marlon said:
Well, the QM-picture does not ignore symmetries...It is because of symmetry that quantumnumbers are born...
For example, look at the way the quantumnumbers (eg : l) are inserted in QM because of the symmetry-properties of spherical harmonics...These properties need to be respected because otherwise these wave-functions are not physical etc...
regards
marlon
You're right about the relevance of symmetries in QM.A symmetry group for a QM system admtis an irreductible representation whose operators commute with the (time independent) hamiltonian,and by means of the Noether theorem for QM,all these operators describe at QM level conserved quantities.
But I'm afraid your example is not the greatest when talking abot symmetries.Because the notion of angular momentum in QM is much,much more general,than symmetries.I shall prove my assertion.
We refer at QM operators for angular momentum as at a set of 3 operators which are operators of an irreductible representation of the Lie algebra of angular momentum obtained through quantization via the second principle and the classical definiton of the angular momentum.The angular momentum operators Jsquared and Jz act in an irreductible linear space of the representation,where the quantum numbers "l" and "m" have the significance of merely hbar units eigenvalues.These thing are very abstract and mathematized.But that's QM at essence.
So,who said anything about symmetries/Noether theorem wrt to angular momentum,SO FAR??
To resume this necessary mathematical divagation:to state that symmetries give birth to Q numbers for angular momentum,that's bull****.
These Q numbers are hbar unit eigenvalues of those operators.Yet symmetries of the Hamiltonian of the H atom are responsible for their values,but not for their fundamental significance.
AAAAAAAAA,when discussing specific exmples (i have a hunch yours refers to H atom),then,yes,symmetries become essential,for 2 reasons:1)the possibilities induced by the Noether theorem;2) the possibility of chosing specific realizations of the irreductible space of the representation.Actually the basis in that space becomes essential.
For the H atom,my friend,u're only given these 3 relations:
\hat{H} =\frac{\hat{p}^{2}}{2m} -\frac{e^{2}}{4\pi\epsilon_{0}r}\hat{1}
\hat{L}^{2} |\psi> =\hbar ^{2} l(l+1) |\psi>
\hat{L}_{z} |\psi> =\hbar m |\psi>
Solving the H atom from these 3,would require a chain of deductions,forrmulas,integrations,to find what u were saying:"l" and "m" have the values they have (for the H atom) because of symmetries.
Sorry for repeating myself,but to state that "l" and "m" are symmetry related in general,just doesn't sound right to me.
Daniel.
PS But you're right about the relevance of symmetries in quantum physics.They guide our steps to knowledge.But mathematical rigurosity,logics and definitons are somethig else.I think so.I might be wrong,though.You're free to say your point of view.
PPS.Sorry,again,Mr.Integral,for putting too much soul in my exposures,but i guess it's just another thing people on PF should become familiar with,as long as I'm still here.When i won't,things would return to normal.
1.Can anyone explain the physical interpretation of the Quantum numbers?
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