Let $H_0$ be some initial time-independent hamiltonian, and let $V$ be a scattering potential, such that the hamiltonian of a scattering process is:
$$H=H_0+V$$
Define the quantum states $|\psi_i\rangle,|\psi_o\rangle$ such that:
$$H_0|\psi_i\rangle=E|\psi_i\rangle \\ H|\psi_o\rangle=E|\psi_o\rangle$$
In other words, the scattering is elastic. We shall define the scattering operator $S$ such that:
$$S|\psi_i\rangle=|\psi_o\rangle$$
Prove that $S$ is unitary.
In order to prove that $S$ is unitary, I must show that $S^\dagger S=I$ where $I$ is the identity operator. This is somewhat confusing - It seems like the operator $S$ is defined using merely $2$ quantum states; rather than a basis of quantum states. How is it possible to show that $S$ is unitary? I thought, maybe I should just prove $S$ is unitary considering that the hilbert space contains only the two given quantum states. In that case, I have to show $2$ matrix elements of $S^\dagger S$ are equal to $1$, and the other two are equal to $0$. I was able to show that:
$$\langle\psi_i|S^\dagger S|\psi_i\rangle=\langle\psi_o|\psi_o\rangle=1$$
But in order to compute the other $3$ matrix elements, I have to know how the operator $S$ operates on $|\psi_o\rangle$, it seems. Unfortunately, I couldn't figure how to continue.
Thank you
I read about this a-lot, and found out that this $S$ operator is connected to something called in the literature The $S$-Matrix. I've seen some explanations on why $S$ is unitary, most of them show this through conservation of probability current. But it seems that they assume the given quantum states have a very specific form - they tend to assume that $|\psi_i\rangle$ is an eigenstate of the momentum operator (fair enough), but they also assume that $|\psi_o\rangle$ is a simple superposition of two eigenstates of the momentum operator, both have the same wavevector (therefore they are only differentiated in sign). This is much less trivial.
Please not that $H_0$ isn't necessarily equal to the kinetic energy, and therefore, it cannot be assumed that $|\psi_i\rangle$ is a superposition of plane or spherical waves.
