radial probability density
{title2=$P(r)=r^2\int|\psi(r,\Omega)|^2\,d\Omega$}
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 $P(r)=r^2\int|\psi(r,\Omega)|^2d\Omega$. 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 $a_0/Z$.
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