We have described time evolution in the Schrödinger picture, in which a state |ψ(t) evolves according to the unitary operator U(t) but observables A are typically time-independent. An equally...

We have described time evolution in the Schrödinger picture, in which a state |ψ(t) evolves according to the unitary operator U(t) but observables A are typically time-independent. An equally meaningful view is the Heisenberg picture, which redefines states and operators by

(Assume uniform dynamics. What is Hˆ ?)

(a) Show that   ψˆ " is independent of time.

(b) The change from one picture to the other preserves the mathematical relations among operators. Sums and scalar multiples are clearly unaffected. You prove that AB = C if and only if AˆBˆ = Cˆ.

(c) Show that observable properties – i.e. the expectations of all observables at all times – are the same in the two pictures.

(d) Since all of the dynamics now resides in the observables, derive a “Schrödinger equation” governing Aˆ(t).

(e) What does it mean for Aˆ(t1) and Aˆ(t2) not to commute with each other? Explain this in physical (rather than mathematical) terms. (f) Show that