- Rotational invariance
In
mathematics , a function defined on aninner product space is said to have rotational invariance if its value does not change when arbitraryrotation s are applied to its argument. For example, the function "f"("x","y") = "x"2 + "y"2 is invariant under rotations of the plane around the origin.For a function from a space "X" to itself, or for an
operator that acts on such functions, rotational invariance may also mean that the function or operator commutes with rotations of "X". An example is the two-dimensionalLaplace operator Δ "f" = ∂"xx" "f" + ∂"yy" "f": if "g" is the function "g"("p") = "f"("r"("p")), where "r" is any rotation, then (Δ "g")("p") = (Δ "f")("r"("p")) -- i.e., rotating a function merely rotates its Laplacian.See also
isotropic ,Maxwell's theorem ,rotational symmetry .Application to quantum mechanics
In
quantum mechanics , rotational invariance is the property that after arotation the new system still obeysSchrödinger's equation . That is: ["R", "E" − "H"] = 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 ["R", "H"] = 0.
Since ["R", "E" − "H"] = 0, and because for
infinitesimal rotation s (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 + "J""z" dθ,
: [1 + "J""z" dθ, d/d"t"] = 0;
thus
:d/d"t"("J""z") = 0,
in other words
angular momentum is conserved.ee also
*
Isotropy References
*Stenger, Victor J. (2000). "Timeless Reality". Prometheus Books. Especially chpt. 12. Nontechnical.
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