  Learning advanced physics, in general, is challenging not only due to the
increased mathematical sophistication but also because one must continue to
build on all of the prior knowledge acquired at the introductory and
intermediate levels. In addition, learning quantum mechanics can be especially
challenging because the paradigms of classical mechanics and quantum mechanics
are very different. Here, we review research on student reasoning difficulties
in learning upper-level quantum mechanics and research on students'
problem-solving and metacognitive skills in these courses. Some of these
studies were multi-university investigations. The investigations suggest that
there is large diversity in student performance in upper-level quantum
mechanics regardless of the university, textbook, or instructor and many
students in these courses have not acquired a functional understanding of the
fundamental concepts. The nature of reasoning difficulties in learning quantum
mechanics is analogous to reasoning difficulties found via research in
introductory physics courses. The reasoning difficulties were often due to
over-generalizations of concepts learned in one context to another context
where they are not directly applicable. Reasoning difficulties in
distinguishing between closely related concepts and in making sense of the
formalism of quantum mechanics were common. We conclude with a brief summary of
the research-based approached that take advantage of research on student
difficulties in order to improve teaching and learning of quantum mechanics.
