Do Commuting Observables in Quantum Physics Share Common Eigenfunctions?

[IHateMayonnaise]

[SOLVED] Silly Quantum Physics questions

I'm afraid to say that I have a test in my undergraduate Quantum Physics course tomorrow. I feel prepared for the most part - but I am trying to tie everything we're learning together. In hopes of doing this, I have a couple questions that I would like some feedback on (I will probably have more, and when I do I will update this thread).

1) When two observables commute (say, $$\hat{L_z}$$ and $$\hat{L^2}$$), does this imply that they have common Eigenfunctions?

2) (Tell me if this is right, probably not going to be on the test but I would still like to know) Dirac proposed that particles must have an intrinsic spin incorporated into them so that Quantum Mechanics would not contradict relativity - thus requiring that particles have a finite structure, even though experimental data does not agree. Therefore electrons "orbiting" the nucleus are not in fact transversing space as we know it (with a calculatable velocity), rather they are taking "quantum jumps" - as to not violate relativity and travel faster than the speed of light. This regards spin as a purely quantum-mechanical effect, and there is no macroscopic analogue.

Thanks Yall

IHateMayonnaise

 

[lbrits]

1) Two commuting Hermitian matrices have common eigenvectors.

2) I'm not sure if Dirac proposed that. I think it was Pauli. Dirac gave an explanation for it in terms of relativity. And I don't think he considered spin to be a result of finite structure.

You use "therefore" and "thus" pretty loosely. In any event, it's not really meaningful to ask what the electron is doing. It's wavefunction is a smeared blob around the nucleus, and electronic transitions are continuous evolutions of one state/blob into another.

Ordinary (1 particle) quantum mechanics is not relativistic and violates relativity quite explicitly. But it does pretty well regardless.

 

[Pere Callahan]

1) When two observables commute (say, $$\hat{L_z}$$ and $$\hat{L^2}$$), does this imply that they have common Eigenfunctions?

Yes, this is a general mathematical fact.

2)... rather they are taking "quantum jumps" - as to not violate relativity and travel faster than the speed of light.

What do you mean by "quantum jumps"? Transitions from one quantum state into another?
Godd luck for you exam tomorrow!

 

[peter0302]

Therefore electrons "orbiting" the nucleus are not in fact transversing space as we know it (with a calculatable velocity), rather they are taking "quantum jumps" - as to not violate relativity and travel faster than the speed of light. This regards spin as a purely quantum-mechanical effect, and there is no macroscopic analogue.

That part is right (the frequent use of "therefore"'s notwithstanding). Another motivation for electron spin was the fact that if the electron were orbiting like a planet, it would lose energ and collapse into the positvely charged nucleus. Since this doesn't happen, the only conclusion is what you said above. That has as much to do with relativity as it does with Newtonian mechanics. And it's also right that there is no classical analogue. So what is "spinning?" It probably wasn't the best word choice, but between the Danish, German, and English that was being thrown around back then, you can't really blame them. :)