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does m tell us anything about the magnetic properties of the system since it is called magnetic quantum number? and if it does, what information exactly does it give us regarding magnetic properties?
<math>m</math> refers to the orientation of particular orbitals in each subshell. I believe <math>m</math> is called the magnetic quantum number because the effect of the different orbital positions were first studied through the Zeeman effect, the splitting of a spectral line when in the presence of a static magnetic field.
Magnetic angular momentum is responsible for spin-orbit coupling which is, I believe, the interaction between the magnetic field of the particles with eachother… I think M has to do with this as well as the Zeeman effect… Oops! Wikipedia just informed me that L and S are responsible for spin-orbit coupling, while M is involved with the Zeeman effect… I am going to read these articles more thoroughly and come back and edit this post. I am having a tough time seeing how M doesn't effect spin-orbit coupling while it does effect the shape of the probability orbitals.
In spherical coordinates, are we most interested in the Volume, Surface Area, both, or neither? What is the interpreted significance if any? ie Is the volume related to the expectation value, allowed energies, ect?
The volume is important. In the 1D case we dealt with integrals like <math>\int|\psi|^{2}dx</math>, but since we are now working in 3D we need to integrate over a volume <math>\int\int\int|\psi|^{2}dx dy dz</math> (or <math>\int\int\int|\psi|^{2}r^{2} sin\theta dr d\theta d\phi</math> in spherical coordinates). I'm not sure exactly what the surface area gives us.
My understanding of why we did spherical coordinates was mostly to be able to have the value r – which is significant because the potential dissipates mostly as a function of R – so that's why we want the to translate the 3D wave into spherical, not necessarily because we're always interested in the volume - (or whatever surface area is).
Anybody else use a Mac to edit these pages? If so do you have issues?