# Question about the units of an operator related to the time evolution operator

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.)

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