# Hamilton Equation and Heisenberg Equation of Motion

1. Feb 8, 2014

### Ang Han Wei

I was given the (general) following form for the Hamilton and Heisenberg Equations of motion

$\dot{A}$ = $(A, H)_{}$, which can represent the Poisson bracket (classical version)

or

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

I was given the general solutin for A(t) = $e^{tL}$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 = $(f, H)_{}$

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

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

$\frac{dA(t)}{dt}$ = L$e^{tL}$$\frac{∂L}{∂T}$A + $e^{tL}$$\frac{∂A}{∂t}$

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 $\dot{A}$ = $(A, H)_{}$

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