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I want to be clear on the difference between radial distribution function ${P}_{nl}(r)dr$ and probability density function in the context of hydrogen wave function.

Definition of ${P}_{nl}(r)dr$ is as following $$ {P}_{nl}(r)dr = \left \{\left \{\int_{0}^{\pi }{sin\theta d\theta \int_{0}^{2\pi }{d}} \right .\Phi [{Y}_{l}^{{m}_{l}}(\theta ,\Phi ){]}^{*}[{Y}_{l}^{{m}_{l}}(\theta ,\Phi )] \right \}{r}^{2}{R}_{nl}^{2}(r)dr={r}^{2}{R}_{nl}^{2}(r)dr\\ $$ (just leave out the first curly bracket).

Definition of probability density is as following ${\psi }_{nl{m}_{l}}^{*}{\psi }_{nl{m}_{l}}{r}^{2}drsin\theta d\theta d\Phi$

My interpretation is that the radial distribution function is the probability of finding e- described by a specific wave function in a spherical shell of radius r and thickness dr (This is because the function integrates over the whole interval of (theta) and (phi) so that the distribution function depends only on (r).

The probability density, on the other hand, gives the probability of finding e- described by a specific wave function in a chosen volume element. Have i correctly understood the difference of the 2 terms?

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  • $\begingroup$ Yes, but how you described the probability density isn’t quite right. The probability of finding the electron in an infinitesimal volume element $dV$ is the probability density times $dV$. That’s why it’s a density. $\endgroup$ – G. Smith Jan 11 at 2:24

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