# Time-Evolution Operators (Sakurai)

For the infinitesimal time-evolution operator, [Sakurai] has the following equation [2.1.21]:

$$\mathcal U\left( t_0 + dt, t_0 \right) = 1 - \frac{iHdt}{\hbar},$$

where $$H$$ is the Hamiltonian. Now they derive the Schrödinger equation for the time-evolution operator [2.1.25] as follows:

The composition property they're referring to is [2.1.12] $$\mathcal U\left( t_2, t_0\right) = \mathcal U\left( t_2, t_1 \right)\mathcal U\left( t_1, t_0\right).$$

Does anybody see why $$\mathcal U\left( t+dt, t_0 \right) = 1-\frac{iHdt}{\hbar},$$ as it indirectly says in their Eq. [2.1.23]? According to my very first formula, this this would hold if the first argument were $$t_0 + dt$$ and not $$t+dt$$?

[Sakurai] J.J. Sakurai, Jim Napolitano, "Modern Quantum Mechanics", 2nd Edition, Pearson Education

Does anybody see why $$\mathcal U\left( t+dt, t_0 \right) = 1-\frac{iHdt}{\hbar}$$, as it indirectly says in their Eq. [2.1.23]?
That's not what the equation says. The second argument of the first time-evolution operator in the middle part of the equation is $$t$$, not $$t_0$$. They are using the the formula for the infinitesimal time-evolution operator given in (2.1.21) to say $$\mathcal U\left( t + dt, t \right) = 1 - \frac{iHdt}{\hbar}$$
Note that there is still a $$\mathcal U(t,t_0)$$ on the right in (2,1,23).