Graduate student understanding of quantum mechanical spin

A framework of cyclic observation and triangulation was applied over a period of 4 years to graduate student difficulties related to quantum spin, in which numerous in-class observations and interviews were used to identify common, persistent difficulties. Written items were iteratively developed over two years to add a quantitative component. Items were administered to graduate students at two collaborating institutions, over three years. We find that students generally obtained scores or correct proportions ranging from 30%-70% on the written items, and answering patterns were similar across all institutions. All items were identified by the course instructors as being relevant to instructional goals of the course. We report on a number of graduate student difficulties with spin, including orthogonality of spin-1/2 states, projections of spin states, spin addition, and exchange symmetry. We briefly discuss possible theoretical frameworks through which to interpret these results.