In all of the above, notice that we have formulated the postulates of quantum
mechanics such that the state vector evolves in time but the
operators corresponding to observables are taken to be stationary. This
formulation of quantum mechanics is known as the *Schrödinger picture*.
However, there is another, completely equivalent, picture in which the
state vector remains stationary and the operators evolve in time. This
picture is known as the *Heisenberg picture*. This particular picture will
prove particularly useful to us when we consider quantum time correlation functions.

The Heisenberg picture specifies an evolution equation for any operator
*A*, known as the Heisenberg equation. It states that the time
evolution of *A* is given by

While this evolution equation must be regarded as a postulate, it has a very
immediate connection to classical mechanics. Recall that any function of
the phase space variables *A*(*x*,*p*) evolves according to

where is the Poisson bracket. The suggestion is that in the classical limit ( small), the commutator goes over to the Poisson bracket. The Heisenberg equation can be solved in principle giving

where *A* is the corresponding operator in the Schrödinger picture. Thus,
the expectation value of *A* at any time *t* is computed from

where is the stationary state vector.

Let's look at the Heisenberg equations for the operators *X* and *P*. If *H*
is given by

then Heisenberg's equations for *X* and *P* are

Thus, Heisenberg's equations for the operators *X* and *P* are just Hamilton's equations
cast in operator form. Despite their innocent appearance, the solution of such
equations, even for a one-particle system,
is *highly* nontrivial and has been the subject of a considerable
amount of research in physics and mathematics.

Note that any operator that satisfies [*A*(*t*),*H*]=0 will not evolve in
time. Such operators are known as constants of the motion. The Heisenberg
picture shows explicitly that such operators do not evolve in time. However,
there is an analog with the Schrödinger picture: Operators that
commute with the Hamiltonian will have associated probabilities for obtaining
different eigenvalues that do not evolve in time. For example, consider the
Hamiltonian, itself, which it trivially a constant of the motion.
According to the evolution equation of the state vector in the
Schrödinger picture,

the amplitude for obtaining an energy eigenvalue at time *t* upon measuring
*H* will be

Thus, the squared modulus of both sides yields the probability for obtaining , which is

Thus, the probabilities do not evolve in time. Since any operator that
commutes with *H* can be diagonalized simultaneously with *H* and will
have the same set of eigenvectors, the above arguments will hold for
any such operator.

Wed Mar 10 13:14:21 EST 1999