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.

## Leave a Reply