1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

I don't understand operators

  1. Nov 12, 2007 #1
    So I was reading from my quantum book (Gasiorowicz) and I ame across this sentence:

    [tex] [p^2, x] = p [p, x] + [p, x] p = \frac{2\hbar}{i} p [/tex]

    I don't understand this. I know that [tex] p = -i \hbar \frac{\partial}{\partial x} [/tex], but I can't see how to get that expression...I just come up with something like [tex] x {\hbar}^2 \frac{{\partial}^2}{{\partial x}^2} [/tex] when I try multiplying it out.
  2. jcsd
  3. Nov 12, 2007 #2
    try to derive [tex][AB,C] = ?[/tex]
    then use [tex][x,p] = i\hbar[/tex]
    to find [tex][p^2,x][/tex]
  4. Nov 12, 2007 #3


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    All those expressions only make sense if you imagine applying them to some "test function" f(x). For example,

    [tex] [x,p_x] f(x) = -i \hbar x \partial_x f(x) + i \hbar \partial_x (x f(x)) [/tex] apply the product rule on the second term and then something will cancel out. At the very end of the calculation (and only then) you may drop the test function f(x).

    Using a test function is not the fastest way to prove complicated commutation relations, however. But it's the only way to make sense of these commutation relations. after you have done a few with a test function you will be able to do the more complex cases without the crutch of a test function.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?