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 "n"-dimensional systems, and most generally, to any symplectic vector space. In the three-dimensional case, the product of two Heisenberg matrices is given by: Since the multiplication is not commutative, the group is non-abelian. The neutral element of the Heisenberg group is the identity matrix, and inverses are given by The group is a subgroup of the 2-dimensional affine group Aff(2): formula_2 acting on formula_6 corresponds to the affine transform formula_7. There are several prominent examples of the three-dimensional case. If , are real numbers (in the ring R) then one has the continuous Heisenberg group H(R). It is a nilpotent