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Can you recover the quantum numbers from just the shape of the spherical harmonic?

Physics Asked by Tim van schaik on July 9, 2021

So I was wondering, in quantum physics beautiful graphs are introduced displaying spherical harmonics relying on the quantum numbers of $m$ and $l$. But is it possible to recover these quantum numbers given the shape of the spherical harmonic they produce?

For example, see the picture I attached. How would one go about finding the quantum numbers back from this? I have heard people find the number of nodes/planes but that looks to be extremely difficult in the below picture.

Absolute real part of the harmonic.

One Answer

Yes, is it possible to read off $ell$ and $m$ from the shape of the spherical harmonic $Y_{ell m}$. The rules are not complicated:

  • Number of plane nodes (containing the $z$-axis): $|m|$
  • Number of cone-like nodes (around the $z$-axis): $ell -|m|$

There is a nice interactive web page for Visualization of Spherical Harmonics. You can choose values $ell$ and $m$ to get an image showing positive $Y_{ell m}$ as red bumps, negative $Y_{ell m}$ as blue bumps, and also the nodes as thin lines.

Here is the image (with $ell=6, m=3$) corresponding to the example image from your question:
enter image description here
(image generated using Visualization of Spherical Harmonics, published by ICGEM Potsdam)

Answered by Thomas Fritsch on July 9, 2021

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