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Hey guys,
I'm studying some quantum physics at the moment and I'm having some problems with understanding the principles behind the necessary lineair algebra:
1) If two operators do NOT commutate, is it correct to conclude they don't have a similar basis of eigenvectoren? Or are there more conditions to be verified to conclude this?
2) If two operators DO commutate, you MAY find a similar basis of eigenvectoren? If this is correct, what are the conditions for it to be absolutely true?
The context to which I am asking these question are the ladder operators for angular momentum states and especially the fact that Lx and Ly do not commutate and don't have a similar basis of eigenvectors, whereas L^2 and Lx/Ly/Lz do commutate and do have a basis of eigenvectors.
Thanks in advance!
I'm studying some quantum physics at the moment and I'm having some problems with understanding the principles behind the necessary lineair algebra:
1) If two operators do NOT commutate, is it correct to conclude they don't have a similar basis of eigenvectoren? Or are there more conditions to be verified to conclude this?
2) If two operators DO commutate, you MAY find a similar basis of eigenvectoren? If this is correct, what are the conditions for it to be absolutely true?
The context to which I am asking these question are the ladder operators for angular momentum states and especially the fact that Lx and Ly do not commutate and don't have a similar basis of eigenvectors, whereas L^2 and Lx/Ly/Lz do commutate and do have a basis of eigenvectors.
Thanks in advance!