1. Limited time only! Sign up for a free 30min personal 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!

Homework Help: 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
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted



Loading...
Similar Threads for Relating Laplacian Quantum Date
Define a relation Tuesday at 5:02 PM
General Relativity - geodesic - affine parameter Apr 11, 2018
QFT, more a QM Question, Hamiltonian relation time evolution Apr 4, 2018
Null curve coordinate system Apr 3, 2018
Laplacian in [0,L] x [0, ∞] Jul 9, 2017