We therefore have . P i are the components of the momentum of the closed system. Homogeneity of space implies momentum conservation . Let us now generalize this result. I, then is a Hermitian operator.
P is the Hermitian generator of the unitary family of translation operators .
The evolution operator is . The Hamiltonian is the Hermitian generator of time translation operators . The rotation operator Assume that the basic laws of physics are the same in all directions. Then the mechanical properties of a closed system are unchanged by any rotation of the entire system in space. This is the assumption of the isotropy of space .