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But what I am not clear about is what happens when you apply the commutator to a wave function (or in terms of an observation of a wave function). Let's take the example of angular momentum operators:

[tex]\left[\hat{L}_z , \hat{L}^{2} \right][/tex]

The fact that they commute, means they share no uncertainty relation between them. But what does this mean in terms of wavefunction and measurement? We can find the simultaneous eigenvalues of a wavefunction (what does this mean)? ... If I was to go and measure the angular momentum squared of (something?) and then the projected angular momentum on a given axis of the same (thing?), I would know these values to 100% certainty within, of course, the confines of experimental error. In terms of a visual experiment, does this mean I can find the angular momentum squared of, say, an electron, and -- " -- " -- with complete certainty? I ve been bombarded with theoretical frameworks of commutators, operators, wavefunctions and yet no appreciation of what is going on!

I hope someone can help.