1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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)


    [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
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted