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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.
     
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