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 connection with the theta functions. If , are integers (in the ring Z) then one has the discrete Heisenberg group H(Z). It is a non-abelian nilpotent group. It has two generators, and relations where is the generator of the center of H. (Note that the inverses of "x", "y", and "z" replace the 1 above the diagonal with −1.) By Bass's theorem, it has a polynomial growth rate of order 4. One can generate any element through If one takes "a, b, c" in Z/"p" Z for an odd prime "p", then one has the Heisenberg group modulo p. It