Impact of problem context on students’ concept definition of an expectation value

As part of ongoing research on student thinking about quantum mechanical concepts and formalism, we explored how students defined and made sense of expectation values. Previous research has focused on student difficulties when defining the expectation value for a generic operator and found that students conflate other quantum mechanical ideas with expectation values. In this study, we analyzed survey data collected from two universities over a number of years and interviews carried out at two points during the semester. With a focus on underlying student thinking, we used a concept image perspective to categorize students’ concept definitions of the expectation values in the context of measuring spin in a spin-$1/2$ system and in the contexts of measuring energy and position for a particle in an infinite square well potential. Analysis of interview data showed that students invoke many different ideas when explaining their reasoning. The two most common definitions for the expectation value were weighted average and most probable value. In interviews, students’ definitions were influenced by whether the problem context involves continuous or discrete observables.

Figure 1

Survey questions analyzed as part of this project. Correct responses to Q2 are circled.

Figure 2

Graphs that were provided to students for the wave functions ${\psi }_2(x)$ and ${\psi }_3(x)$.

Figure 3

Student explanations for their responses on questions 2a and 2b. Student is from class SF3.

Figure 4

Raife’s depiction of a histogram for spin. He identifies that an equal probability will yield an expectation value of 0 and a higher probability of $-\hslash /2$ will mean the expectation value will be shifted to the left (thick vertical dash).

Figure 5

A histogram Wayne drew to describe the expectation value of energy. He labeled the vertical axis using a square of a function. On the horizontal axis he labeled $E_2$ and $E_3$ and plotted the probabilities. He then drew a tick mark closer to $E_3$ because there is a higher probability of measuring $E_3$.

Figure 6

Leanne’s attempt to make sense of the wave function as a histogram. She draws bars for different points of the wave function, but is not able to connect this to the expectation value of position.