I am still confused that L=0 represents the s-wave, which is a spherical dispersion. From this lecture (as well as the previous one) I can understand that a higher L should be more spherical. How can I understand the s-wave contradiction? Thanks
Hi, nice to see someone is going through these along with me :-) I'm wondering the same. I think the sphericity/ellipticity of orbits is just as he says semiclassical intuition. Notice that both r- and r+ are to the right of the origin i.e. the center of the potential, also the expected value of r, is right between them. So I think that the r- and r+ delimit support (or something close to it) of the wave function. If they are close to each other the wave function is confined to narrow region between two spheres and so is the electron. If they are further apart the region is larger and electron can get closer to the center of potential. For example for L=0 the r+ is zero and wave function is positive at zero, meaning the electron can get arbitrary close to the center. Also notice that highly elliptical orbits, again L=0, have the largest possible . So the sphericity/elipticity tells you how elliptical orbits can fit into wave function. I'm not sure if this intuition is right or useful though.
@@Stohlas Hi, thank you for your detailed reply and nice to study together:-) You are right, and now I think I understand it a little more: I think our confusion comes from the misconception between R(r) and Y(θ,φ). When I thought about "spherical dispersion", my first impression was the shape of Y(θ,φ). However, here we are talking about R(r), which is a scalar function. [The l(l+1) term in the R(r) should be viewed as a centrifugal barrier and do not have angle dependence.] As you said, when L goes larger, the r+ and r- get closer. So the R(r) dispersion for large L looks more like a shell (and L=0 looks like a filled ball). Please correct me if I made some mistakes. Thanks.
I am still confused that L=0 represents the s-wave, which is a spherical dispersion. From this lecture (as well as the previous one) I can understand that a higher L should be more spherical. How can I understand the s-wave contradiction? Thanks
Hi, nice to see someone is going through these along with me :-) I'm wondering the same. I think the sphericity/ellipticity of orbits is just as he says semiclassical intuition. Notice that both r- and r+ are to the right of the origin i.e. the center of the potential, also the expected value of r, is right between them. So I think that the r- and r+ delimit support (or something close to it) of the wave function. If they are close to each other the wave function is confined to narrow region between two spheres and so is the electron. If they are further apart the region is larger and electron can get closer to the center of potential. For example for L=0 the r+ is zero and wave function is positive at zero, meaning the electron can get arbitrary close to the center. Also notice that highly elliptical orbits, again L=0, have the largest possible . So the sphericity/elipticity tells you how elliptical orbits can fit into wave function. I'm not sure if this intuition is right or useful though.
@@Stohlas Hi, thank you for your detailed reply and nice to study together:-) You are right, and now I think I understand it a little more: I think our confusion comes from the misconception between R(r) and Y(θ,φ). When I thought about "spherical dispersion", my first impression was the shape of Y(θ,φ). However, here we are talking about R(r), which is a scalar function. [The l(l+1) term in the R(r) should be viewed as a centrifugal barrier and do not have angle dependence.]
As you said, when L goes larger, the r+ and r- get closer. So the R(r) dispersion for large L looks more like a shell (and L=0 looks like a filled ball).
Please correct me if I made some mistakes. Thanks.
@@yyc3491 smart guy, thanks
Sir what is the difference between degeneracy of multi electron and single electron spices