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Bra-ket multiplication

  1. Oct 17, 2009 #1
    hey guys,

    <n|(a-+a+)^4|n> = 5n2+5n+3

    I am wondering if some one can show me why left and right side are equal

        Thanks in advance
     
  2. jcsd
  3. Oct 17, 2009 #2

    Pengwuino

    User Avatar
    Gold Member

    If it is true, it's simply a matter of doing the work.

    [tex](a^ - + a^ + )^2 = (a^ - a^ - + a^ - a^ + + a^ + a^ - + a^ + a^ + )[/tex] for example, so you do the case for the 4th power, you can work out what the result is knowing
    [tex]\[
    \begin{array}{l}
    a^ + \left. {|n} \right\rangle = \sqrt {n + 1} \left. {|n + 1} \right\rangle \\
    a^ - \left. {|n} \right\rangle = \sqrt n \left. {|n - 1} \right\rangle \\
    \end{array}[/tex]

    and any term that doesn't return the ket to |n> can be ignored since it'll go away.
     
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