Dynamical pictures in the context of "Heisenberg picture"

In physics, the Schrödinger picture or Schrödinger representation is a formulation of quantum mechanics in which the state vectors evolve in time, but the operators (observables and others) are mostly constant with respect to time (an exception is the Hamiltonian which may change if the potential ${\displaystyle V}$ changes). This differs from the Heisenberg picture which keeps the states constant while the observables evolve in time, and from the interaction picture in which both the states and the observables evolve in time. The Schrödinger and Heisenberg pictures are related as active and passive transformations and commutation relations between operators are preserved in the passage between the two pictures.

In the Schrödinger picture, the state of a system evolves with time. The evolution for a closed quantum system is brought about by a unitary operator, the time evolution operator. For time evolution from a state vector ${\displaystyle |\psi (t_{0})\rangle }$ at time t₀ to a state vector ${\displaystyle |\psi (t)\rangle }$ at time t, the time-evolution operator is commonly written ${\displaystyle U(t,t_{0})}$, and one has