phy-4600/lecture_notes/1-27/Expectation Value Time Dependence

Computed Derivative d/dt <Â>:

	d <Â> = d 〈Ψ(t)|Â|Ψ(t)〉
	dt      dt
	
	⎛d 〈Ψ(t)⎞ Â|Ψ(t)〉 + 〈Ψ(t)|Â ⎛d Ψ(t)〉⎞ + 〈Ψ(t)|d Â|Ψ(t)〉
	⎝dt     ⎠                   ⎝dt     ⎠         dt
						↓
	ι 〈Ψ(t)|Ĥ Â|Ψ(t)〉 + 〈Ψ(t)|Â Ĥ -ι|Ψ(t)〉 + 〈Ψ(t)|d Â|Ψ(t)〉
	ħ                              ħ               dt 
						↓
	ι 〈Ψ(t)|Ĥ Â - Â Ĥ|Ψ(t)〉 + 〈Ψ(t)|d Â|Ψ(t)〉
	ħ                               dt 


Computed Derivative (pic) d/dt <Â> (pic)

Discovered/Introduced Commutator
Cpmpatible Observables occur when commutator is equal to 0.

[Ĥ,Â] = ĤÂ - 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