Graduate student misunderstandings of wave functions in an asymmetric well

Quantum mechanics is a notoriously counterintuitive subject within physics and has been the subject of a number of studies at the undergraduate level, and a few pioneering studies at the graduate level. The sketching of wave functions in a confining well is in one sense one of the most basic activities in quantum mechanics. But in another sense, it may be viewed as a rather advanced skill, as it requires the coherent inclusion of a number of details of the wave function, such as wavelength, probability amplitude, and boundary conditions, among others. Although sketching a wave function is not a common activity at the graduate level, a great deal of graduate work is concerned with the aforementioned details, especially boundary conditions. Whether it is seen as a basic skill, or as linked to higher-level understanding, sketching a wave function is an ability that physics graduate students should have. Here, we report on graduate students’ ability to sketch wave functions in an asymmetric potential well. We find that the frequency of many errors is not significantly reduced from pretest to posttest, meaning that many errors persist through to the end of graduate quantum mechanics instruction. We find that only 5% of graduate students tested can sketch the 2nd excited state wave function without errors. We include quantitative and diagrammatic descriptions of student errors covering a broader range of misunderstandings than has previously been identified, and we include interview information that speaks to the persistence of some of the errors.

Figure 1

Question used in this study. Part (a) was slightly different for the first cohort’s pretest, in which the ground state energy was not specified to be above $V_a$. The change was made for ease of grading and did not significantly affect student performance.

Figure 2

Examples of acceptable answers to the question parts (a) and (b).

Figure 3

Pre- and postcomparison of student errors on the drawing of ground state wave functions and 2nd excited state wave functions. In general, errors are not mutually exclusive. The bars for the categories “Sym.” and “Wrong sym.” are not shaded in for the ground state, indicating that they were not considered in the grading of student responses for the ground state. Error bars represent binomial error, treating each student’s answer as a separate Bernoulli trial, and treating pretest groups and posttest groups entirely separately. The data correspond to all pretest data and all posttest data, not restructured to within-student paired pretests and posttests.