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Homework Help: Time evolution operator

  1. May 8, 2012 #1
    If you have some Hamiltonian represented by a 2x2 matrix

    ## H = \left(
    0 & \Delta \\
    \Delta & 0
    \right) ##

    And you want to use the time evolution operator

    ## U = \exp ( - \frac{i}{\hbar} H t ) ##

    it says that

    ## U = \exp (- \frac{i \Delta}{\hbar} t) ##

    Why is this?
    How did the ##\Delta## get out?
    Last edited: May 8, 2012
  2. jcsd
  3. May 8, 2012 #2
    The was an error in the latex.

    Is it just true that

    ## e^A | \psi_\alpha > e^{\alpha} | \psi_{\alpha} > ##


    ## A | \psi_\alpha>= \alpha |\psi_{\alpha}> ##?
  4. May 8, 2012 #3
    The exponential of a matrix is really just short hand for the power series representing the exponential function, [tex]e^A = \sum_{n=0}^\infty \frac{A^n}{n!}.[/tex]
    So if you apply this power series operator to the wave function, you should see that in fact yes, what you've said is true.
  5. May 8, 2012 #4


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    Science Advisor
    Homework Helper

    Just a few LaTex hints: use \left( and \right) before brackets containing fractions and \langle and \rangle for bra-ket notation. (No bold in the LaTex code, of course)

    As for

    in your initial post, it's incomplete, it misses the unit matrix 2x2 in the RHS.
  6. May 8, 2012 #5
    Thanks that's so obvious now
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