Enhancing student visual understanding of the time evolution of quantum systems

Time dependence is of fundamental importance for the description of quantum systems, but is particularly difficult for students to master. We describe the development and evaluation of a combined simulation-tutorial to support the development of visual understanding of time dependence in quantum mechanics. The associated interactive simulation shows the time dependence of an energy eigenstate and a superposition state, and how the time dependence of the probability density arises from that of the wave function. In order to assess transitions in student thinking, we developed a framework to characterize student responses in terms of real and complex mathematical reasoning and classical and quantum visual reasoning. The results of pre-, mid-, and post-tests indicate that the simulation-tutorial supports the development of visual understanding of time dependence, and that visual reasoning is correlated with improved student performance on a question relating to the time evolution of the wave function and the probability density. The results also indicate that the analogy of a classical standing wave for the infinite well energy eigenfunctions may be problematic in cueing incorrect ideas of time dependence.

Figure 1

A screenshot of the “Time-development of infinite well quantum states” simulation showing a superposition of the ground and first-excited states.

Figure 2

The timeline for the study.

Figure 3

Abbreviated versions of the three pre- and midtest questions administered before and after the activity discussed in Sec. 3a. All three questions also included the instruction: “If it does not depend on time, state so explicitly.” Examples of three different student responses are also shown. Using the framework in Sec. 2d, these responses were coded as incorrect classical visual (1a), incorrect complex mathematics (2a), and correct quantum visual (2b).

Figure 4

The post-test question used as part of the end-of-course exam.

Figure 5

Response transitions from pre- and midtest question 1a. Circles are pretest codes and triangles are the midtest codes for mathematical and visual. Note that the data presented are for the type of reasoning used, not correctness. Response patterns with less than three students are in gray to help illustrate overall trends.

Figure 6

Response transitions from pre- and midtest question 2a.

Figure 7

Response transitions from pre- and midtest question 2b.