Difference on the invariance of operators and their transformations under unitary operators

Question

I am confused about, what I believe, refers to passive and active transformations in QM. What I have understood so far is that the matrix elements $\langle \psi| \hat{H}|\phi\rangle$ should remain unchanged under transformations; this implies that $\hat{H} = \hat{U}^\dagger \hat{H} \hat{U}$.

On the other hand, under certain transformation $\hat{U}$, an arbitrary operator $\hat{\Omega}$ transforms as $\hat{U} \hat{\Omega} \hat{U}^\dagger$.

1) Why should the matrix elements of the hamiltonian remain unchanged under a transformation? Is that the same as saying: "The evolution of the state is governed by the hamiltonian; under a transformation the system should still evolve in the same way, hence the matrix elements should be conserverd"?

2) What does $\hat{H} = \hat{U}^\dagger \hat{H} \hat{U}$ mean? Is that talking about how the hamiltonian transforms? Or is it only a condition on the transformation $\hat{U}$?

3) What is $\hat{\Omega} \to \hat{U} \hat{\Omega} \hat{U}^\dagger$ opposed $\hat{U}|\phi\rangle$. Does an active transformation mean applying $\hat{U}$ to every state, while a passive transformation is leaving vectors as they are and transforming the opeartors as $\hat{\Omega} \to \hat{U} \hat{\Omega} \hat{U}^\dagger$? Or does applying a transformation mean doing both of the above?

Answer 1

$ \langle \psi | \hat{H} | \phi \rangle$ should remain unchanged under transformations.

This is certainly not true in general for an arbitrary transformation. I think perhaps what you meant is that $ \langle \psi | \hat{H} | \phi \rangle$ should remain unchanged under a basis transformation.

This implies that $H = U H U^\dagger$.

This is not true. What you are asserting here is that $[H, U]=0$, which has no reason to be true without additional assumptions.

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I think your confusion lies in what actually constitutes a unitary basis transformation. If we go to a different basis, the transformation prescription is

\begin{align*} &|\psi\rangle \mapsto U|\psi \rangle\\ &\langle \psi | \mapsto\langle \psi | U^\dagger\\ & \hat{H} \mapsto U\hat{H}U^\dagger. \end{align*}

Plug these transformations into $\langle \psi | \hat{H} | \phi \rangle$ and you can clearly see that that it is invariant under a basis transformation.