PHYSICAL REVIEW SPECIAL TOPICS - PHYSICS EDUCATION RESEARCH 8, 010117 (2012) Improving students’ understanding of quantum measurement. I. Investigation of difficulties Guangtian Zhu1,2 and Chandralekha Singh2 1School of Education Science, East China Normal University, Shanghai, China, 200062 2Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, Pennsylvania, 15260, USA (Received 3 October 2011; published 19 April 2012) We describe the difficulties that advanced undergraduate and graduate students have with quantum measurement within the standard interpretation of quantum mechanics. We explore the possible origins of these difficulties by analyzing student responses to questions from both surveys and interviews. Results from this research are applied to develop research-based learning tutorials to improve students’ understanding of quantum measurement. DOI: 10.1103/PhysRevSTPER.8.010117 PACS numbers: 01.40.Fk I. INTRODUCTION measured and the measured value is the corresponding eigenvalue. For example, in an ideal measurement, if we Quantum mechanics is a particularly challenging sub- measure the position of a quantum particle in a one- ject, even for advanced students [1–4]. These difficulties dimensional (1D) infinite square well, its wave function have been described in a number of investigations [5–12]. will collapse to a position eigenfunction which is a delta Based on these findings, we are developing a set of function in the position representation. If we measure its research-based learning tools to reduce students’ difficul- energy instead, the wave function of the system will col- ties and help them develop a solid grasp of quantum lapse into an energy eigenfunction, which is a sinusoidal mechanics [13–15]. This paper (Part I) is the first of two function inside the 1D well and goes to zero at the two papers in which we discuss the investigation of students’ boundaries (and is zero everywhere outside the well). difficulties with quantum measurement. This investigation The eigenvalue spectrum of an operator can either be was conducted with the undergraduate and graduate stu- discrete or continuous or a combination of the two. In an N dents at the University of Pittsburgh (Pitt) and other uni- dimensional Hilbert space, an operator Q^ corresponding to versities by administering written tests and by conducting a physical observable Q with discrete eigenvalue spectrum in-depth individual interviews with a subset of students. has N eigenvalues qn and corresponding eigenstates jqni. The development of the research-based learning tools and The state of the system at a given time, jÉðtÞi, can be the preliminary evaluation of students’ performance after written as a linear superposition of a complete set of using the learning tools is described in Part II [16]. eigenstates of jqni. By projecting the wave function of The standard formalism of quantum measurement the system jÉðtÞi at time t onto an eigenstate jqni of the (which is taught to the undergraduate and graduate students ^ 2 operator Q, we can find the probability jhqnjÉðtÞij of universally) is quite different from classical mechanics, obtaining qn when the observable Q is measured at time t. where position and momentum of a particle evolve in a After the measurement of the observable Q, the time deterministic manner based upon the interactions [1]. In evolution of the state of the system, which is an eigenstate quantum mechanics, position, momentum, and other ob- of Q^ right after the measurement, is again governed by the servables are in general not well defined for a given state of TDSE. Right after the measurement of energy, the state of a quantum system. The time-dependent Schro¨dinger equa- the system collapses into the same energy eigenstate, and tion (TDSE) governs the time evolution of the state which the probability density does not change with time (we only can be written as a linear superposition of a complete set of focus on time-independent Hamiltonians) since the only eigenstates of any Hermitian operator corresponding to a physical observable. The state of the system evolves in a change in the wave function with time is an overall time- deterministic manner depending on the Hamiltonian of the dependent phase factor. If the system is initially in an energy eigenstate at time t ¼ 0 and we measure an arbi- system. According to the Copenhagen interpretation, quan- Q t tum measurement would instantaneously collapse the wave trary physical observable after a time , the probability of obtaining an eigenvalue will be time independent since the function (or the state of the system) to an eigenstate of t the operator corresponding to the physical observable system was still in an energy eigenstate at time at the instant the measurement of Q was performed. Therefore, the energy eigenstates are called the stationary states. On the other hand, a measurement of position would Published by the American Physical Society under the terms of the Creative Commons Attribution 3.0 License. Further distri- collapse the system into a position eigenstate at the instant bution of this work must maintain attribution to the author(s) and the measurement is made. However, a position eigenstate is the published article’s title, journal citation, and DOI. a linear superposition of the energy eigenstates and the 1554-9178=12=8(1)=010117(8) 010117-1 Published by the American Physical Society GUANGTIAN ZHU AND CHANDRALEKHA SINGH PHYS. REV. ST PHYS. EDUC. RES. 8, 010117 (2012) different energy eigenstates in the linear superposition will TABLE I. The choice distribution of 10 students answering the evolve with different time-dependent phase factors. question about stationary state and eigenstate after traditional Therefore, the probability density after position measure- instruction. ment will change with time. In this case, the probability of Answers A B (correct) C D E No answer measuring a particular value of energy will be time inde- pendent, but the probability of measuring another physical Percentage 20% 0% 0% 10% 50% 20% observable whose operator does not commute with the Hamiltonian will depend on time. The correct answer is B (3 only). In statement (1), the stationary states should refer to the energy eigenstates only. II. INVESTIGATION OF Q^ STUDENTS’ DIFFICULTIES A complete set of eigenstates of an arbitrary operator cannot be stationary states if Q^ does not commute with the The goal of the investigation was to examine students’ Hamiltonian operator H^ . However, out of 10 students difficulties with quantum measurement after traditional randomly selected from a junior-senior level quantum instruction. The topics included in the investigation, such mechanics class, none of them gave the correct answer as the measurement outcomes, probability of obtaining an after traditional instruction. The distribution of students’ eigenvalue, stationary states, and eigenstates, etc., were all answers is shown in Table I. The most common incorrect covered in the traditional instruction of quantum mechan- choice was E (all of the above). Nearly half of the students ics. The investigation was carried out over several years. thought that all three statements were correct because they For example, students were given the questions as part of had difficulty in differentiating between the related con- the concept tests, quizzes, or tests depending upon the cepts of stationary states and eigenstates of other observ- instructor’s preference. Therefore, the number of students ables. Some students selected choice A (1 only), which is who answered a particular question varies. To simplify the interesting because one may expect that students who mathematics and focus on the concepts related to quantum claimed statement (1) was correct and understood why a measurement, we often used the model of a 1D infinite stationary state is called so may think that statement (2) is square well during the investigation. Both open-ended correct as well. In particular, for students who claimed questions and multiple-choice questions were administered statement (1) is correct, statement (2) may be considered to probe students’ difficulties. We also had informal dis- ‘‘a system in a stationary state stays in that state unless an cussions with a subset of students who took the written test external perturbation is applied,’’ which described the and formally interviewed some students to get a better property of stationary state. However, students who se- understanding of students’ reasoning process. lected choice A did not relate the stationary state with the special nature of the time evolution in that state. A. Difficulty in distinguishing between eigenstates of operators corresponding to different observables B. Difficulty with possible outcomes of a measurement The measurement of a physical observable collapses the and the expectation value of the measurement result wave function of the quantum system into an eigenstate of the corresponding operator. Many students have difficulties The following is an example of a multiple-choice ques- distinguishing between energy eigenstates and the eigen- tion which was administered to investigate students’ under- states of other physical observables. To investigate the standing of the possible outcomes of a measurement for a pervasiveness of this difficulty in distinguishing between given state of a particle in a 1D infinite square well when c ðxÞ c ðxÞ the eigenstates of different physical observables, one of the the measurement is performed. 1 and 2 are the multiple-choice questions administered to the students was ground state and first excited state wave functions. the following: c 1ðxÞþpffiffic 2ðxÞ An electron is in the state given by 2 . Which Choose all of the following statements that are one of the following outcomes could you obtain if you correct: measure the energy of the electron? (1) The stationary states refer to the eigenstates of any A. E1 þ E2 operator corresponding to a physical observable. B. ðE1 þ E2Þ=2 (2) If a system is in an eigenstate of any operator that C. Either E1 or E2 corresponds to a physical observable, it stays in that state D.