- #1

Ang Han Wei

- 9

- 0

[itex]\dot{A}[/itex] = [itex](A, H)_{}[/itex], which can represent the Poisson bracket (classical version)

or

[itex]\dot{A}[/itex] = -i/h[A,H] (Quantum Mechanical commutator).

I was given the general solutin for A(t) = [itex]e^{tL}[/itex]A. In addition, L is an operator acting on functions of initial conditions (q,p) in the classical case or quantum mechanical operator in the Hilbert Space such that Lf = [itex](f, H)_{}[/itex]

I am asked to show that A(t) = [itex]e^{tL}[/itex]A is the formal solution to [itex]\dot{A}[/itex] = [itex](A, H)_{}[/itex]

I started with the Heisenberg picture in which the obervables are time dependent and differentiated A(t) wrt time t. I got the following:

[itex]\frac{dA(t)}{dt}[/itex] = L[itex]e^{tL}[/itex][itex]\frac{∂L}{∂T}[/itex]A + [itex]e^{tL}[/itex][itex]\frac{∂A}{∂t}[/itex]

This is where I am stuck as I am unable to find ways to introduce the Hamiltonian H into the equation and show that this is a formal solution to [itex]\dot{A}[/itex] = [itex](A, H)_{}[/itex]

Any help on how i should continue from here will be appreciated!