An earlier paper2 introduces quantum physics by means of four experiments: Youngs double-slit interference experiment using (1) a light beam, (2) a low-intensity light beam with time-lapse photography, (3) an electron beam, and (4) a low-intensity electron beam with time-lapse photography. It's ironic that, although these experiments demonstrate most of the quantum fundamentals, conventional pedagogy stresses their difficult and paradoxical nature. These paradoxes (i.e., logical contradictions) vanish, and understanding becomes simpler, if one takes seriously the fact that quantum mechanics is the nonrelativistic limit of our most accurate physical theory, namely quantum field theory, and treats the Schroedinger wave function, as well as the electromagnetic field, as quantized fields.2 Both the Schroedinger field, or “matter field,” and the EM field are made of “quanta”—spatially extended but energetically discrete chunks or bundles of energy. Each quantum comes nonlocally from the entire space-filling field and interacts with macroscopic systems such as the viewing screen by collapsing into an atom instantaneously and randomly in accordance with the probability amplitude specified by the field. Thus, uncertainty and nonlocality are inherent in quantum physics. This paper is about quantum uncertainty. A planned later paper will take up quantum nonlocality.
An earlier paper2 introduces quantum physics by means of four experiments: Youngs double-slit interference experiment using (1) a light beam, (2) a low-intensity light beam with time-lapse photography, (3) an electron beam, and (4) a low-intensity electron beam with time-lapse photography. It's ironic that, although these experiments demonstrate most of the quantum fundamentals, conventional pedagogy stresses their difficult and paradoxical nature. These paradoxes (i.e., logical contradictions) vanish, and understanding becomes simpler, if one takes seriously the fact that quantum mechanics is the nonrelativistic limit of our most accurate physical theory, namely quantum field theory, and treats the Schroedinger wave function, as well as the electromagnetic field, as quantized fields.2 Both the Schroedinger field, or “matter field,” and the EM field are made of “quanta”—spatially extended but energetically discrete chunks or bundles of energy. Each quantum comes nonlocally from the entire space-filling field and interacts with macroscopic systems such as the viewing screen by collapsing into an atom instantaneously and randomly in accordance with the probability amplitude specified by the field. Thus, uncertainty and nonlocality are inherent in quantum physics. This paper is about quantum uncertainty. A planned later paper will take up quantum nonlocality.
The nature of the problem The quantum world is inexplicable in classical terms. The predictions pertaining to the interaction of matter and light embodied in Newton's laws for the motion of particles and Maxwell's equations governing the propagation of electromagnetic fields are in flat contradiction with the experimental facts at the microscopic scale. A key feature of quantum effects is their apparent indeterminism, that individual atomic events are unpredictable, uncontrollable and literally seem to have no cause. Regularities emerge only when one considers a large ensemble of such events. This indeed is generally considered to constitute the heart of the conceptual problem posed by quantum phenomena, necessitating a fundamental revision of the deterministic classical world view. Some of the principal phenomena, among those discussed in this book, are as follows: ( a ) Self-interference: a beam of electrons sent one at a time through a barrier containing two apertures builds up through a series of localized detection events on a screen an interference pattern characteristic of waves (cf. the frontispiece and §5.1). ( b ) Tunnelling: an α-particle trapped in a nucleus can pass through a potential barrier in a manner forbidden to a classical particle (§5.3). ( c ) The stability of matter: atoms and molecules are found to exist only in certain discrete, or ‘stationary’, energy states. For this reason they do not ‘collapse’, the result predicted by classical electrodynamics. During transitions between stationary states (quantum jumps) an atom exchanges a discrete quantity of energy with the electromagnetic field (§§4.5, 7.6). […]
Nonlocality arises from the unified “all or nothing” interactions of a spatially extended field quantum such as a photon or an electron.2 In the double-slit experiment with light, for example, each photon comes through both slits and arrives at the viewing screen as an extended but unified energy bundle or “field quantum.” When the photon interacts (randomly2) with the screen, field quantization requires it to alter its state instantaneously rather than gradually. Thus if the photon is absorbed, it must vanish or “collapse” nonlocally and instantaneously across a macroscopic portion of the screen, even across many kilometers in the case of interference patterns of light from a small distant star. The interaction instantly transfers the photons energy to a single atom of the screen. But a quantized field can contain any whole number of “excitations” (particles such as photons or electrons). If a single field quantum contains, say, two excitations, then generally the unified all-or-nothing character of quanta implies that any interaction of one excitation must also instantaneously affect the other excitation, regardless of the distance between them. The particles are then said to be “entangled” (see the “Background” section for a more precise definition of this term). Particles can become entangled by being created together in a single microscopic process, or by interacting with each other. Quantum entanglement is at least as fundamental as quantum uncertainty but is seldom mentioned in physics courses, although it has received broad attention recently in a wonderful book by Louisa Gilder.3 A recent paper in this journal presents entanglement in a manner that is useful for high school and college physics teachers.4 This paper builds on that presentation and looks at a different, more intuitive entanglement experiment that should be accessible to both scientists and nonscientists.
There are two ways to teach quantum mechanics. The first way – which for most physicists today is still the only way – follows the historical order in which the ideas were discovered. So, you start with classical mechanics and electrodynamics, solving lots of grueling differential equations at every step. Then, you learn about the “blackbody paradox” and various strange experimental results, and the great crisis these things posed for physics. Next, you learn a complicated patchwork of ideas that physicists invented between 1900 and 1926 to try to make the crisis go away. Then, if you're lucky, after years of study, you finally get around to the central conceptual point: that nature is described not by probabilities (which are always nonnegative), but by numbers called amplitudes that can be positive, negative, or even complex. Look, obviously the physicists had their reasons for teaching quantum mechanics that way, and it works great for a certain kind of student. But the “historical” approach also has disadvantages, which in the quantum information age are becoming increasingly apparent. For example, I’ve had experts in quantum field theory – people who've spent years calculating path integrals of mind-boggling complexity – ask me to explain the Bell inequality to them , or other simple conceptual things like Grover's algorithm. I felt as if Andrew Wiles had asked me to explain the Pythagorean Theorem.
Recent advances in quantum technology – from quantum computers and simulators to communication and metrology – have not only opened up a whole new world of applications but also changed the understanding of quantum theory itself. This text introduces quantum theory entirely from this new perspective. It does away with the traditional approach to quantum theory as a theory of microscopic matter, and focuses instead on quantum theory as a framework for information processing. Accordingly, the emphasis is on concepts like measurement, probability, statistical correlations, and transformations, rather than waves and particles. The text begins with experimental evidence that forces one to abandon the classical description and to re-examine such basic notions as measurement, probability, and state. Thorough investigation of these concepts leads to the alternative framework of quantum theory. The requisite mathematics is developed and linked to its operational meaning. This part of the text culminates in an exploration of some of the most vexing issues of quantum theory, regarding locality, non-contextuality, and realism. The second half of the text explains how the peculiar features of quantum theory are harnessed to tackle information processing tasks that are intractable or even impossible classically. It provides the tools for understanding and designing the pertinent protocols, and discusses a range of examples representative of current quantum technology.
Quantum interference is traditionally interpreted as evidence of intrinsic wave–particle duality, invoking self-interference and wavefunction superposition. Here, we propose an alternative causal framework in which interference arises from interactions between a classical electron and a quantized electromagnetic field. In this picture, electrons follow definite, localized trajectories and produce discrete detection events, while the observed interference pattern emerges statistically from stochastic, nonlocal momentum transfers mediated by quantized field modes. Numerical simulations of the double-slit experiment show that dot-like interference fringes arise without invoking wavefunction collapse or intrinsic matter waves. Doubling the electron mass narrows the fringe spacing in agreement with de Broglie scaling, indicating that the effective wavelength emerges dynamically from particle–field coupling rather than representing an inherent property of the particle. The formulation is based on a rigorously defined Hamiltonian and derived equations describing electromagnetic coupling between particles and the slit environment. Although neutral particles such as neutrons or atoms carry no net charge monopoles, they possess electric or magnetic dipole and higher multipole moments that enable electromagnetic interactions with the slit apparatus, allowing discrete momentum transfer through the quantized field. Our simulations show that interference fringes arise from intermittent singleparticle detection events caused by stochastic momentum impulses from quantized electromagnetic modes. These results suggest a reinterpretation of wave–particle duality: the apparent wave-like behavior of matter may arise from interaction with the inherently wave-like quantized electromagnetic field—whose quanta are photons, the massless U(1)-gauge bosons—rather than from an intrinsic wave nature of the particles themselves. Our analysis is rigorous and is not speculative, because the proposed Hamiltonian for this system is physical. Within this nonlocal hidden-variable framework, the dot-like interference pattern observed in the double-slit experiment emerges from individual electrons traversing one slit at a time while receiving discrete momentum impulses from stochastic, spatially extended quantized electromagnetic fields.
I propose a conceptual change in the way we teach nonrelativistic quantum physics in introductory survey courses and general modern physics courses. Traditional instruction treats radiation as a quantized electromagnetic wave that, because it is quantized, is observable only as discrete field quanta, while treating matter as particles that are accompanied by a wave function. In other words, traditional instruction views radiation as fundamentally a field phenomenon, and matter as fundamentally a particle phenomenon. But quantum field theory has a more unified view, according to which both radiation and matter are continuous fields while both photons and material particles are quanta of these fields. The quantum field theory view of radiation and matter clarifies particle identity issues, dispels students’ Newtonian misconceptions about matter, arguably resolves the wave-particle paradox, is the accepted view of contemporary physics, and might be the simplest and most effective teaching approach for all students. I propose that we make this field-theory viewpoint the conceptual basis for teaching non-relativistic quantum physics.