Approaches for improving students' understanding of quantum mechanics

In their article “Improving Students’ Understanding of Quantum Mechanics,” the authors present the following survey question: “By definition, the Hamiltonian acting on any allowed state of the system Ψ will give the same state back, i.e., HΨ = EΨ…. Explain why you agree or disagree.” This wording appears to be ambiguous, since an “allowed state of the system” seems to connote an eigenstate. Perhaps better wording would be “the Hamiltonian acting on any wavefunction Ψ,” or even better, “acting on a wavefunction Ψ in Hilbert space,” rather than referring to the state as “allowed.”