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Operators and order

  1. Feb 18, 2007 #1

    Lee

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    In my question I have to find what the commutation of a electrons kinetic and potentials energys are, in 3 Dimensions. I have started by finding the kinetic operator T and the potential energy from coloumbs law. I have then applied commutation brackets and I'm at the stage where I'm solving the commutation bracket for the x-direction. (and then apply symmetry for my 2 other axis) My question is, as we have to retain order when dealing with operators, how do I 'deal' with my

    [tex]
    \newcommand{\pd}[3]{ \frac{ \partial^{#3}{#1} }{ \partial {#2}^{#3} } }

    xi \hbar \pd {} {x} {}
    [/tex]

    I presume I can't just differentate the x as I need to preserver order, does this just sit like this till I can 'deal' with it?
     
  2. jcsd
  3. Feb 18, 2007 #2

    nrqed

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    In calculating commutators of differntial operators, it is convenient to apply th commutator on a "test function", which is is just som arbitrary function of x, y and z that must be removed at the very end of the calculation.

    So if you have two operators A and B (which are differential operators) and you want to compute their commutator, just consider
    [tex] [A,B] f(x,y,z) = AB f(x,y,z) - BA f(x,y,z) [/tex]
    Apply all the derivatives and at the very end, remove the test function.
     
  4. Feb 18, 2007 #3

    Hurkyl

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    You can apply a commutation relation if you wanted to reverse the order.

    But remember that "x" is an operator, not a function. So

    [tex]
    \frac{\partial}{\partial x} x \neq 1
    [/tex]

    Instead, it's supposed to be the operator

    [tex]
    \psi(x, y, z, t) \rightarrow \frac{\partial (x \psi(x, y, z, t))}{\partial x}
    [/tex]
     
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