Besides rotating the system itself, we can also rotate the instruments with which we observe it. We now define the law for the transformation of the observables representing the various measurements that can be made on the system. Let A be an observable and let A' be its transform in the rotation, i.e. A describes what is measured with a certain apparatus, and A' describes what is measured after the apparatus has been rotated. An example of a scalar observable is J 2 . An example of a vector observable is J itself.
Conservation laws are intrinsically related to the invariance of a physical system.
Show classically and quantum mechanically that the conservation of angular momentum is a consequence of isotropy of space . Properties of an isolated physical system do not vary when it is rotated as a whole in any manner in space. This is the assumption of isotropy of space. Assume a system is rotated about the axis by an infinitesimal amount d a . Therefore the Lagrangian must remain unchanged by the rotation.