probability density

The probability density is the quantity |ψ|² — the squared magnitude of the wavefunction — that tells you how thickly probability is spread at each point in space. It is not itself a probability, but a density: to get an actual probability you must multiply it by a little volume, or add it up over a region. Where |ψ|² is high the particle is likely to be found; where it dips to zero the particle is never seen.

Thinking of it as a density is the right instinct. Just as mass per unit volume must be multiplied by a volume to give a mass, the probability density must be integrated over a stretch of space to give the probability of finding the particle there. In one dimension |ψ(x)|² has units of one over length, so |ψ(x)|² dx is a pure, dimensionless probability for the interval dx.

Because total probability must be one — the particle is somewhere — the probability density summed over all of space must equal exactly one, the condition called normalization. The shape of |ψ|² is what experiments map out when they record many particles prepared the same way: each lands somewhere unpredictable, but the cloud of many landings traces the probability density precisely.