Symmetry in quantum mechanics

Remarks

By Wigner's theorem, every symmetry transformation of rays is induced by a unitary or an antiunitary operator, unique up to a phase. A unitary $U$ commuting with $H$ commutes with the time evolution: $U e^{-iHt/\hbar} = e^{-iHt/\hbar} U$. If $U(s) = e^{-isA}$ is a one-parameter group of such symmetries, with $A$ self-adjoint, then $A$ commutes with $H$ and its expectation value is constant in time: $A$ is a conserved quantity.

Example: on $L^2(\R^3)$ let $H = -\frac{\hbar^2}{2m}\Delta + V(\abs{x})$ with a radial potential $V$. For a rotation $R \in SO(3)$, the operator $(U_R \psi)(x) = \psi(R^{-1}x)$ is unitary and commutes with $H$, since both $\Delta$ and $\abs{x}$ are rotation invariant. So every rotation is a symmetry of the system, and the generators of these rotations, the components of the orbital angular momentum, are conserved.

Non-example: in the same setting take $V(x) = x_3$, a constant force along the third axis. A rotation $R$ about the first axis changes $V$ into another linear function. So $U_R$ does not commute with $H$, and it is not a symmetry of the system. It is still a symmetry transformation of the rays.