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Introductory quantum mechanics problem

  1. Feb 2, 2016 #1
    1. The problem statement, all variables and given/known data
    Consider A(x) is an arbitrary function of x, and px is the momentum operator. Show that they satisfy the following condition:
    [px,A(x)] = (-i/ħ)*d/dx(A(x))

    where [px,A(x)] = pxA(x) - A(x)px

    2. Relevant equations
    ħ = h/2π
    px = (-iħ)d/dx

    3. The attempt at a solution
    Starting with the expression given at the bottom of the question and subbing in the expression for the momentum operator and factoring out -iħ, I get:

    [px,A(x)] = -iħ*(d/dx(A(x)) - A(x)d/dx)

    and I'm stuck there for the time being... I don't really understand how to get ħ into the denominator or what to do with the term in brackets. Any help would be appreciated.
     
  2. jcsd
  3. Feb 2, 2016 #2

    TSny

    User Avatar
    Homework Helper
    Gold Member

    There must be a misprint in the problem as given to you. The ħ should be in the numerator. You can verify this by dimensional analysis.

    When unraveling commutators like this, it is usually a good idea the think of the commutator as acting on some arbitrary function f(x).

    Thus, try to simplify [px, A(x)]f(x).
     
  4. Feb 2, 2016 #3
    h-bar should not be in the denominator. Do a simple unit analysis to see that's not the case.

    As for evaluating the commutator, I think it might help if you put the commutator in front of function. Remember p and A are operators, they need to operate on something.

    [tex] [p,A]\psi = (pA - Ap)\psi = pA\psi - Ap\psi [/tex]

    Now A and psi are just functions of position. But p is a derivative operator acting on everything to its right. Recall the product rule.

    [tex] pA\psi = p(A)\psi + Ap(\psi) [/tex]

    I'll let you do the rest. Hopefully this makes sense.
     
  5. Feb 2, 2016 #4
    Thanks to both of you for the help. I've done as you suggested and solved the commutator and found that it was equal to dA(x)/dx, now the answer I have is:

    [px,A(x)] = -iħ*(dA(x)/dx)

    Another source I've seen online seems to suggest that the first term should be ħ/i (which is equal to my answer -iħ) I will ask my professor whether there is a typo in the assignment tomorrow.
     
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