Overview
The Heisenberg Picture is a formulation of quantum mechanics that is equivalent to the Schrodinger Picture, but using a differnt mathematical formalism.
In particular, in the Schrodinger picture, the quantum state is represented by an element in a Hilbert Space, and the Hamiltonian operator specifies how that state changes over time. The Hamiltonian operator is a fixed operator, that is, it is not time dependent.
In the Heisenberg picture, the state is a fixed object in the Hilbert space, and the operators (including the Hamiltonian) evolve over time. This difference is just a matter of formalism, and the two pictures have been shown to be equivalent.
Formalism
In the Schrodinger picture, the quantum state is a function of time {% |\psi,t\rangle %} that returns an element of the Hilbert space. This can be written as
{% |\psi,t\rangle = U(t) | \psi,0 \rangle %}
where {% | \psi,0 \rangle %} is the quantum state at time zero. Here, there is a function (operator) ({% U(t) %})
that takes the current time and the initial state, and returns the state at time {% t %}. This operator has effectively
moved the dynamics of time evolution from the state itself, into the operator.
Observables
{% Q_t = U^{\dagger}(t) Q U(t) %}
{% \langle \psi, t|Q|\psi, t \rangle = \langle \psi, 0 | U^{\dagger}(t) Q U(t) | \psi, 0 \rangle =
\langle \psi, 0 | Q_t | \psi, 0 \rangle
%}