commutator of L and H

mmwave

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].)

heumpje

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.

mmwave

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?

heumpje

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

mmwave

Thanks for the suggestion! Worked great.

dr/dx = x *(x^2+y^2+z^2)^(-1/2), etc.