# Peer Instruction For Quantum Mechanics

**Guangtian Zhu and Chandralekha Singh**

Quantum Mechanics is one of the most widely taught topics at the college/university level in the physical sciences. Undergraduates see aspects of quantum mechanics in their introductory physics, modern physics, physical chemistry, statistical and thermal physics, and quantum mechanics courses. However, the subject matter makes instruction in quantum mechanics quite challenging—able students constantly struggle to master basic concepts. Effective teaching of quantum mechanics to undergraduate students is an intellectually important task; if it is not done well, students will be handicapped in their later careers.

One way to immerse students actively in the learning process is to have them interact with each other. In introductory physics instruction, integration of peer interaction (PI) with lectures has been popularized by Mazur from Harvard University [1]. In this PI approach, the instructor poses conceptual problems in the form of multiple-choice questions (ConcepTests) to students periodically during the lecture [1]. The students reflect with their peers about the answers to the conceptual multiple-choice questions. After the students discuss their reasoning with peers, they are polled about their choices on the ConcepTests. The focal point of the PI method is the discussion among students, which is based on conceptual questions; the lecture component is limited and intended to supplement the self-directed learning.

To help students develop a solid grasp of the fundamental principles of quantum mechanics, we have been developing and evaluating resource material for “Peer Instruction” in quantum mechanics. The resource material includes ConcepTests for formative assessment, standardized assessment tools for summative assessment and “Reflective Homework” problems for Just-In-Time Teaching (JITT) [2]. Instant feedback on ConcepTests from students provides a “reality check” to the instructors about the extent to which students have actually learned to apply the concepts discussed. This can help instructors adjust the pace of the class appropriately. Peer interaction also keeps students alert during the lectures because they know they must discuss the questions with peers, and it also helps students organize and extend their knowledge. Articulating one's opinion requires attention to logic and organization of thought process. Moreover, there is often a mismatch between the instructor and students' expectations about the level of understanding that is desired related to a concept. Peer instruction helps convey the instructor's expectations explicitly and concretely to the students so that they are on the same wavelength.

The following features of the Peer Instruction material and approach make it particularly suited for the challenging task of teaching quantum mechanics: (1) Formative assessment by polling students about their responses provides feedback to the instructors which is critical for bridging the gap between teaching and learning. (2) The material is being developed based upon prior research by us and others on student difficulties and misconceptions related to quantum mechanics (for example see [3-5]). (3) The material strives to bridge the gap between the abstract quantitative formalism of quantum mechanics and the qualitative understanding necessary to explain and predict diverse physical phenomena. (4) The method consistently keeps students actively engaged in the learning process because not only must the students answer the questions, they must also discuss it with their peers. (5) The method provides a mechanism to convey the goals of the course and the level of understanding that is desired of students. It can also help students monitor their own learning.

The development of ConcepTests goes through an iterative process to ensure that they are pedagogically valuable. Several ConcepTests being developed are related to each other to help students build a robust knowledge structure. Similar to the introductory physics courses, ConcepTests for quantum mechanics can be integrated with lecture after every 10-15 minutes or at the beginning of a lecture to reinforce material from the previous lecture. Posing research-based review questions at the beginning of the lectures ensures that students are the ones who do the thinking, organizing, repairing and extending their knowledge structure.

Figure 1. Simulation following ConcepTests related to time evolution of wavefunction after the measurement of position or energy.

Students are also given ConcepTests in which they discuss with their peers what should happen in a given situation and then they observe simulations to help them visualize the situation [6]. For example, students are asked a sequence of four ConcepTests dealing with the evolution of a wave function after the measurement of different physical observables. These ConcepTests also help students understand the difference between stationary state wave functions and position eigenfunctions, a topic about which students have many common misconceptions [3]. One ConcepTest asks about what happens when we start with a general wave function for a system and perform a measurement of position. The next question asks about the time-development of the wave function after the measurement of position. The next two ConcepTests in the set ask similar questions about the stationary state wave functions. After discussion with the peers, students observe the corresponding simulations and a class discussion ensues. For example, the top row in Figure 1 shows that starting with a general wave function, measurement of position makes the wave function very peaked (theoretically a delta function but in the simulation this collapsed wave function has a width) but then the system evolves with time and the wave function does not remain peaked for all future times (contrary to a common misconception that if the system is in an eigenstate of the position operator, it will remain in that state for all future times). The second row of Figure 1 shows that if the measurement of energy is made and the system “collapses” to an energy eigenstate, the isolated system remains in the same energy eigenstate (the only change is the overall phase that is represented by colors in the plots shown).

Just-in-Time Teaching (JITT) is a strategy for keeping students actively engaged in the learning process via web-based assignments which are typically conceptual [2]. The web-based conceptual assignments are carefully developed and can be submitted by students via course website and the instructor can browse over the student submissions just-in-time to incorporate the needs of the students and adjust the classroom lessons accordingly. Thus, students' responses can be fed back into the in-class discussions. As part of the JITT material, we have been developing “Reflective Homework” problems that strive to bridge the gap between conceptual and quantitative learning. These “Reflective homework” assignments can be effective tools for classroom discussions after students turn them in via the course website.

For example, one set of Reflective Homework questions asks students to compare a stationary state wave function with a non-stationary state wave function which is an equal superposition of the ground state and the first excited state wave functions. Students are asked to compare the probability density, wave function after a time *t*, the expectation value of position and momentum at the initial time and as a function of time. They are asked to perform these comparisons for a one dimensional infinite square well and a simple harmonic oscillator. Students can later discuss in the class why the probability density should not depend on time for the stationary state wave function but it should depend on time for the non-stationary state wave function. Students can also discuss why the same formalism is applicable whether these questions are about the infinite square well or simple harmonic oscillator.

Our preliminary evaluation suggests that the Peer Instruction tools developed so far are helping students, but further development and refinement is necessary. For example, Figure 2 shows students' performance on a study in which one class did not use ConcepTests related to the time-development of wave function whereas two classes did. One of the two classes that used the ConcepTests used the modified version that took into account the feedback from prior year's administration. The performance of the classes that used the ConcepTests is better than the class that covered the same material using more traditional methods. We have been refining the tools based upon the feedback obtained from the students. Surveys administered to students about the effectiveness of Peer Instruction tools suggest that students themselves value these tools.

Figure 2: Post test results for infinite square well.

This work is supported in part by the National Science Foundation.

**References**

[1] E. Mazur, *Peer Instruction: A User's Manual*, Series title: Prentice Hall series in educational innovation. Upper Saddle River, Prentice Hall, N.J., 1997.

[2] G. Novak, A. Garvin, W. Christian, E. Patterson, *Just-in-time teaching: Blending active learning with web technology*, Prentice hall, 2001.

[3] C. Singh, Student understanding of quantum mechanics at the beginning of graduate instruction, *Am. J. Phys.*, **76(3)**, 277-287, (2008); C. Singh, M. Belloni, W. Christian, Improving student's understanding of quantum mechanics, *Physics Today*, 8, 43-49, August (2006); C. Singh, Student understanding of quantum mechanics, Am. J. Phys., **69(8)**, 885-896, (2001).

[4] M. Wittmann, R. Steinberg, E. Redish, Investigating student understanding of quantum physics: spontaneous models of conductivity, *Am. J. Phys.* **70(3)**, 218-226, (2002).

[5]P. Jolly, D. Zollman, S. Rebello, A. Dimitrova, Visualizing potential energy diagrams, *Am. J. Phys.*, **66(1)**, 57, (1998).

[6] For example, see http://www.opensourcephsyics.org, M. Belloni, W. Christian and A. Cox, *Physlet Quantum Physics*, Pearson Prentice Hall, Upper Saddle River, NJ, 2006.

*Guangtian Zhu is a graduate student and Chandralekha Singh is an Associate Professor in the Department of Physics and Astronomy at the University of Pittsburgh.*

Disclaimer - The articles and opinion pieces found in this issue of the APS Forum on Education Newsletter are not peer refereed and represent solely the views of the authors and not necessarily the views of APS.