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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?

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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 ($\text{C/m}^3$).

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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 $[L]^{-3}$.

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