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Commutator of two operators

  1. Mar 12, 2013 #1
    1. The problem statement, all variables and given/known data
    Hello.

    I am supposed to find the commutator between to operators, but I can't seem to make it add up.
    The operators are given by:
    [tex]\hat{A}=\alpha \left( {{{\hat{a}}}_{+}}+{{{\hat{a}}}_{-}} \right)[/tex]
    and
    [tex]\hat{B}=i\beta \left( \hat{a}_{+}^{2}-\hat{a}_{-}^{2} \right),[/tex]
    where alpha and beta are real numbers, i being the irrational number, and a+ and a- are the ladder operators.

    Now, I just have to find the commutator [A, B]


    2. Relevant equations



    3. The attempt at a solution

    By attempt is given by the following

    [tex]\left[ \hat{A},\,\hat{B} \right]=\hat{A}\hat{B}-\hat{B}\hat{A}=\alpha \left( {{{\hat{a}}}_{+}}+{{{\hat{a}}}_{-}} \right)i\beta \left( \hat{a}_{+}^{2}-\hat{a}_{-}^{2} \right)-i\beta \left( \hat{a}_{+}^{2}-\hat{a}_{-}^{2} \right)\alpha \left( {{{\hat{a}}}_{+}}+{{{\hat{a}}}_{-}} \right)[/tex]
    [tex]=i\alpha \beta \left[ \begin{align}
    & -{{{\hat{a}}}_{+}}{{{\hat{a}}}_{+}}{{{\hat{a}}}_{-}}+{{{\hat{a}}}_{+}}{{{\hat{a}}}_{-}}{{{\hat{a}}}_{+}}-{{{\hat{a}}}_{-}}{{{\hat{a}}}_{+}}{{{\hat{a}}}_{-}}+{{{\hat{a}}}_{-}}{{{\hat{a}}}_{-}}{{{\hat{a}}}_{+}} \\
    & -{{{\hat{a}}}_{+}}{{{\hat{a}}}_{+}}{{{\hat{a}}}_{-}}+{{{\hat{a}}}_{+}}{{{\hat{a}}}_{-}}{{{\hat{a}}}_{+}}-{{{\hat{a}}}_{-}}{{{\hat{a}}}_{+}}{{{\hat{a}}}_{-}}+{{{\hat{a}}}_{-}}{{{\hat{a}}}_{-}}{{{\hat{a}}}_{+}} \\
    \end{align} \right]
    [/tex]
    [tex]=2i\alpha \beta \left[ \left( -\hat{a}_{+}^{2}{{{\hat{a}}}_{-}}+{{{\hat{a}}}_{+}}{{{\hat{a}}}_{-}}{{{\hat{a}}}_{+}}-{{{\hat{a}}}_{-}}{{{\hat{a}}}_{+}}{{{\hat{a}}}_{-}}+\hat{a}{{_{-}^{2}}_{-}}{{{\hat{a}}}_{+}} \right) \right][/tex]
    Now, according to the answer I have gotten from my teacher, it is supposed to be:
    [tex]\left[ \hat{A},\hat{B} \right]=2i\alpha \beta \hat{A}[/tex]

    But I am kinda lost in how to end up with the operator A in the end, and even another alpha constant, since A operator is equal to alpha and some ladder operators.

    So, what am I missing ? :)


    Thanks in advance.
     
    Last edited: Mar 12, 2013
  2. jcsd
  3. Mar 12, 2013 #2
    First of all, do not use directly the definition of commutator... remember that there are properties of the commutator you can use to do things simpler:
    1) Commutator is linear, i.e. ##[A+B,C]=[A,C]+[B,C]## and ##[\alpha A,B]=\alpha[A,B]##
    2) Multiplication is treated like ##[AB,C]=A[B,C]+[A,C]B##

    Using these two properties, you can simplify a lot what you wrote. Then use also the definitions of commutations of ladder operators ##[a_+,a_+]=[a_-,a_-]=0##, ##[a_-,a_+]=1=-[a_+,a_-]##.
    Now you are done. Anyway you're right as for the ##\alpha##, you get only one so you have to put it into the definition of ##A## and you will get as result ##2i\beta A##
     
  4. Mar 12, 2013 #3
    Ahhh yes.

    Haven't thought of re-writing it that way. Thank you :)
     
  5. Mar 13, 2013 #4
    R: Commutator of two operators

    Always try to use this method in exercises like that, it's often much simpler than just splitting all up :)
    And anyway, you're welcome
     
  6. Mar 13, 2013 #5
    zero
     
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