Students’ difficulties with partial differential equations in quantum mechanics

Upper-division physics students solve partial differential equations in various contexts in quantum mechanics courses. Separation of variables is a standard technique to solve these equations. We investigated students’ solutions to midterm exam questions and utilized think-aloud interviews. We also applied a framework that organizes students’ problem-solving process into four stages: activate, construct, execute, and reflect. Here we focused on students’ problem-solving process for two typical problems in the context of quantum mechanics: an energy eigenfunction problem in two spatial dimensions and a time evolution problem in one spatial dimension. We found that the students encountered various difficulties when they used the separation of variables technique to solve these partial differential equations. Common difficulties included recognizing when separation of variables is the appropriate method, deriving the correct separated equations from the original equation, deciding the signs of the separation constants, justifying when using the summation form of the wave function, and using an effective reflecting tool for their final solutions. In addition, we observed qualitatively and quantitatively different errors in students’ solutions to the two problems. Finally, we discussed the possible implications of our findings for instruction.

Figure 1

An example of the exam problem about the energy eigenfunction.

Figure 2

An example of the exam problem about the time evolution of the wave function.

Figure 3

The correctness of each step as students progressed through the energy eigenvalue problem. In the graph, the histogram denotes the number of correct solutions, and the dots label the correct percentage with respect to the total solutions.

Figure 4

The correctness of each step as students progressed through the time evolution problem. In the diagram, the histogram denotes the number of correct solutions, and the dots label the correct percentage with respect to the total solutions.