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Hamilton Equation and Heisenberg Equation of Motion

  1. Feb 8, 2014 #1
    I was given the (general) following form for the Hamilton and Heisenberg Equations of motion

    [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!
     
  2. jcsd
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