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Using the Baker-Campbell-Hausdorff Identity

  1. Feb 29, 2012 #1
    1. The problem statement, all variables and given/known data
    Given that U = e[itex]^{-i*θ*\hat{n}*\vec{J}/hbar}[/itex]

    Show that U[itex]^{†}[/itex][itex]\vec{J}[/itex]U = [itex]\hat{n}(\hat{n}*\vec{J}) - \hat{n}\times(\hat{n}\times\vec{J})cos(θ) + \hat{n}\times\vec{J}sin(θ)[/itex]

    2. Relevant equations
    The Baker-Campbell-Hausdorff Identity

    3. The attempt at a solution
    U[itex]^{†}[/itex][itex]\vec{J}[/itex]U = e[itex]^{-i*θ*\hat{n}*\vec{J}/hbar}[/itex][itex]\vec{J}[/itex]e[itex]^{i*θ*\hat{n}*\vec{J}/h_bar}[/itex]

    U[itex]^{†}[/itex][itex]\vec{J}[/itex]U = [itex]\vec{J} + [i*θ*\hat{n}*\vec{J}/hbar,\vec{J}]+[i*θ*\hat{n}*\vec{J}/hbar,[i*θ*\hat{n}*\vec{J}/hbar,\vec{J}]]/2! + ...[/itex]

    I guess I just need some help on whether or not I am headed in the right direction.
    Last edited: Feb 29, 2012
  2. jcsd
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