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:(adsbygoogle = window.adsbygoogle || []).push({});

[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].)

**Physics Forums - The Fusion of Science and Community**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Commutator of L and H

Loading...

Similar Threads - Commutator | Date |
---|---|

I Commuting Operators and CSCO | Feb 20, 2018 |

A Commutator vector product | Jan 29, 2018 |

Commutation relation | Jan 18, 2018 |

I Solving the Schrödinger eqn. by commutation of operators | Jan 8, 2018 |

I Commutator of p and x/r | Dec 8, 2017 |

**Physics Forums - The Fusion of Science and Community**