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Let $U(t, t_0)$ be the unitary time evolution operator. Define an operator $\Lambda (t):= \partial_t [U(t, t_0)] U^\dagger (t, t_0)$. This operator turns out to be independent of $t_0$. Now, the notes I'm reading use this operator to derive the Schrodinger equation from unitary time evolution, and they state that this operator has units $[\Lambda(t)] = s^{-1}$. Why is this? (This is a crucial step in the derivation, because the operator $H(t) := i \hbar \Lambda(t) $ has units of energy, and we define this to be the Hamiltonian.)

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$U$ and $U^\dagger$ are mutual inverses so their product is dimensionless. The only dimension comes from the $d/dt$, which has units of s$^{-1}$.

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