Heisenberg picture

From Physics wiki

In the Heisenberg picture, the state $\left|\Psi \right\rangle\,$ of a system is considered fixed in time, while the time dependence of the system lies in the operators corresponding to observables of the system. The Heisenberg picture is perhaps the most natural counterpart to classical mechanics, in which the time dependence of some function $f(p_i,q_i,t)\,$ is given by

$\frac{df}{dt} = \{f, H\}+ \frac{\partial f}{\partial t}\,$ ,

where

$\{f,g\} = \sum_{i=1}^{N} \left( \frac{\partial f}{\partial q_{i}} \frac{\partial g}{\partial p_{i}} - \frac{\partial f}{\partial p_{i}} \frac{\partial g}{\partial q_{i}} \right)$ ,

is the Poisson bracket of $f\,$ and $g\,$ , and where $H\,$ is the Hamiltonian of the system. In going from classical mechanics to quantum mechanics, we have seen that we should make the replacement $\left\{ f, g\right\} \to \tfrac{1}{i\hbar} [F, G]\,$ . Therefore, it is reasonable to guess that for some operator $\mathcal{O}(p_i, q_i, t)\,$ ,

$\frac{d\mathcal{O}}{dt} = \frac{1}{i\hbar}[\mathcal{O}, H]+ \frac{\partial \mathcal{O}}{\partial t}\,$ .

This statement is known as Heisenberg's equation of motion, and its validation lies both in the success of quantum mechanics as well as the equivalence between the Heisenberg and Schrödinger pictures, which we shall now prove.