# Prove that the scattering operator is unitary [duplicate]

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.

• You need some additional assumptions. $S=|\psi_o\rangle\langle \psi_i|$ satisfies all of your assumptions, but it is not unitary. Commented Dec 17, 2020 at 13:11
• Possible duplicate: Why are scattering matrices unitary? Commented Dec 17, 2020 at 13:34

The unitarity of this operator follows from the current conservation. It is necessary to keep in mind that here we are dealing with a scattering problem, rather than with an eigenvalue problem - i.e. we are considering how a system with a given energy evolves from a state specified at $$t\rightarrow -\infty$$ to a state at $$t\rightarrow +\infty$$ (although this remains somewhat hidden in the introductory treatments of the scattering theory). This means that the only accessible states are those having the specified energy, i.e., the states on the same energy shell - in the simplest case these are the states that have the same momentum, but more general situations are possible - e.g., scattering which results in particle generation or ejecting an electron from atom.