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.
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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;
x_{i} \partial_{k}V(x) = x_{k} \partial_{i}V(x)
Proof
<br /> \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]<br />
Ok, you finish off the proof.
sam