[Physics] Why does the Hamiltonian define symmetry/invariance

definitionhamiltonianhamiltonian-formalismquantum mechanicssymmetry

In Sakurai's Modern Quantum Mechanics, in Chapter 4, he effectively states that the operation of rotation or translation, represented by a unitary operator $U$, is customarily called a symmetry operator regardless of whether the physical system itself possesses the symmetry corresponding to $U$. It's a symmetry or invariance of the system only when $U^\dagger H U=H$.

Why are symmetries defined with respect to invariance of the Hamiltonian?

Best Answer

In classical mechanics, a conserved quantity has vanishing Poisson bracket with the Hamiltonian. Such quantities become "good quantum numbers" in QM: they commute with $H$, so simultaneous eigenstates from a complete basis. The evolution operator $e^{-iHt/\hbar}$ also commutes with good quantum numbers, so their probability distribution is unchanged. (Thus the connection from Noether's theorem of conservation laws to continuous symmetries survives in QM.) For unitary $U$ commuting with $H$, $U^\dagger=U^{-1}$ obtains your equation.

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