In physics, the Heisenberg picture or Heisenberg representation[1] is a formulation (largely due to Werner Heisenberg in 1925) of quantum mechanics in which observables incorporate a dependency on time, but the states are time-independent.
It stands in contrast to the Schrödinger picture in which observables are constant and the states evolve in time.
In the Heisenberg picture of quantum mechanics the state vectors |ψ⟩ do not change with time, while observables A satisfy
where "H" and "S" label observables in Heisenberg and Schrödinger picture respectively, H is the Hamiltonian and [·,·] denotes the commutator of two operators (in this case H and A).
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 some sense, the Heisenberg picture is more natural and convenient than the equivalent Schrödinger picture, especially for relativistic theories.
Lorentz invariance is manifest in the Heisenberg picture, since the state vectors do not single out the time or space.
This approach also has a more direct similarity to classical physics: by simply replacing the commutator above by the Poisson bracket, the Heisenberg equation reduces to an equation in Hamiltonian mechanics.
According to Schrödinger's equation, the quantum state at time
is the time-evolution operator induced by a Hamiltonian
refers to time-ordering, ħ is the reduced Planck constant, and i is the imaginary unit.
in the Schrödinger picture, which is a Hermitian linear operator that could also be time-dependent, in the state
In the Heisenberg picture, the quantum state is assumed to remain constant at its initial value
, so the same expectation value can be obtained by working in either picture.
The Schrödinger equation for the time-evolution operator is
where differentiation was carried out according to the product rule.
An important special case of the equation above is obtained if the Hamiltonian
The equation is solved by use of the standard operator identity,
A similar relation also holds for classical mechanics, the classical limit of the above, given by the correspondence between Poisson brackets and commutators:
In classical mechanics, for an A with no explicit time dependence,
In effect, the initial state of the quantum system has receded from view, and is only considered at the final step of taking specific expectation values or matrix elements of observables that evolved in time according to the Heisenberg equation of motion.
A similar analysis applies if the initial state is mixed.
Commutator relations may look different than in the Schrödinger picture, because of the time dependence of operators.
The time evolution of those operators depends on the Hamiltonian of the system.
the evolution of the position and momentum operators is given by:
are the position and momentum operators in the Heisenberg picture.
Differentiating both equations once more and solving for them with proper initial conditions,
Direct computation yields the more general commutator relations,
, one simply recovers the standard canonical commutation relations valid in all pictures.