How does adding an identity to an Hamiltonian affect the corresponding time-evolution in the Bloch sphere?

Question

For the Hadarmard Hamiltonian, $\hat H = (\hat X+\hat Z)/\sqrt 2$, where $\hat X$ and $\hat Z$ are Pauli matrices. The time evolution of a state under this Hamiltonian could be visualized by a rotation on the Bloch sphere with an axis

$$ \hat n = \frac{1}{\sqrt2}\begin{bmatrix} 1 \\ 0 \\ 1 \\ \end{bmatrix} $$

However, I'm wondering if I have another Hamiltonian defined as

$$ \hat H_1 = \frac{1}{\sqrt3}(\hat X +\hat Z +\hat I) $$

where $\hat I$ is the identity operator. Then what the role $\hat I$ would have on this Hamiltonian? If I still want to visualize the time-evolution rotation on the Bloch sphere, what the 'new' axis would be?

Thanks:)

Answer 1

The identity operator does not change the axis of rotation, because it does not change the eigenvectors of your Hamiltonian.

The only thing that the identity operator changes is the global phase of your state, but since the global phase does not alter the Bloch sphere picture (because you're looking at $|\psi\rangle\langle\psi|$ not $|\psi\rangle$), nothing changes in the Bloch sphere picture.