Visualizing motion in potential wells

Abstract

The concept of potential energy diagrams is of fundamental importance in the study of quantum physics. Yet, students are rarely exposed to this powerful alternative description in introductory classes and, thus, have difficulty comprehending its significance when they encounter it in beginning level quantum courses. We describe a learning unit that incorporates a sequence of computer interfaced experiments using dynamics or air track systems. This unit is designed to make the learning of potential energy diagrams less abstract. Students begin by constructing the harmonic or square well potential diagrams using either the velocity data and assuming conservation of energy or the force-displacement graph for the elastic interaction of an object constrained by springs or bouncing off springy blocks. Then, they investigate the motion of a rider magnet interacting with a configuration of field magnets, directly and plot the potential energy diagrams using a magnetic field sensor. The ease of measurement allows exploring the motion in a large variety of potential shapes in a short duration class.

Surveying Students’ Understanding of Quantum Mechanics

“…Our prior research shows [13] that different instructors categorize a given QM question in different ways so the categorization shown in Table 1 is only one of them that we found convenient. The group "Other" includes questions about the uncertainty principle, the concept of degeneracy in the context of a free particle, and the Ehrenfest theorem that says that the expectation value of a physical observable obeys the classical laws [1][2][3][4][5][6]. Below, we summarize the common difficulties found via the QMS in the sub-category of the time dependence of expectation values.…”

“…In Question (2), the initial state is an energy eigenstate, so the expectation value of any timeindependent operator is time-independent. The most common mistake in Question (2) was that the students believed that the expectation values of position and momentum depend on time in a stationary state. The initial state in Question (23) is a linear superposition of the stationary states 2 / ) (…”

“…In developing good alternative choices for the multiple-choice questions, we took advantage of the prior investigations of advanced students' difficulties with various topics at the junior/senior level QM, e.g., the set of possible wavefunctions, quantum measurements, time dependence of the wavefunction and expectation values [1][2][3][4][5][6]. The alternative choices for each question often had distractors which reflected students' common misconceptions to increase the discriminating properties of the questions.…”

“…The QM formalism is not very intuitive and it is very different from classical mechanics that students are used to from their everyday experiences and previous courses [1][2][3][4][5][6]. Moreover, a good understanding of the QM formalism requires a good grasp of mathematics including linear algebra, differential equations and special functions.…”

Investigating Student Difficulties with Dirac Notation

“…Learning quantum mechanics concepts can be challenging even for advanced undergraduate and graduate students in physics [1][2][3][4][5][6]. We have been examining student difficulties in learning quantum mechanics.…”