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The time evolution operator (QM) Algebraic properties

  1. May 19, 2012 #1
    1. The problem statement, all variables and given/known data

    The hamiltonian for a given interaction is

    [itex] H=-\frac{\hbar \omega}{2} \hat{\sigma_y}[/itex]

    where

    [itex]\sigma_y = \left( \begin{array}{cc} 0 & i \\ -i & 0 \end{array} \right)[/itex]

    the pauli Y matrix

    2. Relevant equations


    3. The attempt at a solution

    So from the time dependant schrodinger equation we, can take the time dependance and put it into the time evolution operator U(t)

    [itex]HU(t)\left|\Psi(r,0)\right>=i\hbar \frac{d}{dt}U(t)\left|\Psi(r,0)\right>[/itex]

    becomes

    [itex]i\hbar\frac{d}{dt}U(t) = HU(t)[/itex]

    so for a non time dependant Hamiltonian H, this means:

    [itex]U(t) = e^{-\frac{i}{\hbar}H t}[/itex]

    so we have then:

    [itex]U(t) = e^{\frac{i\omega t}{2}\hat{\sigma_y}}[/itex]

    How do you treat this? Is there any particular identity that allows you to move the operator out of the exponent?
     
    Last edited: May 19, 2012
  2. jcsd
  3. May 19, 2012 #2
    edit: changed the matrix to the correct form
     
  4. May 19, 2012 #3

    dextercioby

    User Avatar
    Science Advisor
    Homework Helper

    Do you know how the exponential of a finite matrix is defined? If so, use the definition.
     
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