Computed Derivative d/dt : d = d 〈Ψ(t)|Â|Ψ(t)〉 dt dt ⎛d 〈Ψ(t)⎞ Â|Ψ(t)〉 + 〈Ψ(t)| ⎛d Ψ(t)〉⎞ + 〈Ψ(t)|d Â|Ψ(t)〉 ⎝dt ⎠ ⎝dt ⎠ dt ↓ ι 〈Ψ(t)|Ĥ Â|Ψ(t)〉 + 〈Ψ(t)| Ĥ -ι|Ψ(t)〉 + 〈Ψ(t)|d Â|Ψ(t)〉 ħ ħ dt ↓ ι 〈Ψ(t)|Ĥ  -  Ĥ|Ψ(t)〉 + 〈Ψ(t)|d Â|Ψ(t)〉 ħ dt Computed Derivative (pic) d/dt (pic) Discovered/Introduced Commutator Cpmpatible Observables occur when commutator is equal to 0. [Ĥ,Â] = Ĥ - ÂĤ Constant of Motion ------------------ if d/dt  = 0 and [Ĥ,Â] = 0, then  is a constant of motion Thm: Two operators such that [Â,B̂] always have common Eigenstates