The Exercise

The evolution of a closed quantum system over a fixed time is a unitary operator $U$ on its state space, and a state that $U$ only rescales is an eigenvector of $U$. The exercise asks what the scale factor can be: every eigenvalue of a unitary operator has modulus one, so such a state changes by a phase and nothing more.

Unitary Operators and Eigenvalues

The library states the claim on any Hilbert space $H$, over the real or the complex numbers, of finite or infinite dimension.

A unitary operator $U$ on $H$ is a bounded linear operator on $H$ whose adjoint is its inverse: $$U^\dagger U = U U^\dagger = I .$$(1)

An eigenvalue $\mu$ of a linear operator $U$ on $H$ is a scalar for which some nonzero vector $|\psi\rangle$ of $H$ satisfies $$U|\psi\rangle = \mu\,|\psi\rangle .$$(2)

The Claim

What Is Checked

The claim is stated over a unitary operator $U$ on a Hilbert space $H$, as in (1), and an eigenvalue $\mu$ of $U$, as in (2). What the library proves about these constructions is the theorem itself, with no hypothesis beyond the two definitions.