# Time Evolution and the Pictures of Quantum Mechanics

### From FSUPhysicsWiki

In this chapter, we discuss different "pictures" in which one may treat the dynamics of a quantum mechanical system. So far, we have been employing what is known as the Schrödinger picture, in which we treat operators as time-independent and place all of the time dependence of the system in the state vector. We will show that one may also consider the state vectors to be constant in time and the operators to be changing; this is known as the Heisenberg picture. We also consider the interaction, or Dirac, picture, in which the time dependence is split between the state vector and the operators. Both of these pictures are very useful. The Heisenberg picture allows one to make an even closer connection between quantum and classical mechanics via the equations of motion satisfied by observables, which resemble the classical equations of motion for the same system. The interaction picture is useful in describing the response of an exactly solvable system to external perturbations.

We also discuss Ehrenfest's theorem, which gives us yet another way to make a connection with classical mechanics, this time through the expectation values of operators, as well as the virial theorem, which gives us a relation between the expectation values of the kinetic and potential energies of a particle.