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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) = x^{2} + y^{2} is invariant under rotations of the plane around the origin.## Application to quantum mechanics

## See also

## References

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.

See also isotropic, Maxwell's theorem, rotational symmetry.

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.

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

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Last updated on Saturday September 15, 2007 at 17:22:34 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Saturday September 15, 2007 at 17:22:34 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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