Back to our series on QM: one thing to remember about observables: they are operators with a set collection of eigenvectors and eigenvalues (allowable values that can be observed; “quantum levels” if you will). These do not change with time. So . One can work this out by expanding if one wants to.
So with this fact, lets see how the expectation of an observable evolves with time (given a certain initial state):
Now apply the Hamiltonian to account for the time change of the state vector; we obtain:
Now use the fact that both and are Hermitian to obtain:
So, we see the operator once again; note that if commute then the expectation of the state vector (or the standard deviation for that matter) does not evolve with time. This is certainly true for itself. Note: an operator that commutes with is sometimes called a “constant of motion” (think: “total energy of a system in classical mechanics).
Note also that
If does NOT correspond with a constant of motion, then it is useful to define an evolution time where This gives an estimate of how much time must elapse before the state changes enough to equal the uncertainty in the observable.
Note: we can apply this to and to obtain
Consequences: if is small (i. e., the state changes rapidly) then the uncertainty is large; hence energy is impossible to be well defined (as a numerical value). If the energy has low uncertainty then must be large; that is, the state is very slowly changing. This is called the time-energy uncertainty relation.