20 Mar '10 05:43

For two commuting operators, one can find a common eigenbasis. For simplicity, assume the eigenvalues here are nondegenerate. In this case, after applying the first operator, the state collapses into one of the basis eigenvectors. Since this is also an eigenvector of the second operator, we essentially know which eigenvalue will be returned when the second operator is applied to this collapsed state. This much makes sense to me.

But what about for non-commuting operators, where no common eigenbasis can be found? Lets say the original, uncollapsed state vector is a linear combination of the eigenvectors for the first operator. After applying the first operator to the state, it collapses into one of its eigenvectors. What exactly happens after applying the second, non-commuting operator into this collapsed state? Do we now find some basis comprised of eigenvectors for the second operator, which superpose into the collapsed state, and then make the measurement as usual? Or is there some deeper, more intricate process going on?

But what about for non-commuting operators, where no common eigenbasis can be found? Lets say the original, uncollapsed state vector is a linear combination of the eigenvectors for the first operator. After applying the first operator to the state, it collapses into one of its eigenvectors. What exactly happens after applying the second, non-commuting operator into this collapsed state? Do we now find some basis comprised of eigenvectors for the second operator, which superpose into the collapsed state, and then make the measurement as usual? Or is there some deeper, more intricate process going on?