Prove the time evolution operator is unitary

[Happiness]

How is (5.240b) derived? I get ${U^{-1}}^\dagger(t, t_0)\,U^{-1}(t, t_0)=I$ instead.

My steps:
$$\begin{align}<\psi(t_0)\,|\,\psi(t_0)>&=\,<U(t_0, t)\,\psi(t)\,|\,U(t_0, t)\,\psi(t)>\\ &=\,<U^{-1}(t, t_0)\,\psi(t)\,|\,U^{-1}(t, t_0)\,\psi(t)>\\ &=\,<\psi(t)\,|\,{U^{-1}}^\dagger(t, t_0)\,U^{-1}(t, t_0)\,|\,\psi(t)>\end{align}$$

Also, to get (5.240a), do we use the fact that $<\psi(t_0)\,|\,\psi(t_0)>\,=\,<\psi(t_0)\,|\,U^\dagger(t, t_0)\,U(t, t_0)\,|\,\psi(t_0)>$is true for any $\psi(t_0)$?

 

[mfb]

How is (5.240b) derived? I get ${U^{-1}}^\dagger(t, t_0)\,U^{-1}(t, t_0)=I$ instead.

Multiply both sides by appropriate matrices and you should get the result you want.

Also, to get (5.240a), do we use the fact that $<\psi(t_0)\,|\,\psi(t_0)>\,=\,<\psi(t_0)\,|\,U^\dagger(t, t_0)\,U(t, t_0)\,|\,\psi(t_0)>$is true for any $\psi(t_0)$?

Right.