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. 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)); that is, rotating a function merely rotates its Laplacian.

In physics, if a system behaves the same regardless of how it is oriented in space, then its Lagrangian is rotationally invariant. According to Noether's theorem, if the action (the integral over time of its Lagrangian) of a physical system is invariant under rotation, then angular momentum is conserved.

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

[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 + Jz dθ,
[1 + Jz dθ, d/dt] = 0;

thus

d/dt(Jz) = 0,

in other words angular momentum is conserved.

See also

References