Quantum Mechanics Courses Research Articles

Quantum mechanics concept assessment: Development and validation study

As part of an ongoing investigation of students’ learning in first semester upper-division quantum mechanics, we needed a high-quality conceptual assessment instrument for comparing outcomes of different curricular approaches. The process of developing such a tool started with converting a preliminary version of a 14-item open-ended quantum mechanics assessment tool (QMAT) to a multiple-choice (MC) format. Further question refinement, development of effective distractors, adding new questions, and robust statistical analysis has led to a 31-item quantum mechanics concept assessment (QMCA) test. The QMCA is used as post-test only to assess students’ knowledge about five main topics of quantum measurement: the time-independent Schrödinger equation, wave functions and boundary conditions, time evolution, and probability density. During two years of testing and refinement, the QMCA has been given in alpha (N=61) and beta versions (N=263) to students in upper division quantum mechanics courses at 11 different institutions with an average post-test score of 54%. By allowing for comparisons of student learning across different populations and institutions, the QMCA provides instructors and researchers a more standard measure of effectiveness of different curricula or teaching strategies on student conceptual understanding of quantum mechanics. In this paper, we discuss the construction of effective distractors and the use of student interviews and expert feedback to revise and validate both questions and distractors. We include the results of common statistical tests of reliability and validity, which suggest the instrument is presently in a stable, usable, and promising form.3 MoreReceived 22 September 2014DOI:https://doi.org/10.1103/PhysRevSTPER.11.010110This article is available under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.Published by the American Physical Society

The “accidental” degeneracy of the hydrogen atom is no accident

The Schrödinger equation does not account for the 2n2degeneracy of the hydrogen atom, which it dismisses as an “accidental” degeneracy. The factor of “2” in the 2n2degeneracy is well-accounted-for in the relativistic formulation by the two spin states of the electron. The n2degeneracy is nevertheless not quite an “accident”; it is due to the SO(4), rather than SO(3), symmetry of the hydrogen atom. This result is well known, but is inadequately commented upon in most courses in quantum mechanics and atomic physics, leaving the student wondering about the origins of the n2degeneracy of the hydrogen atom. A pedagogical analysis of this interesting aspect, which highlights the fundamental principles of quantum mechanics, is presented in this article. While doing so, not only is the n2degeneracy of the hydrogen atom explained, but its energy spectrum and eigenfunctions are obtained without even using the Schrödinger equation, employing only the fundamental principles of quantum mechanics rather than the Schrödinger equation.

Position–momentum correlations in matter waves double-slit experiment

We present a treatment of the double-slit interference of matter-waves represented by Gaussian wavepackets. The interference pattern is modelled with Green's function propagator which emphasizes the coordinate correlations and phases. We explore the connection between phases and position–momentum correlations in the intensity, visibility and predictability of the wavepacket interference. This formulation will indicate some aspects that can be useful for theoretical and experimental treatment of particle, atom or molecule interferometry and can be discussed in introductory quantum mechanics courses.

Capture of a quantum particle by a deep accelerating potential well

We investigate the capture of a quantum particle in free space by a deep, accelerating potential well. The dependence of the probability of capture on the depth of the well is investigated. We take the potential well to move with an initial speed, v0, and to decelerate with constant acceleration, ac, until the well stops moving. We obtain and explain some unexpected results. Our work may be of interest to instructors of and students in upper-year undergraduate quantum mechanics courses.

The Place of Learning Quantum Theory in Physics Teacher Education: Motivational Elements Arising From the Context

Quantum theory is one of the most successful theories in physics. Because of its abstract, mathematical, and counter-intuitive nature, many students have problems learning the theory, just as teachers experience difficulty in teaching it. Pedagogical research on quantum theory has mainly focused on cognitive issues. However, affective issues about student learning are just as important as cognitive issues. The purpose of this study was to investigate pre-service physics teachers’ motivation towards learning quantum theory by examining motivational constructs such as expectancies, values, ability beliefs, and goals. The participants (n = 6) of this case study were fourth-year pre-service physics teachers at a state university in Turkey. Through semi-structured interviews, the participants were asked seventeen questions that focused on motivational constructs. Analysis of the qualitative data indicated that the instructor, nature of the content, and previous performance were three motivational elements that originated in the context of the quantum mechanics course. Furthermore, these context-dependent elements interacted with the other elements of motivational constructs in both direct and indirect ways. Because unsuccessful learning situations are explained by low motivation, context-dependent affective elements and their interactions should be considered in the teaching and learning of quantum theory.

Identical distinguishable gas particles in the real world

In undergraduate courses in statistical or quantum mechanics, students are exposed to quantum and classical particles that are described as being identical or indistinguishable. It is difficult to appreciate this terminology unless real-world examples are given to illustrate the different possibilities. In particular, while it is easy to think of systems of identical indistinguishable particles or of nonidentical distinguishable particles, it requires more care to identify a gas of identical distinguishable particles.

Elementos de mecânica quântica da partícula na interpretação da onda piloto

Apresenta-se uma introdução elementar à interpretação da mecânica quântica conhecida como "interpretação da onda piloto", inicialmente proposta por Louis de Broglie e posteriormente elaborada por David Bohm. Com o objetivo de adequar o nível do tratamento a um primeiro curso de mecânica quântica, considera-se apenas o caso de uma única partícula, ignorando aspectos associados ao spin. Os assuntos tradicionalmente abordados em tal curso, quais sejam, a partícula livre, a partícula ligada, a reflexão e a transmissão por uma barreira de potencial, o átomo de hidrogênio e o experimento de duas fendas, são discutidos do ponto de vista dessa interpretação, focando em especial a visualização das trajetórias da partícula.

Quantum wells and the generalized uncertainty principle

The finite and infinite square wells are potentials typically discussed in undergraduate quantum mechanics courses. In this paper, we discuss these potentials in the light of the recent studies of the modification of the Heisenberg uncertainty principle into a generalized uncertainty principle (GUP) as a consequence of attempts to formulate a quantum theory of gravity. The fundamental concepts of the minimal length scale and the GUP are discussed and the modified energy eigenvalues and transmission coefficient are derived.

Quantum mechanics teaching resources from the Institute of Physics

In December 2013, the Institute of Physics (IOP) launched a set of freely available resources at quantumphysics.iop.org for the teaching and learning of quantum mechanics. The website includes about 80 short articles written by experts in the field and 17 interactive simulations with accompanying activities for an introductory course in quantum mechanics starting from two-level systems. The articles are arranged according to five themes, including a focus on quantum information, interpretations of quantum mechanics, the mathematical structure of the theory, physics applications and historical experiments. The resources make topics such as entanglement, hidden variables and quantum information theory accessible to introductory-level students. They can be used flexibly for a variety of instructional aims at both the introductory and more advanced level. The website includes links to pre-readings, suggestions for further reading, a glossary of technical terms and allows users to rate their understanding of articles.Sharing of these resources is encouraged, with all usage under the Creative Commons CC BY-NC-ND licence. Solutions to problems and activities are available for instructors by emailing quantumphysics@iop.org. Instructors interested in evaluating these resources with their students in order to help us further develop and optimize the site are requested to contact the corresponding author.

Parametric instability in the Watt governor with periodic loading

In this work we examine a potential source of instability in the Watt governor, which can occur when the governor is subjected to periodic variation of the load torque. Floquet analysis is used to obtain the condition for existence of the unstable solutions, and results are obtained which supplement the usual Maxwell–Vyshnegradskii conditions for governor stability. Pedagogically, this paper is relevant for senior undergraduate students. It presents a detailed yet simple exposition of classical Floquet theory, a subject often ignored in undergraduate education even though the related concept of Bloch's theorem is popular in quantum mechanics courses. At the same time, it touches on certain practical engineering applications of physical concepts, with its discussion on mechanical governors.

Variable phase equation in quantum scattering

This paper presents the derivation and applications of the variable phase equation for single channel quantum scattering. The approach was first presented in 1933 by Morse and Allis and is based on a modification of the Schrödinger equation to a first order differential equation, appropriate to the scattering problem. The dependence of phase shift on angular momentum and energy, together with Levinson's theorem, is discussed. Because the variable phase equation method is easy to program it can be further explored in an introductory quantum mechanics course.

Using Quantum Games To Teach Quantum Mechanics, Part 1

The learning of quantum mechanics is contingent upon an understanding of the physical significance of the mathematics that one must perform. Concepts such as normalization, superposition, interference, probability amplitude, and entanglement can prove challenging for the beginning student. Several class activities that use a nonclassical version of tic-tac-toe are described to introduce several topics in an undergraduate quantum mechanics course. Quantum tic-tac-toe (QTTT) is a quantum analogue of classical tic-tac-toe (CTTT) and can be used to demonstrate the use of superposition in movement, qualitative (and later quantitative) displays of entanglement, and state collapse due to observation. QTTT can be used to aid student understanding in several other topics with the aid of proper discussion.

Chiral symmetries associated with angular momentum

In quantum mechanics courses, symmetries of a physical system are usually introduced as operators which commute with the Hamiltonian. In this paper we will consider chiral symmetries which anticommute with the Hamiltonian. Typically, introductory courses at the (under)graduate level do not discuss these simple, useful and beautiful symmetries at all. The first time a student encounters them is when the Dirac equation is discussed in a course on relativistic quantum mechanics, or when particle–hole symmetry is studied in the context of superconductivity. In this paper, we will show how chiral symmetries can be simply elucidated using the theory of angular momentum, which is taught in virtually all introductory quantum mechanics courses.

Simultaneous Stark and Zeeman effects in atoms with hyperfine structure

A quantum model for calculating the combined Stark and Zeeman effects of simultaneously applied electric and magnetic fields is presented. Our focus here is on atoms with hyperfine structure, such as Cesium. Matrix representations of the Stark, Zeeman, and hyperfine interaction operators are constructed using angular momentum theory and spherical tensor algebra. Matrix elements are evaluated in order to determine the energy-level dependence on the applied fields and reveal intriguing state dynamics in both parallel and orthogonal electric and magnetic fields. The fundamental physics is relevant for an advanced undergraduate or graduate quantum mechanics course.

A new introductory quantum mechanics curriculum

The Institute of Physics New Quantum Curriculum consists of freely available online learning and teaching materials (quantumphysics.iop.org) for a first course in university quantum mechanics starting from two-level systems. This approach immediately immerses students in inherently quantum-mechanical aspects by focusing on experiments that have no classical explanation. It allows from the start a discussion of the interpretive aspects of quantum mechanics and quantum information theory. This paper gives an overview of the resources available from the IOP website. The core text includes around 80 articles which are co-authored by leading experts, arranged in themes, and can be used flexibly to provide a range of alternative approaches. Many of the articles include interactive simulations with accompanying activities and problem sets that can be explored by students to enhance their understanding. Much of the linear algebra needed for this approach is included in the resource. Solutions to activities are available to instructors. The resources can be used in a variety of ways, from being supplemental to existing courses to forming a complete programme.

Science Academies’ Refresher Course in Quantum Mechanics at Physics Department, Indian Institute of Technology, Roorkee, Uttarkhand 28 November–December 12, 2013

Science Academies’ Refresher Course in Quantum Mechanics at Physics Department, Indian Institute of Technology, Roorkee, Uttarkhand 28 November–December 12, 2013

Capture of a quantum particle by a moving trapping potential

We investigate the capture of a quantum particle in free space by a moving trapping potential. The capture is investigated for various initial conditions. We examine the dependence of the probability of capture on the shape and depth of the trapping potential, initial speed and mass of the particle, and other parameters. We take the trapping potential to move with an initial speed v0 and to decelerate with constant acceleration ac until the well stops moving. We study the motion during the time the well is moving and after the well has stopped. We compare the probability of capture to the probability that the particle is in a stationary state (while the well is moving) and the probability the particle is in a bound state (after the well's motion has stopped). Our work could be of interest to instructors and students in upper-year undergraduate quantum mechanics courses.

Science Academies’ Refresher Course in Quantum Mechanics 1–14 May 2013

Science Academies’ Refresher Course in Quantum Mechanics 1–14 May 2013

Quantum mechanical Earth: where orbitals become orbits

Macroscopic objects, although quantum mechanical by nature, conform to Newtonian mechanics under normal observation. According to the quantum mechanical correspondence principle, quantum behavior is indistinguishable from classical behavior in the limit of very large quantum numbers. The purpose of this paper is to provide an example of the correspondence principle appropriate for an undergraduate quantum mechanics course by showing that Earth conforms to quantum mechanics while it orbits the Sun. It is argued that the planarity of Earth's orbit implies that the quantum numbers l and ml are equal and very large. It is additionally shown that for Earth in its orbit, the quantum number n is relatively indistinguishable from l and ml. Furthermore, it is demonstrated that the very large common value for the three quantum numbers implies a spatial probability distribution for Earth that is consistent with its actual distribution. In other words, the wavefunction of Earth, whose orbit will be assumed for simplicity to be circular, is consistent with a planar orbit with all angles within the plane equally probable and with a precisely defined Earth–Sun distance.

Surveying students’ understanding of quantum mechanics in one spatial dimension

We explore the difficulties that advanced undergraduate and graduate students have with non-relativistic quantum mechanics of a single particle in one spatial dimension. To investigate these difficulties we developed a conceptual survey and administered it to more than 200 students at 10 institutions. The issues targeted in the survey include the set of possible wavefunctions, bound and scattering states, quantum measurement, expectation values, the role of the Hamiltonian, and the time-dependence of the wavefunction and expectation values. We find that undergraduate and graduate students have many common difficulties with these concepts and that research-based tutorials and peer-instruction tools can significantly reduce these difficulties. The findings also suggest that graduate quantum mechanics courses may not be effective at helping students to develop a better conceptual understanding of these topics, partly because such courses mainly focus on quantitative assessments.