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Thread: Commutators

  1. #1
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    Commutators

    What's up I am working on some quantum chemistry problems and I am somewhat confused about commutators.

    The question asks to calculate the commutators, the questions are:

    [x,d/dx]
    [sqrt,()^2]

    There is also one concerning nuclear spins that you guys may not be able to help me with, but if I can see how to do these two I think I can figure that one out on my own.

    Thanks for the help
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  2. #2
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    Quote Originally Posted by csuiter View Post
    The question asks to calculate the commutators, the questions are:

    [x,d/dx]
    [sqrt,()^2]
    First, you have to ask what sort of objects these commutators act on.

    The commutator [x,d/dx] has to act on a function f(x). Then [x,\tfrac d{dx}]f(x) = x\tfrac d{dx}f(x) - \tfrac d{dx}\bigl(xf(x)\bigr) = xf'(x) - \bigl(f(x) + xf'(x)\bigr) = {\color{red}-}f(x). So [x,d/dx] is actually the negative of the identity operator: it takes the function f(x) to –f(x).

    The commutator [√,^2] has to act on a nonnegative real number. You should be able to work out for yourself that this is the zero operator: it takes each number to 0.
    Last edited by mr fantastic; Dec 12th 2008 at 01:28 AM. Reason: Small correction of typo (in red)
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    thank you very much for your help!
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