Rotational invariance

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In mathematics, a function defined on an inner product space is said to have rotational invariance if its value does not change when arbitrary rotations are applied to its argument. For example, the function f(x,y) = x² + y² 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-dimensional Laplace 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.

In quantum mechanics, rotational invariance is the property that after a rotation the new system still obeys Schrödinger'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 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 + J_z dθ,

[1 + J_z dθ, d/dt] = 0;

thus

d/dt(J_z) = 0,

in other words angular momentum is conserved.

Reference

Stenger, Victor J. (2000). Timeless Reality. Prometheus Books. Especially chpt. 12. Nontechnical.

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Rotational invariance

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