Undergrad Definition of a symmetry transformations in quantum mechanics

[Lebnm]

By the Wigner theorem, symmetries transformations are implemented by operators $\hat{U}$ that are unitary or antiunitary. This is what is written in most books. But I have read somewhere that, to $\hat{U}$ represent a symmetrie, it's necessary that $\hat{U}^{\dagger} \hat{H} \hat{U} = \hat{H}$, that is, the hamiltonian has to be invariant under the transformation. Is it true? Are these definitions equivalents?

 

[Truecrimson]

They are different. Wigner proved that the most general transformation that preserves the total probability has to be linear and unitary or anti-unitary. On the other hand, such $U$ is a symmetry of $H$ (i.e. of a system with Hamiltonian $H$) if $U^{\dagger}HU = H$.

 

[haushofer]

Or: not every transformation (described by U) is necessarily a symmetry transformation (described by [H,U]=0) ;)

 

[haushofer]

Or even different: being unitary is necessary but not sufficient to be a symmetry transformation.

 

[Lebnm]

Ok, thank you!