  The expectation value of an observable is an important concept in quantum
mechanics. However, upper-level undergraduate and graduate students in physics
have both conceptual and procedural difficulties when determining the
expectation value of physical observables, especially when using Dirac
notation. To investigate these difficulties, we administered free-response and
multiple-choice questions and conducted individual interviews with students.
Here, we discuss the analysis of data on student difficulties when determining
the expectation value.
