Let $$H := \sigma_z \otimes \sum_{k=1}^{n}\mathrm{id} \otimes ...\otimes g_k \sigma_z \otimes...\otimes \mathrm{id}$$ be the Hamiltonian ($\sigma_z$ is the Pauli-matrix of couse)
and $$|\psi_0 \rangle:=(a|\!\uparrow\rangle+b |\!\downarrow\rangle ) \otimes_{k=1}^{n} (\alpha_k |\!\uparrow \rangle + \beta_k |\!\downarrow\rangle).$$
Then my script says that the time-evolution of this state is
\begin{align} |\psi(t)\rangle &=a|\!\uparrow\rangle \otimes_{k=1}^{n} (\alpha_k e^{i g_kt}|\!\uparrow \rangle + e^{-i g_k t} \beta_k |\!\downarrow\rangle) \\ &\quad +b |\!\downarrow\rangle\otimes_{k=1}^{n} (\alpha_k e^{-i g_kt}|\!\uparrow \rangle + e^{i g_k t} \beta_k |\!\downarrow\rangle). \end{align}
Does anybody have an explanation for this?
My concern about this equation arises from the fact that for $n=0$, we would have $|\psi(t)\rangle= |\psi(0)\rangle$ which should not be case, but rather $|\psi(t)\rangle= e^{-it}a|\!\uparrow\rangle+ e^{it} b|\!\downarrow\rangle.$
What do you think?
If anything is unclear, please let me know.