Examining student ideas about energy measurements on quantum states across undergraduate and graduate levels

[This paper is part of the Focused Collection on Upper Division Physics Courses.] Energy measurements play a fundamental role in the theory of quantum mechanics, yet there is evidence that the underlying concepts are difficult for many students, even after all undergraduate instruction. We present results from an investigation into student ability to determine the possible energies that can be measured for a given wave function and Hamiltonian, to determine the probabilities of each energy measurement and how they depend on time, and to recognize how a measurement of energy affects the state. By analyzing student responses to open-ended questions, we identify five broad, interrelated sets of conceptual and reasoning difficulties related to energy measurements. Data are drawn from sophomore-, junior-, and graduate-level quantum mechanics courses. Particular attention is paid to incorrect ideas that persist across all levels.

Figure 1

A sequence of questions given after all instruction in the sophomore quantum mechanics course. Similar questions have been given to students at the junior and graduate levels.

Figure 2

A sequence of questions that are given after approximately one month of instruction in the junior-level quantum mechanics course at UW. Students are also asked to explain their reasoning after each question.

Figure 3

Question 3 has been given after lecture instruction on perturbation theory at the junior level. Questions testing similar ideas have been asked at different points in the junior year and at the graduate level.