← Atomic Physics

Radial Probability Densities

Sunday, March 6, 2022

Why Radial Probability?

The full three-dimensional probability function can be useful, but to analyze the probability an electron is a certain distance from the nucleus, the radial probability density is more useful.

Imagine a spherical shell of radius rr and thickness drdr. The radial probability function, P(r)P\left(r\right), is the probability of finding the electron in that shell. We define this function by integrating over the polar and azimuthal angles:

P(r)=Rn,l(r)2r2 dr0πΘl,ml(θ)2sinθ dθ02πΦml(ϕ)2 dϕP\left(r\right)=\left|R_{n,l}\left(r\right)\right|^2r^2~dr\int_0^\pi\left|\Theta_{l,m_l}\left(\theta\right)\right|^2\sin{\theta}~d\theta\int_0^{2\pi}\left|\Phi_{m_l}\left(\phi\right)\right|^2~d\phi

Since the polar and azimuthal functions are normalized, however, the integrals of Θ\Theta and Φ\Phi are equal to unity. This means

P(r)=r2Rn,l(r)2 drP\left(r\right)=r^2\left|R_{n,l}\left(r\right)\right|^2~dr

Note that as r0r\rightarrow 0, P(r)0P\left(r\right)\rightarrow 0 even if R(r)R\left(r\right) does not go to 0 since the volume becomes zero even if the probability density is positive.

Average Distance

Analyzing the radial probability densities for different principle quantum numbers, it appears that the quantum number nn largely determines the average distance much more so than the quantum number ll.

For any value of nn, the smaller the ll, the more time the electron spends close and far from the nucleus (just like a planet orbiting with small angular momentum values).