The expectation value of an observable is an important concept in quantum mechanics since
measurement outcomes are, in general, probabilistic and we only have information about the
probability distribution of measurement outcomes in a given quantum state of a system. However, we
find that upper-level undergraduate and PhD students in physics have both conceptual and procedural
difficulties when determining the expectation value of a physical observable in a given quantum
state in terms of the eigenstates and eigenvalues of the corresponding operator, especially when
using Dirac notation. Here we first describe the difficulties that these students have with
determining the expectation value of an observable in Dirac notation. We then discuss how the
difficulties found via student responses to written surveys and individual interviews were used as a
guide in the development of a quantum interactive learning tutorial (QuILT) to help students develop
a good grasp of the expectation …