Heisenberg picture

The Heisenberg picture assigns a closed system's unitary time evolution to its observables while keeping the state fixed at a reference time. If $U(t,t_0)$ is the evolution operator, a Schrödinger-picture linear operator $A_S(t)$ becomesThe transformation preserves expectation values:It is a change of description, not a different physical evolution.

For a possibly time-dependent Hamiltonian operator $H_S(t)$, let $H_H(t)=U^{†}{H}_S(t)U$. Differentiating the transformation givesIf $H_S=H$ is time independent and $A_S$ has no explicit time dependence, this reduces to $d{A}_H/dt=i[H,A_H]/\hslash$. Using an untransformed time-dependent Hamiltonian without checking the picture would generally be incorrect.

For a spin one-half with $H=\hslash \omega {\sigma }_z/2$, a fixed Schrödinger-picture ${\sigma }_x$ becomesThe Pauli matrix commutator identity verify its initial derivative $-\omega {\sigma }_y$, illustrating uniform-field spin precession with a fixed Heisenberg-picture state.

Basically the same as matrix mechanics it seems, just a bit more generalized.