Heisenberg picture formulation of matrix mechanics in an arbitrary basis, in which the Hamiltonian is not necessarily diagonal. It further serves to define a third, hybrid, picture, the interaction picture. In the Heisenberg picture of quantum mechanics the state vectors, |"ψ("t")"〉, do not change with time, while observables satisfy where is the Hamiltonian and <nowiki>[•,•]</nowiki> denotes the commutator of two operators (in this case and ). Taking expectation values automatically yields the Ehrenfest theorem, featured in the correspondence principle. By the Stone–von Neumann theorem, the Heisenberg picture and the Schrödinger picture are unitarily equivalent, just a basis change in Hilbert space. In
Heisenberg picture line above is the Heisenberg Hamiltonian "H"("t"), which may differ from the Schrödinger Hamiltonian. An important special case of the equation above is obtained if the Hamiltonian does not vary with time. Then the time-evolution operator can be written as Therefore, and, Here ∂"A"/∂"t" is the time derivative of the initial "A", not the "A"("t") operator defined. The last equation holds since commutes with . If the Hamiltonian does not vary with time, then the time-evolution operator can be written as where is the Hamiltonian and is the reduced Planck constant. Therefore, Thus and hence emerges the above Heisenberg equation