Adjoint of the time-evolution operator

Question

The time-evolution operator $hat U$ is defined so that $Psi(x,t)=hat U(t)Psi(x,0)$. In terms of the Hamiltonian, it is expressed as $hat{U}(t)=exp left(-frac{i t}{hbar} hat{H}right)$. I'm trying to calculate the adjoint conjugate $hat U^dagger(t)$.

My attempt at a solution
It must satisfy $langle hat{U}(t) Psi(x,0) | Phi(x,0) rangle=langle Psi(x,0) | hat{U}^dagger(t)Phi(x,0) rangle$, so
$$int _{-infty}^{+infty} hat{U}^star(t) Psi^star(x,0) Phi(x,0) dx=int _{-infty}^{+infty} Psi^star(x,0)hat{U}^dagger(t)Phi(x,0)dx$$
I know that $hat U$ is unitary, so $hat U^dagger(t)=hat U^{-1}(t)=hat U^{star}(t)$, but, without using this information, could the expression of $hat U^dagger(t)$ be deduced from the expression above?

user2820579 · Accepted Answer

Probably it is cleaner to do it by series.
begin{equation}
begin{split}
U^dagger(t)&=left(sum_{n=0}^infty frac{1}{n!}left(frac{-it}{hbar} right)^n H^nright)^dagger
&=sum_{n=0}^infty frac{1}{n!}left(left(frac{-it}{hbar} right)^n right)^dagger (H^n)^dagger
&=sum_{n=0}^infty frac{1}{n!}left(frac{it}{hbar} right)^n H^n
&=exp(it H/hbar),
end{split}
end{equation}
since $H$ is Hermitian.
Another possibility is to start with Schrödinger equation, compute the adjoint and finally derive and solve an equation for $U^dagger$ provided that $<Psi(t)| = <Psi(t=0)|U^dagger$.

Adjoint of the time-evolution operator

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