Application To Quantum Mechanics
In quantum mechanics, rotational invariance is the property that after a rotation the new system still obeys Schrödinger's equation. That is
- = 0 for any rotation R.
Since the rotation does not depend explicitly on time, it commutes with the energy operator. Thus for rotational invariance we must have = 0.
Since = 0, and because for infinitesimal rotations (in the xy-plane for this example; it may be done likewise for any plane) by an angle dθ the rotation operator is
- R = 1 + Jz dθ,
- = 0;
thus
- d/dt(Jz) = 0,
in other words angular momentum is conserved.
Read more about this topic: Rotational Invariance
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