# Why is $[U(\alpha),H]=0$ is considered to be the definition of symmetry in quantum mechanics?

In quantum mechanics, if a family of unitary operators $$\{U(\alpha)\}$$ depending on a bunch of continuous parameters $$\{\alpha\}$$, commute with the Hamiltonian of a system, we say that the system has a continuous symmetry. I think that we define symmetry in this manner because we want the Schrodinger equation to remain unchanged after the symmetry transformation. I think this is true but I don't have any source to validate my impression. Please tell me if my reasoning below is correct.

Under the symmetry, $$|\psi\rangle\to |\psi'\rangle=U|\psi\rangle$$ and $$H\to H'= UHU^{-1}=H$$, then $$i\hbar\frac{\partial|\psi\rangle}{\partial t}=H|\psi\rangle\to i\hbar\frac{\partial|\psi'\rangle}{\partial t}=H|\psi'\rangle$$

One advanced note I wish someone had told me explicitly when I was learning quantum mechanics: when you get to study more complex systems (and in particular more complex spacetime symmetries), sometimes it is useful to generalize the definition of symmetry from the one you have given. In particular, rather than equating the set of operators that commute with the Hamiltonian with the set of symmetries, sometimes it is useful to think of the Hamiltonian operator as being one of a number of operators in an algebra, with prescribed commutation relationships. For example, we want Galilean relativity to be a symmetry of non-relativistic quantum mechanics. It turns out that the operator which represents boosts is the position of the center of mass, $$K=x+ p t / m$$. This operator does not commute with the Hamiltonian, but the Hamiltonian and boost operator together are part of the Galilean algebra, which ensures that states form unitary representations of the Galilean group (which then ensures that the quantum theory has the Galilean symmetry). This is a more advanced notion of symmetry and may not be directly relevant for your work, but in my opinion it's at least useful to be aware that sometimes it is useful to generalize the idea that "symmetries must commute with the Hamiltonian".
• You may have run across the angular momentum operators. These are actually representations for the generators of the group $SO(3)$, which is the group corresponding to rotations in 3 dimensions. You can classify the states by the $\ell,m$ quantum numbers. In particle physics, to obey special relativity, you need to form representations of the Poincaire group; following the same logic leads to classifying particles by mass and spin. Mar 4, 2021 at 11:29