I'm a little confused as to why anyone would want to use the Heisenberg picture of time evolution instead of the Schrodinger picture, beyond showing that the equations of motion are similar to those of classical mechanics. For example, consider a free particle. Using the Heisenberg equations of motion, you can show that the position and momentum operators obey equations similar to their classical analogs (i.e., the position is linear in time and the momentum is constant). But what does this really give you? It doesn't seem to me that it has any concrete meaning outside of the fact that it is mathematically equivalent to the Schrodinger picture. I guess what I'm really asking is this: what physical meaning does a time-varying operator have?