Hi! So let's say we measured the angular momentum squared of a particle, and got the result ##2 \hbar^2##, so ##l=1##. Now we have the choice of obtaining a sharp value of either ##L_z, L_y## or ##L_x##. Okay, fair enough. But I have two questions:(adsbygoogle = window.adsbygoogle || []).push({});

1) The degeneration degree is ##3## because the eigenvalues squared of ##L_z## must be smaller than or equal to ##l^2##. So ##m \in [-1,0,1]##.

However, if you were to use ##L_x## or ##L_y## to find the degeneration degree, you'd get the same resultbecause what is defined as ##L_x## and ##L_y## is only dependant on the coordinate system, and the physics doesn't care about your coordinate systems. Is this correct reasoning?

2) The complete set of eigenfunctions for ##l=1## can be found by adding together all the functions that are simultaneously eigenfunctions for both ##L_z## and ##L##. So The complete set for ##l=1## are: ##Y_{10}, Y_{1-1},Y_{11}##, where ##Y_{lm}## is an eigenfunction to both ##L_z## and ##L##.

However, why is 2) true? I mean this means you can write eigenfunctions of both ##L_y## and ##L_x## as linear combinations of those ##Y##s. Can somebody post the relevant theorem explaining this? Thanks.

PS: I got my exam tomorrow so pls help

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# Complete set of eigenfunctions

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