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Verify the commutation relations for x and p by definition.

  1. Jun 13, 2013 #1
    1. The problem statement, all variables and given/known data
    Verify ##\left[ x^{i} , p_{k}\right] = i \hbar \delta^{i}_{k}##

    2. Relevant equations
    ## p_{j} = -i \hbar \partial_{j}##

    3. The attempt at a solution

    Writing it out i get
    $$ i \hbar \left( \partial_{k} x^{j} - x^{j} \partial_{k} \right)$$
    The Kronecker makes perfect sense, it's identically zero unless k=j. Assuming it does, I arrive at:
    $$ i \hbar \delta^{j}_{k} \left( \partial_{k} x^{j} - x^{j} \partial_{k} \right)$$

    I assume I am missing something obvious, because most of the problem in this book are pretty straight forward, but this one's been a pain. I'm not doing any coursework, I already did my undergrad and am in limbo.

    *Note Einstein convention is in use*
    cheers.
     
  2. jcsd
  3. Jun 13, 2013 #2

    CompuChip

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    You should apply the commutator to a test function f(xk) (which is shorthand for f(x1, x2, x3) because k is a free index) and work out what the partials do with it.
     
  4. Jun 13, 2013 #3
    I don't follow, do you mean what the term in parenthesis becomes when i=J? (## 1- x \bullet \nabla##). The book's question specifically says verify the commutation relation using the definition of momentum given. Sorry, I should have been more precise maybe?
     
  5. Jun 13, 2013 #4

    CompuChip

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    I mean that "##(\partial_k x^j - x^j \partial_k)##" by itself does not make sense. You should consider a test function f and work out what
    $$(\partial_k x^j - x^j \partial_k)f$$
    is, taking into account things like the product (Leibniz) rule.

    I assume that by "use the definition of momentum" they just mean you should use ## p_{j} = -i \hbar \partial_{j}## as you have already done.
     
  6. Jun 15, 2013 #5
    I understand that it works, the problem comes from a chapter titled "Lie Groups and Lie Algebras", so I assumed it was something more fundamental. Thank you.
     
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