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Angular Probability Densities

Sunday, March 6, 2022

What are the Angular Probability Densities?

The angular parts of the probability density function from the wave function is

P(θ,ϕ)=Θl,ml(θ)Φml(ϕ)2P\left(\theta, \phi\right)=\left|\Theta_{l,m_l}\left(\theta\right)\Phi_{m_l}\left(\phi\right)\right|^2

Looking at the l=0l=0 and l=1l=1 angular probability density functions, we see that all are cylindrically symmetric, and the l=0l=0 wave function is spherically symmetric:

Angular probability densities

For the l=1l=1 wave functions, when ml=0m_l=0, the density function forms two regions along the zz-axis. This makes sense given the fact that the direction of the angular momentum vector is in the xyxy-plane. Additionally, for the ml=±1m_l=\pm 1 wave functions, the maximum projection of the angular momentum vector is along the zz-axis, making the electrons spend most of their time near the xyxy-plane.