So, in QM making a measurement collapses the state into an eigenstate of that observable. Thus, if the system is properly isolated, then the same measurement should return the same value. But the eigenvalue for that state is degenerate, then does that mean the state might actually collapse to a different state in the same eigenspace after the second measurement? Here is what I mean, if that's not clear: Suppose I make a measurement associated with the operator "O" and get a result "o". My system is now in the eigenstate |ψ0>, where O|ψ0> = o|ψ0> So far so good. But let's suppose the eigenvalue "o" is actually degenerate with multiplicity two. Thus, there exist linearly independent |ψ1> and |ψ2>, both having eigenvalue o under O, such that: |ψ0> = c1|ψ1> + c2|ψ2> Therefore, by the postulates of QM, a second measurement of O will collapse the state of the system to |ψ1> with probability |c1|2 and to |ψ2> with probability |c2|2. Aside from the fact that it seems strange that a repeated measurement could change the system's state (even if it yields the same value for the observable), it's even more confusing because of the arbitrariness of the basis for the eigenspace. I could just as well choose |ψ'1> and |ψ'2> such that: |ψ0> = c'1|ψ'1> + c'2|ψ'2> and now there are four different states my system could collapse to and the probabilities sum to 200% ...and you could, of course, repeat this process infinitely many times, choosing a new eigenbasis each time. So... clearly I'm missing something.