Heisenberg picture

In the Heisenberg picture, the quantum states are constant and do not evolve with time. Instead, the operators transform as

$${\hat{A}}_H(t)={\hat{U}}^{†}{\hat{A}}_S\hat{U}.$$

Because we often work in the Heisenberg picture, we often write the transformation as $\hat{A}\to {\hat{U}}^{†}\hat{A}U$, using the notation for the same operator. By differentiating the above equation, an operator ${\hat{A}}_H$ transforms as

$$\frac{d}{dt}{\hat{A}}_H(t)=i[{\hat{H}}_H(t),{\hat{A}}_H(t)]+{\left(\frac{\partial {\hat{A}}_S}{\partial t}\right)}_H,$$

which is known as Heisenberg’s equation.