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!

Commutator of L and H

  1. Nov 9, 2003 #1
    I'm trying to demonstrate that angular momentum and the Hamiltonian commute provided that V is a function of radius r only. Using L = Lx + Ly + Lz components, the definition of Lx = yPz - zPy, etc, the commutator definition and the fact that H = p^2/2m + V. I can reduce [Lx,H] to the following:

    [Lx,H] = [yPz,V] - [ZPy,V] and by expanding using and the relation [AB,C] = A[B,C] + [A,C]B and the Pz = hbar/i * d/dz, etc. get

    [Lx,H] = hbar/i * ( yf dV/dz - zf dV/y) where f is a test function.

    I have not used the fact that V is a function of r only and I can't take this any further.

    I know that ( yf dV/dz - zf dV/y) must equal zero but have no idea how to prove it. Please help with a suggestion.

    (I believe that H commutes with each component Lx, Ly and Lz and not just L so I don't think the terms above will cancel with terms from [Ly,H] and [Lz,H].)
     
  2. jcsd
  3. Nov 10, 2003 #2
    You should change to a coordinate system with r and two angles. That is , to spherical coordinates. You will have to work out what d/dx is in these coordinates and so on. A lot of writing but that is physics. Don't forget to fill in the spherical coordinates for x, y and z as well.
     
  4. Nov 10, 2003 #3
    This I now think is wrong. You need to look at all 3 components of L to get it to commute.

    The book sometimes makes problems hard but it would have a whole section done in rectangular coordinates and then give a problem that must be done spherical coordinates. That's in the next section!

    Any other suggestions?
     
  5. Nov 11, 2003 #4
    Hmmm...that would be a bit strange indeed. However, if you have a spherical symmetric potential it would be easiest to use spherical coordinates. What book are you using? In my QM classes we used Griffiths and I can remember that sometimes he gave a sort of sneak preview....You can also write d/dx=d/dr*dr/dx and same for d/dy and d/dz. You can write out dr/dx=x*(x^2+y^2+z+2)^3/2....I think that will work (just gave it a quick glance).

    Good luck
     
  6. Nov 11, 2003 #5
    Thanks for the suggestion! Worked great.

    dr/dx = x *(x^2+y^2+z^2)^(-1/2), etc.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?