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}&lt;\psi(t_0)\,|\,\psi(t_0)&gt;&amp;=\,&lt;U(t_0, t)\,\psi(t)\,|\,U(t_0, t)\,\psi(t)&gt;\\<br /> &amp;=\,&lt;U^{-1}(t, t_0)\,\psi(t)\,|\,U^{-1}(t, t_0)\,\psi(t)&gt;\\<br /> &amp;=\,&lt;\psi(t)\,|\,{U^{-1}}^\dagger(t, t_0)\,U^{-1}(t, t_0)\,|\,\psi(t)&gt;\end{align}

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

Right.