This builds on our previous example. We start with a state and we will make three successive observations of observables which have operators and in the following order: . The assumption is that these observations are made so quickly that no time evolution of the state vector can take place; all of the change to the state vector will be due to the effect of the observations.
A simplifying assumption will be that the observation operators have the following property: no two different eigenvectors have the same eigenvalues (e. g., the eigenvalue uniquely determines the eigenvector up to multiplication by a constant of unit modulus).
First of all, this is what “compatible observables” means: two observables are compatible if, upon three successive measurements the first measurement of is guaranteed to be the second measurement of . That is, the state vector after the first measurement of is the same state vector after the second measurement of .
So here is what the compatibility theorem says (I am freely abusing notation by calling the observable by the name of its associated operator):
The following are equivalent:
1. are compatible observables.
2. have a common eigenbasis.
3. commute (as operators)
Note: for this discussion, we’ll assume an eigenbasis of for and for .
1 implies 2: Suppose the state of the system is just prior to the first measurement. Then the first measurement is . The second measurement yields which means the system is in state , in which case the third measurement is guaranteed to be (it is never anything else by the compatible observable assumption). Hence the state vector must have been which is the same as . So, by some reindexing we can assume that . An argument about completeness and orthogonality finishes the proof of this implication.
2 implies 1: after the first measurement, the state of the system is which, being a basis vector for observable means that the system after the measurement of stays in the same state, which implies that the state of the system will remain after the second measurement of . Since this is true for all basis vectors, we can extend this to all state vectors, hence the observables are compatible.
2 implies 3: a common eigenbasis implies that the operators commute on basis elements so the result follows (by some routine linear-algebra type calculations)
3 implies 2: given any eigenvector we have which implies that is an eigenvector for with eigenvalue . This means that where has unit modulus; hence must be an eigenvector of . In this way, we establish a correspondence between the eigenbasis of with the eigenbasis of .
Ok, what happens when the observables are NOT compatible?
Here is a lovely application of conditional probability. It works this way: suppose on the first measurement, is observed. This puts us in state vector . Now we measure the observable which means that there is a probability of observing eigenvalue . Now is the new state vector and when observable is measured, we have a probability of observing eigenvalue in the second measurement of observable .
Therefore given the initial measurement we can construct a conditional probability density function
Again, this makes sense only if the observations were taken so close together so as to not allow the state vector to undergo time evolution; ONLY the measurements changes the state vector.
Next: we move to the famous Heisenberg Uncertainty Principle, which states that, if we view the interaction of the observables and with a set state vector and abuse notation a bit and regard the associated density functions (for the eigenvalues) by the same letters, then
Of course, if the observables are compatible, then the right side becomes zero and if for some non-zero scalar (that is, for all possible state vectors ), then we get which is how it is often stated.