Proof of expectation value for a dynamic observable

[digogalvao]

Homework Statement

Show that:
d<A(q,p)>/dt=<{A,H}>, where {A,H} is a Poisson Bracket

Homework Equations

Liouville theorem

The Attempt at a Solution

<A>=Tr(Aρ)⇒d<A>/dt=Tr(Adρ/dt)=Tr(A{H,ρ})
So, in order to get the correct result, Tr(A{H,ρ}) must be equal to Tr({A,H}ρ), but I don't think I can do that substitution. Is it valid? How can I prove that?

 

[vela]

Use the cyclic property of the trace: Tr(ABC) = Tr(BCA) = Tr(CAB).

 

[digogalvao]

Use the cyclic property of the trace: Tr(ABC) = Tr(BCA) = Tr(CAB).

But there is a Poisson Bracket in it...

 

[vela]

Good point.

 

[digogalvao]

So? lol

 

[vela]

Try expanding out the Poisson bracket and see if there's something you can do with the derivatives.

 

[digogalvao]

No luck :(

 

[digogalvao]

Bump...