Heisenberg picture once more and solving for them with proper initial conditions, leads to Direct computation yields the more general commutator relations, For formula_26, one simply recovers the standard canonical commutation relations valid in all pictures. For a time-independent Hamiltonian "H", Heisenberg picture In physics, the Heisenberg picture (also called the Heisenberg representation) is a formulation (largely due to Werner Heisenberg in 1925) of quantum mechanics in which the operators (observables and others) incorporate a dependency on time, but the state vectors are time-independent, an arbitrary fixed basis rigidly underlying the theory. It stands in contrast to the Schrödinger picture in which
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