The discussion focuses on proving the rotation invariance of the Schrödinger equation and the commutation relation [Lx, H] = 0. The theorem states that the Schrödinger equation is invariant under O(3)-rotation if and only if the condition x_{i} ∂_{k}V(x) = x_{k} ∂_{i}V(x) holds. The proof involves the commutation relation [L_{i}, H] = ε_{ijk}[x_{j}p_{k}, p^{2} + V(x)], leading to the conclusion that the invariance is tied to the potential's behavior under rotations.
PREREQUISITESStudents and researchers in quantum mechanics, theoretical physicists, and anyone interested in the mathematical foundations of quantum theory and symmetries.
torehan said:Here is my first post,
How can I prove that the rotation invariability of the Sch. equations?
How can I prove that [Lx,H]=0
Theorem: Schrödinger equation is invariant under O(3)-rotation if and only if;
[tex]x_{i} \partial_{k}V(x) = x_{k} \partial_{i}V(x)[/tex]
Proof
[tex] \left[ L_{i} , H \right] = \epsilon_{ijk} \left[x_{j}p_{k} , p^{2} + V(x) \right] = \epsilon_{ijk}x_{j} \left[ p_{k} , V(x) \right][/tex]
Ok, you finish off the proof.
sam