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Physical Review Special Topics - Physics Education Research

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

Gina Passante, Paul J. Emigh, Peter S. Shaffer

Published Sep 23, 2015• 63 citations

Abstract

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.

DOI: 10.1103/physrevstper.11.020111

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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.

We investigate the difficulties of advanced undergraduate students toward the end of a full year upper-level quantum mechanics course with concepts related to quantum measurements and time development. Our analysis is based upon a test administered to 89 students from six universities and interviews with 9 students. Strikingly, most students shared the same difficulties despite variations in background, teaching styles, and textbooks. Concepts related to stationary states, eigenstates, and time dependence of expectation values were found to be particularly difficult. An analysis of written tests and interviews suggests that widespread misconceptions originate from an inability to discriminate between related concepts and a tendency to overgeneralize.

Abstract Quantum mechanics (QM) is one of the core subject areas in the undergraduate physics curriculum and many of the advanced level physics courses involve direct or indirect application of the concepts and ideas taught in QM. On the other hand, proper understanding of QM interpretations requires an optimum level of understanding of fundamental concepts in classical physics such as energy, momentum, force and their role in determining motion of the particle. This study is an attempt to explore a group of undergraduate students’ mental models regarding fundamental concepts in classical physics which are actually the stepping stone for understanding and visualisation of QM. The data and analysis presented here elucidate the challenges students face to understand the classical ideas and how that affects their understanding of QM.

Quantum mechanics is challenging even for advanced undergraduate and graduate students.In the Schrödinger representation, the wave function evolves in time according to the time dependent Schrödinger equation.The time dependence of the wave function gives rise to time dependence of the expectation value of observables.We have been exploring the difficulties that advanced undergraduate and graduate students have with time dependence of expectation values in quantum mechanics.We have developed and administered conceptual free response and multiplechoice questions to students to investigate these difficulties.We also interviewed 23 students individually using a thinkaloud protocol to obtain a better understanding of the rationale behind students' written responses.We find that many students struggle with time dependence of expectation values of observables.We discuss some findings.