Editing Rotational invariance (section)

=== Application to quantum mechanics ===

{{Further|Rotation operator (quantum mechanics)|Symmetry in quantum mechanics}}

In [[quantum mechanics]], '''rotational invariance''' is the property that after a [[rotation]] the new system still obeys the [[Schrödinger equation]].  That is
: <math>[R,E-H] = 0</math> 
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'',&nbsp;''H''] = 0.

For [[infinitesimal rotation]]s (in the ''xy''-plane for this example; it may be done likewise for any plane) by an angle ''dθ'' the (infinitesimal) rotation operator is
: <math>R = 1 + J_z d\theta \,,</math>
then
: <math>\left[1 + J_z d\theta , \frac{d}{dt} \right] = 0 \,,</math>
thus
: <math>\frac{d}{dt}J_z = 0\,,</math>

in other words [[angular momentum]] is conserved.