  We explore the difficulties that advanced undergraduate and graduate students
have with non-relativistic quantum mechanics of a single particle in one
spatial dimension. To investigate these difficulties we developed a conceptual
survey and administered it to more than 200 students at 10 institutions. The
issues targeted in the survey include the set of possible wavefunctions, bound
and scattering states, quantum measurement, expectation values, the role of the
Hamiltonian, and the time-dependence of the wavefunction and expectation
values. We find that undergraduate and graduate students have many common
difficulties with these concepts and that research-based tutorials and
peer-instruction tools can significantly reduce these difficulties. The
findings also suggest that graduate quantum mechanics courses may not be
effective at helping students to develop a better conceptual understanding of
these topics, partly because such courses mainly focus on quantitative
assessments.
