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 group of differential operators on ℝ with polynomial coefficients, since any such operator has a unique representation in the form This algebra is called the Weyl algebra. It follows from abstract nonsense that the Weyl algebra "W" is a quotient of formula_79. However, this is also easy to see directly from the above representations; viz. by the mapping The application that led Hermann Weyl to an explicit realization of the Heisenberg group was the question of why the Schrödinger picture and Heisenberg picture are physically equivalent. Abstractly, the reason is the Stone–von Neumann theorem: there is a unique unitary representation with