Issues

Editor's note: Such a specialized field as quantum mechanics naturally uses many specialized terms. Words such as “spin,” “degeneracy,” and “quantum jump,” have come out of a variety of contexts, and some have even taken on new meanings in the general public. This paper discusses the origins and linguistic history of quantum terminology. Readers will especially enjoy learning how certain words came to be the dominant, established, terms in the field. For example, why do we talk about “eigenvalues” today when we could be speaking of “proper values” or “eigenwerts?”

Editor's Note: This paper helps instructors celebrate the 100th anniversary of Heisenberg, Born, and Jordan's development of matrix mechanics by sharing how modern graduate-level instruction can be informed by their work.

Editor's Note: Introductory quantum mechanics courses typically discuss only the early history of the quantum era, including the experiments that differed from classical predictions, such as black body radiation spectra, and the rough heuristic rules invented to describe these unexpected results, such as the Rydberg formula for atomic emission spectra and Bohr's correspondence principle. This “old quantum theory” built the conceptual foundations that then enabled the development of several complete mathematical formalisms, with matrix mechanics first emerging in 1925, and wave mechanics in 1926. Introductory courses typically skip this history and present only the modern formalism. Here, the arguments of Heisenberg's “magical” 1925 paper that quickly inspired Born and Jordan's matrix mechanics are used to find the energy spectrum of the quantum harmonic oscillator in a presentation appropriate for upper-level undergraduates.

Editor's Note: Recent progress in laser optics, high-speed electronics, and low-noise photon counting has significantly simplified what used to be either thought experiments or state-of-the-art quantum optics experiments. In this paper, the authors take advantage of this progress to present a Mach–Zehnder interferometer operated at the single-photon counting level in order to introduce undergraduate students to quantum concepts, building upon their previous knowledge of wave optics. The simplicity of the Mach–Zehnder configuration allows for an easy introduction to the formalism of quantum states in two-dimensional Hilbert spaces. Anyone teaching introductory quantum mechanics should have a look at this paper, either to perform such experiments or simply to discuss them in class.

Editor's Note: Richard Feynman famously said of the two-slit experiment that it contained “all of the mystery of quantum mechanics.” While such an experiment has been performed with all kinds of quantum systems over the past century, some of the mystery at the heart of the Einstein–Bohr debates still remains. In this paper, the author revisits the famous argument by Einstein of the movable slits used to monitor the paths of the interfering particles, with a tractable model of path monitoring by Coulomb scattering. This example illustrates the relationship between entanglement and the loss of coherence of the interference fringes. Any physicist with an interest in fundamental quantum mechanics will benefit from this discussion, which has profound consequences in many fields of physics, from photoionization to quantum information.

Editor's Note: This paper provides a carefully designed path for teaching Dirac notation, operators, and the Born rule to introductory physics students. The curriculum includes a series of ten classes with no out-of-class exercises. It could easily be extended to teach further aspects of quantum mechanics. Readers may especially wish to adopt the Bloch Cube that provides a tangible system on which rotations act.

Editor's Note: Students face many challenges when studying quantum mechanics at the first time, including how theory and experiment are related. This article explores how expectation values evolve in time, focusing on the Larmor precession of spin-1/2 systems. This choice of system is ideal for a teaching schedule following a classical, history-based approach or a more recent approach where finite-dimensional spin systems are taught first. The authors have designed activities for students and interesting checks for the teachers. Early tests suggest that the interactive tutorials help students apply quantum mechanical concepts and recall the conclusions long after the activity.

Editor's Note: This paper presents a learning sequence for teaching quantum mechanics concepts to 10–15-year-olds, making concepts tangible through tools such as the phasor wheel, which is used to calculate measurement probabilities when multiple paths interfere. Quantum spin is then taught through analogies with tops that can change their orientation or stack and sum together. The curriculum could be used by other instructors at this level, and the manipulatives may be appreciated even by more advanced students.

Editor's note: Quantum cryptography is not only a major application of quantum technologies but also an easy gateway to entanglement and modern quantum mechanics in the classroom. Several protocols based upon the exchange of single photons can easily be demonstrated with faint laser pulses. This paper presents a simple and cost-effective quantum key distribution setup based on the BB84 protocol that can be used for science outreach, in the undergraduate laboratory, or in class demonstrations.

Editor's Note: We are witnessing the second quantum revolution, which exploits the quantum world to perform computation. The paper shows how concepts such as qubits and superposition, along with innovative approaches to quantum computation, can be explored by secondary school students. Grover's algorithm is presented and applied to a classical satisfiability problem. In addition to the theoretical underpinning, practical codes are also provided, which novices can try out on publicly available quantum computers. This set of codes offers an open-ended exploration of this rapidly evolving field.

Editor's Note: This paper describes the curriculum for a QIST summer program for high school students. In a total of 25 h of instruction, students are introduced to essential classical physics concepts such as waves and polarization, and then to a carefully chosen series of quantum ideas leading through superposition, measurement, entanglement, and Bell's inequality. Quantum computation is introduced using IBM's Quantum Composer, and students learn the process of quantum key distribution. The provided course structure and materials could be used to replicate the instruction in a similar program for high school students, but it can also be used as the basis of more formal instruction at the high school or university level.

Editor's Note: Richard Feynman wrote in 1982 that, if we wanted to simulate physics with computers, then we needed quantum computers. The problem, of course, was that they didn't exist then. But now they do! This paper shows how students can use IBM's free quantum hardware to simulate spin systems that we teach about in class, thus reinforcing what they learn in quantum mechanics while simultaneously gaining exposure to the current capabilities of quantum computing.

Editor's Note: The concept of complex numbers is often introduced in first-year physics classes, such as when studying wave phenomena, and we tell students that they provide a convenient tool to do calculations. It is also emphasized that the same results could be derived using only real quantities. Hence students might ask whether quantum mechanics inherently needs complex numbers. The researchers show a case where a real-valued quantum theory predicts identical correlations to complex-valued quantum mechanics in certain Bell tests. However, a small modification to the tests shows differing predictions of the real and complex descriptions, with specific devices able to falsify the real-valued quantum theory. Using IBM's quantum hardware, the authors highlight that, despite the limitations of noisy quantum devices, they can still effectively address foundational issues in quantum mechanics.