  A survey was developed to probe student understanding of quantum mechanics at
the beginning of graduate instruction. The survey was administered to 202
physics graduate students enrolled in first-year quantum mechanics courses from
seven universities at the beginning of the first semester. We also conducted
one-on-one interviews with fifteen graduate or advanced undergraduate students
who had just completed a course in which all the content on the survey was
covered. Although students from some universities performed better on average
than others, we found that students share universal difficulties understanding
the concepts of quantum mechanics. The difficulties were often due to
over-generalizations of concepts learned in one context to other contexts where
they are not directly applicable. Difficulties in distinguishing between
closely related concepts and making sense of the formalism of quantum mechanics
were common. The results of this study can sensitize instructors of first-year
graduate quantum physics to some of the difficulties students are likely to
face.
