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Math Help - Commutators

  1. #1
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    Commutators

    Calculate the following commutators:
    a) [x, i d/dx ]
    b) [x, d/dy ]
    c) [-d^2/dx^2, x ]
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  2. #2
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    Quote Originally Posted by myoplex11 View Post
    Calculate the following commutators:
    a) [x, i d/dx ]
    b) [x, d/dy ]
    c) [-d^2/dx^2, x ]
    The best way I know to calculate these are to operate on an arbitrary function.

    a) for example:
    \left [ x, i \frac{d}{dx} \right ] = x \left ( i \frac{d}{dx} \right ) - \left ( i \frac{d}{dx} \right ) x

    So operate this on some function f(x):
    \left [ x, i \frac{d}{dx} \right ] f(x) = x \left ( i \frac{d}{dx} \right ) f(x) - \left ( i \frac{d}{dx} \right ) ( x f(x) )

    = ixf^{\prime}(x) - if(x) - ixf^{\prime}(x)
    (Remember we need to use the product rule on the last term.)

    = -if(x)

    Thus
    \left [ x, i \frac{d}{dx} \right ] f(x) =-i f(x)

    Thus
    \left [ x, i \frac{d}{dx} \right ] = -i

    -Dan
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