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Using the time evolution operator

  1. Jun 1, 2010 #1
    I hope someone can help me out here,

    I am confused with a line of text I read - it is an example of a 2D Hilbert space with orthonormal basis e1, e2. The Hamiltonian of the system is the Pauli matrix in the y-direction. Given by the matrix:

    [tex]\sigma_{y} = (\frac{0, -i}{i, 0})[/tex]

    The eigenvectors of the Hamiltonian are given by:

    [tex]| \pm >_{y}= \frac{1}{\sqrt{2}}(| e_{1} > \pm i|e_{2}>)[/tex]

    So, applying the time evolution operator to the eigenvectors gives:

    [tex]U| \pm >_{y}=exp(\frac{-i(t-t_{0}) \sigma_{y}}{\hbar})| \pm >_{y}[/tex]
    [tex]U| \pm >_{y}=exp(\frac{\mp i(t-t_{0})}{\hbar})| \pm >_{y}[/tex]

    I don't understand how the last line came about?
  2. jcsd
  3. Jun 1, 2010 #2


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    Science Advisor

    The last line follows because the eigenvalues of [tex]\sigma_y[/tex] are +/- 1.
  4. Jun 1, 2010 #3
    Oh right, of course, so it let's you simplify the expression.
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