We find the
time development operator
by solving the equation
We have been working in what is called the Schrödinger picture in which the
wavefunctions (or states) develop with time.
There is the alternate
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
We use operator methods to compute the
uncertainty relationship between non-commuting variables
Again we use operator methods to calculate the
time derivative of an expectation value.
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