Degeneracy

Dear Members,

in P.29 of Sakurai's Modern Quantum Mechanics, he counts some
difficulties about degeneracy: " In such a case the notation |a> that
labels the eigenket by its eigenvalue alone does not give a complete
description; furthermore, we may recall that our earlier theorem on the
orthogonality of different eigenkets was proved under the assumption of
no degeneracy. Even worse, the whole concept that the ket space is
spanned by {|a>} appears to run into difficulty when the dimensionality
of the ket space is larger than the number of distinct eigenvalues of
A. Fortunately, in practical applications in quantum mechanics, it is
usually the case that in such a situation the eigenvalues of some other
commuting observable, say B, can be used to label the degenerate
eigenkets."
1-why, in principle, these are difficulties?
2-I don't understand the last difficulty, i.e. " even worse, the
whole concept that the ket space ..."
3-I don't understand his " fortunately, in practical applications
... .
4-All these problems and difficulties are resolved by using the
eigenvalues of some other commuting observables, say B? How?
cheers,

Ali

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