VT wrote:Few comments on blackcat007's post:

First, you cannot call a 'basis' to the common eigenfunctions of Lx and Ly. They do not form a 'basis' of the Hilbert space, cuz they don't commute.

Second, the concept of angular momentum is important in problems ( classical and quantum mechanics) with spherically symmetric potential (central force problem). This is because the angular momentum is the generator of rotation but the Hamiltonian is independent of rotation for central potential. Therefore, it follows from Noether's theorem that such a generator is a conserved quantity.

Now in QM, we can always find a ''complete set of operators'' ( which is NOT unique, btw) for a given Hilbert space. It turns out that H,L^2 and Lz form a complete set of operators of the Hilbert space of central force system. H alone is not complete due to the degeneracy in n. Similarly, L^2 and Lz are not complete either due to their n and l dependence respectively. This is the reason why we want common eigenfunctions of L^2 and Lz ( which will be the eigenfunctions for H too), and they will form a ''complete basis'' of the Hilbert space and you can do all sorts of things after this. ( I hope you are not planning to ask why we want complete set of operators. If in case you are: complete set of operators help us to find non-degerate, complete, commom basis and we can label each basis vector with eigenvalues just like in H-atom, Inlm> )

I do not have Griffiths' book with me and I have never read his book before( neither do I plan to read in the future), so I do not exactly understand your first question.

no no..i know that.. they won't form a basis.

i mentioned " and all other combinations" yes i admit it was a bit sloppy.. i meant all combinations including L^2 and since L^2 and

L commute thus we can find a basis in which both of them can be diagonalized..

griffith introduces the concept of angular momentum.. and then goes on to find operators which commute, so that he can find a set of common eigenfunctions to construct a common basis.

my question is what is the significance of finding a common set of eigenfunction to form a basis.. why can't we proceed in a different way.. what induced (whoever did it for the first time) to proceed by searching a common set of eigenfunctions.

another question. since the common eigenfunctions of Lx and Ly won't be complete.. then can't they represent any physically realizable state? is it necessary for the eigenstate to span the L2 space?