Deeper look at student learning of quantum mechanics: The case of tunneling

We report on a large-scale study of student learning of quantum tunneling in four traditional and four transformed modern physics courses. In the transformed courses, which were designed to address student difficulties found in previous research, students still struggle with many of the same issues found in other courses. However, the reasons for these difficulties are more subtle, and many new issues are brought to the surface. By explicitly addressing how to build models of wave functions and energy and how to relate these models to real physical systems, we have opened up a floodgate of deep and difficult questions as students struggle to make sense of these models. We conclude that the difficulties found in previous research are the tip of the iceberg, and the real issue at the heart of student difficulties in learning quantum tunneling is the struggle to build the complex models that are implicit in experts’ understanding but often not explicitly addressed in instruction.

Figure 1

(Color) The standard presentation of quantum tunneling: a plane wave tunneling through a square potential barrier. Total energy, potential energy, and the real part of the wave function are all drawn on the same graph, and the real part of the wave function is labeled as simply wave function.

Figure 2

Common student difficulties reported in the literature: incorrectly drawing the real part of the wave function with (a) an offset between the horizontal axes on the left and right sides of the barrier and (b) a smaller wavelength on the right than on the left. These drawings are taken from student responses to an exam question asking students to draw the real part of the wave function, as discussed in Sec. 6. We have observed physics faculty making drawings similar to both (a) and (b), and a popular introductory quantum mechanics textbook contains a figure similar to (b).[42]

Figure 3

(Color) A concept question designed to elicit student confusion between energy and wave function. The correct answer is C but students who do not understand the meaning of superimposing a wave-function graph on an energy graph may be inclined to answer A. When we ask this question in class, it generates a large amount of discussion. While most students (73%–88%, depending on the semester) eventually answer the question correctly, listening in on student discussions reveals that most do not know the answer right away and only figure it out through vigorous debate with their neighbors. Even after discussion, 9%–19% give answer A.

Figure 4

(Color) Physical contexts for (a) a square well and (b) a square barrier. A square well with width $L$ and height $U_0$ represents a wire with length $L$ and work function $U_0$. A square barrier with width $L$ and height $U_0$ represents two long wires with work function $U_0$ separated by an air gap with length $L$.

Figure 5

(Color) The Quantum Tunneling and Wave Packets simulation provides interactive visual models of tunneling of wave packets and plane waves in a variety of physical situations and removes constraints imposed on curriculum by what problems can be easily calculated.

Figure 6

A tunneling question from an early version of the QMCS. This question was developed to test students understanding of tunneling wave functions. It has been removed from the QMCS because interviews suggest that it tests memorization rather than conceptual understanding.

Figure 7

A tunneling question from an early version of the QMCS. This question was developed to explore the belief that energy is lost in tunneling. It has been removed from the QMCS because interviews suggest that responses from students in transformed courses are not necessarily indicative of whether students think energy is lost in tunneling.

Figure 8

A student drawing of a wave function on top of a potential-energy graph.

Figure 9

A reproduction of a graph drawn by a student to represent the potential (solid line) and total (dotted lines) energies of a particle tunneling through a barrier. This student said the particle was losing energy because only a part of it was transmitted.

Figure 10

(Color) An illustration of how to build up a square well by adding up the Coulomb wells of individual atoms (taken from slides used in lecture in Spring 2006, Fall 2006, and Spring 2007). This is a simplified sketch in which the discontinuities of the potential at the ion cores are smoothed over and irrelevant details such as the zero of potential are not labeled.

Figure 11

(Color) Potential-energy graphs for (a) an alpha particle undergoing alpha decay, (b) an electron in a scanning tunneling microscope, and (c) an electron in your finger when you get shocked by a doorknob. Determining how to draw each of these graphs requires many subtle approximations.

Figure 12

(Color) A case where the amplitude of the transmitted wave is higher than the amplitude of the incident wave.

Figure 13

(Color) (a) A representation showing the real and imaginary parts of a wave function and (b) a representation showing the magnitude and phase of a wave function. In interviews we see that students can make sense of representation (a) but not representation (b).