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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:)