# What are the units of probability density?

Units of probability density? If bound electron is thought of as a cloud of charge, and it's charge density is proportional to the probability density. Then coulombs /m3 proportional to?

The units of probability density in three-dimensional space are inverse volume, $$[L]^{-3}$$. This is because probability itself is a dimensionless number, such as 0.5 for a probability of 50%.

A normalized wavefunction $$\Psi(\vec{r},t)$$ has dimensions $$[L]^{-3/2}$$ because the square of its complex magnitude, integrated over all space, must be 1. There is 100% probability that the electron is somewhere.

The charge density at a point is the particle’s charge times the probability density at that point.

When the wavefunction is for an electron, the charge density is $$\rho(\vec{r},t)=-e|\Psi(\vec{r},t)|^2$$, which could be expressed in units such as Coulombs per cubic meter ($$\text{C/m}^3$$).

You can think about it by looking at what quantities you get from probability density. Namely, probability is dimensionless (it's just a number between 0 and 1), and given by the integral of probability density, so $$P = \int \rho \text{ d}V$$.

The volume element has dimensions $$[L]^3$$, so the probability density must have dimensions $$[L]^{-3}$$.