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Time evolution operator

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

    ## H = \left(
    \begin{array}{cc}
    0 & \Delta \\
    \Delta & 0
    \end{array}
    \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} > ##

    Where

    ## 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

    dextercioby

    User Avatar
    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
    U=exp(−iΔt/ℏ)

    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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