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!

Relating the Laplacian to the Quantum Angular Momentum

  1. Sep 13, 2008 #1
    This is my first post, so if it belongs somewhere else, please help me out. I've got a homework problem that I believe I solved, but I'm not sure if I did it right. (It seems too easy this way.)

    I am given that [tex]\vec{L}[/tex]=-i([tex]\vec{r}[/tex]X[tex]\nabla[/tex]). I have to prove the relation that:

    [tex]\nabla[/tex]2=1/r2([tex]\partial[/tex]/[tex]\partial[/tex]r(r2[tex]\partial[/tex]/[tex]\partial[/tex]r)-L2).

    To solve this, I did the following:

    L2=[tex]\vec{L}[/tex][tex]\bullet[/tex][tex]\vec{L}[/tex]=-i([tex]\vec{r}[/tex]X[tex]\nabla[/tex])[tex]\bullet[/tex]-i([tex]\vec{r}[/tex]X[tex]\nabla[/tex])=-r2[tex]\nabla[/tex]2+([tex]\vec{r}[/tex][tex]\bullet[/tex][tex]\nabla[/tex])([tex]\vec{r}[/tex][tex]\bullet[/tex][tex]\nabla[/tex])

    ([tex]\vec{r}[/tex][tex]\bullet[/tex][tex]\nabla[/tex])([tex]\vec{r}[/tex][tex]\bullet[/tex][tex]\nabla[/tex])-L2=r2[tex]\nabla[/tex]2

    which yields:

    [tex]\nabla[/tex]2=1/r2([tex]\partial[/tex]/[tex]\partial[/tex]r(r2[tex]\partial[/tex]/[tex]\partial[/tex]r)-L2).

    I apologize for the bad formatting. I can't figure out how to fix it. The black dot is the dot product. Assume that the black dot, the del, and the partial d's should all be lowered.
     
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you help with the solution or looking for help too?



Similar Discussions: Relating the Laplacian to the Quantum Angular Momentum
  1. Laplacian help (Replies: 0)

  2. Recurrence Relation (Replies: 0)

Loading...