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.

中文翻译：

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学生在量子力学中偏微分方程的难题

高年级物理学生在量子力学课程中的各种情况下求解偏微分方程。变量分离是解决这些方程式的标准技术。我们调查了学生对期中考试问题的解决方案，并使用了思考型访谈。我们还应用了一个框架，将学生的问题解决过程分为四个阶段：激活，构建，执行和反思。在这里，我们集中讨论了学生在量子力学中解决两个典型问题的过程：两个空间维的能量本征函数问题和一个空间维的时间演化问题。我们发现学生在使用变量分离技术求解这些偏微分方程时遇到了各种困难。常见的困难包括：识别何时分离变量是适当的方法，从原始方程式导出正确的分离方程式，确定分离常数的符号，在使用波动函数的求和形式时证明其合理性以及对其使用有效的反映工具最终解决方案。此外，我们在学生对这两个问题的解决方案中观察到了质和量上不同的错误。最后，我们讨论了我们的发现对指导的可能含义。我们在学生对这两个问题的解决方案中观察到了质和量上不同的错误。最后，我们讨论了我们的发现对指导的可能含义。我们在学生对这两个问题的解决方案中观察到了质和量上不同的错误。最后，我们讨论了我们的发现对指导的可能含义。