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Homework Help: Noncommuting Series Expansion

  1. Mar 16, 2016 #1
    1. The problem statement, all variables and given/known data
    Need to show that [tex][a,f(a,a^\dagger]=\frac{\partial f}{\partial a^\dagger}[/tex]

    2. Relevant equations

    3. The attempt at a solution
    Need to expand [tex]f(a,a^\dagger)[/tex] in a formal power series. However I don´t know how to do it if the variables don´t commute.
    Last edited: Mar 16, 2016
  2. jcsd
  3. Mar 17, 2016 #2


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    ##f(a,a^+)= b_0+ b_1 a+b_2 a^+ b_3 a^2 +b_4 aa^+ +b_5 a^+a+b_6 (a^+)^2+\ldots##
    So the general term is some product of a and ##a^+## in arbitrary order.
  4. Mar 17, 2016 #3
    Thanks for answering, but how would it be in terms of derivatives? Normally It would be:

    [tex]f=\sum_{m,n} \frac{a^m a^{\dagger m}}{n!m!}\frac{\partial^{n+m} f}{\partial a^n \partial a^{\dagger m}}[/tex]

    But in this case I guess I have to take into account that they're noncommuting.
  5. Mar 17, 2016 #4


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    Start with a simpler case, i.e., ##f = (a^\dagger)^n## only. Use induction (on ##n##) to show that the desired formula holds. Once you understand the induction method for this problem, you'll probably work it out for more general ##f## more easily.
  6. Mar 18, 2016 #5
    Thanks, that did the trick!
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