Given a wave function , at a time , the instantaneous change can be found from the Time-Evolution operator , also known as the Hamiltonian;

(1) |

In principle, this equation can be integrated to give the value of for later times.

The non-relativistic Hamiltonian consists of a kinetic part, which is simply the differential operator , and a potential part which is a function of x. The simplest case is the ''free particle``, in which the potential is constant, independent of x.

Here is a video of the time evolution of a momentum eigenstate ;

The time-evolution of a momentum eigenstate consists of nothing more than a steady rotation of the phase. As you can see from the absolute magnitude displayed in the lower half of the frame, this phase rotation does not alter the ``position`` of the particle. It is equally likely to be anywhere in the coordinate space, and that fact is independent of time.

Jerome Berryhill 2013-05-09