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  1. L

    Angular momentum commutator

    fzero, I didn't even think of that! A free particle would have zero potential energy, so for L to be conserved it would still have to commute with the Hamiltonian, and that means L and P^2 must commute (2m factor is irrelevant). I thought for sure I was making a mistake somewhere since my mind...
  2. L

    Angular momentum commutator

    Oh, I found one mistake but it still leads me to the same conclusion. The second equality should be plus, not minus: [A,BC] = [A,B]C+B[A,C]: \left[ \vec{L} , P^2 \right] = \left[ L^k , P_i P_i \right] = \left[ L^k , P_i \right] P_i + P_i \left[ L^k , P_i \right] = \left( - i \hbar...
  3. L

    Angular momentum commutator

    I have been told that L and P^2 do not commute, but I don't see why. It seems like the commutator should be zero. \left[ \vec{L} , P^2 \right] = \left[ L^k , P_i P_i \right] = \left[ L_k , P_i \right] P_i - P_i \left[ L_k , P_i \right] = \left( - i \hbar \epsilon_{i}^{km} P_m \right)...
  4. L

    Vector decomposition (Helmholtz)

    I actually just found an easy way of showing it using projection operators. Thanks for the reply. Consider this question solved.
  5. L

    Vector decomposition (Helmholtz)

    Helmholtz' Theorem starts with the two components in my original post and defines the divergence and curl as: div[V] = s(r) and curl[V] = c(r), where div[c(r)] = 0 But I can't find anything about how we can define a generic vector as two components: V = -grad[phi] + curl[A], where "phi" is...
  6. L

    Vector decomposition (Helmholtz)

    I have to show that a generic vector can be decomposed into an irrotational and solenoidal component: V(r) = -Grad[phi(r)] + Curl[A(r)] I'm not sure how to start. Do I need to take the curl or div of V and use a vector identity? Any help would be greatly appreciated!
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