Interesting problem: Suppose one has 2 hermitian operators, O1,O2 with distinct eigenvalues. Say the first is the Hamiltonian. We measure the energy, get a value, say E_1. So the system is in |1>. Suppose now that the second operator has the property to turn state 1 into state |2> and operating on |2> gives |1> Hence if after having measured the energy to be E_1, we operate on the state with the second op, we get |2>. No uncertainty about the outcome. So should not the uncertainity be 0? So O2|1>=|2>,<1|O2|1>=0 O2**2|1>=O2|2>=|1>. So <1|O2**2|1>=<1|1>=1 But uncertainity is (<1|O2**2|1>-<1|O2|1>**2)=1, not 0.