Students’ difficulties with the Dirac delta function in quantum mechanics

In the context of quantum mechanics, students are often asked to use delta functions to solve problems. Here, we investigate three typical problem-solving processes using delta functions: a delta function potential well problem, a position space delta function problem, and a momentum space delta function problem. We studied students’ solutions in written exams and conducted think-aloud interviews. We use the activation, construction, execution, and reflection framework for our analysis. We find that students encountered various difficulties in solving quantum mechanics problems using delta functions. Common challenges included difficulty with establishing expressions of delta functions for position eigenstates, difficulty with expressing the orthonormality of eigenfunctions with continuous spectra using delta functions, difficulty with boundary conditions for delta function potentials, and difficulty with calculating integrals involved delta functions. In particular, students rarely used effective reflective methods to gain confidence in their solutions. We commonly observed this challenge for students in all questions we investigated. In addition, we compare the similarities and differences in the use of delta functions in electrostatics and in quantum mechanics, and discuss possible explanations for the reasoning mechanisms that cause these difficulties. Finally, we discuss the potential pedagogical implications of our findings.

Figure 1

Problem 1: A particle in a delta function potential well.

Figure 2

Problem 2: A particle in an infinite square well.

Figure 3

Sankey diagram depicting the pathways of students’ solutions as they progressed through the delta function well problem. The diagram shows the flow from the beginning to the different types of solutions that have difficulties at a particular step, where the width of the arrows are proportional to the number of solutions in each flow.

Figure 4

Sankey diagram depicting the pathways of students’ solutions as they progressed through the position space delta function problem.

Figure 5

Sankey diagram depicting the pathways of students’ solutions as they progressed through the momentum space delta function problem.