We find the
time development operator
by solving the equation
.

This implies that is the time development operator. In some cases we can calculate the actual operator from the power series for the exponential.

We have been working in what is called the Schrödinger picture in which the
wavefunctions (or states) develop with time.
There is the alternate
Heisenberg picture
in which the operators develop with time
while the states do not change.
For example, if we wish to compute the expectation value of the operator
as a function
of time in the usual Schrödinger picture, we get

In the Heisenberg picture the operator .

We use operator methods to compute the
uncertainty relationship between non-commuting variables

which gives the result we deduced from wave packets for and .

Again we use operator methods to calculate the
time derivative of an expectation value.

(Most operators we use don't have explicit time dependence so the second term is usually zero.) This again shows the importance of the Hamiltonian operator for time development. We can use this to show that in Quantum mechanics the expectation values for and behave as we would expect from Newtonian mechanics

Any operator
that commutes with the Hamiltonian has a
**time independent** expectation value.
The energy eigenfunctions can also be (simultaneous) eigenfunctions of the commuting operator
.
It is usually a symmetry of the
that leads to a commuting operator and hence an additional
constant of the motion.

Jim Branson 2013-04-22