Rotational invariance
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.
Mathematics
Functions
For example, the function
is invariant under rotations of the plane around the origin, because for a rotated set of coordinates through any angle θ
the function, after some cancellation of terms, takes exactly the same form
The rotation of coordinates can be expressed using matrix form using the rotation matrix,
or symbolically x'′ = Rx'. Symbolically, the rotation invariance of a real-valued function of two real variables is
In words, the function of the rotated coordinates takes exactly the same form as it did with the initial coordinates, the only difference is the rotated coordinates replace the initial ones. For a real-valued function of three or more real variables, this expression extends easily using appropriate rotation matrices.
The concept also extends to a vector-valued function f of one or more variables;
In all the above cases, the arguments (here called "coordinates" for concreteness) are rotated, not the function itself.
Operators
For a function
which maps elements from a
which acts on a function f to obtain another function ∇2f. This operator is invariant under rotations.
If g is the function g(p) = f(R(p)), where R is any rotation, then (∇2g)(p) = (∇2f )(R(p)); that is, rotating a function merely rotates its Laplacian.
Physics
In
Application to quantum mechanics
In
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.
For
then
thus
in other words angular momentum is conserved.
See also
References
- Stenger, Victor J. (2000). Timeless Reality. Prometheus Books. Especially chpt. 12. Nontechnical.