i think it's time to re-indoctrinate the youthpic.twitter.com/tdi0bFAYOA
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Take the 2p orbital, for example. The electron is in a superposition of 2px, 2py and 2pz. |ψ2p> = (|2px>+|2py>+|2pz>)/sqrt(3). Now take the pdf <ψ2p|ψ2p> and you'll see the dependence on the angles cancels out. It only depends on the distance to the nucleus.
That's just not true.. a superposition of the 2ps is a superposition of 3 dumbbells each of which has an angular dependence in an independent direction .. it *can* happen that you measure l=0, but it can also happen that you measure l=1
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That's what you would measure. But it is hard to imagine a free hydrogen atom being anisotropic...
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why is it hard to imagine? .. the point is that the wave function has non-trivial angular dependence and so does the PDF .. i'm sorry if you're upset but it's just the eigensolutions to the schrodinger equation
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It's not the angular momentum we're measuring. If you want a visual representation of the electron cloud, you're interested in the position, which is isotropic around the nucleus. This is what we see in a scanning tunneling microscope, for example. Atoms show up as tiny balls.
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Doesn't it depend on the phase between the 2ps? If you sum |px> + |py> and normalize, you get a 45 degree dumbbell, but |px> + i|py> is a torus. The way I think of it, eigenvectors with a fixed rotating phase are stationary, but phase that changes with time... 1/2
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Contains almost a crystallized motion in the wave-function. The out of phase sum has a "shape" that doesn't change over time, but it represents a rotating electron. So with QM you can have stationary states with an "internal" motion. like the pure linear velocity states 2/2
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@wood_croft's comment sounds like a not-quite-right version of closed shell theorem (@wood_croft summed over the wavefunctions themselves rather than summing over magnitude squareds). https://quantummechanics.ucsd.edu/ph130a/130_notes/node385.html … -
It's both: |x> = |a> + |b> => <x|x> = <a|a> + <a|b> + <b|a> + <b|b> = <a|a> + <b|b> if |a> and |b> are orthogonal.
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