The discussion of particle-wave dualism still plays an important part in teaching quantum theroy. This surely is for historical reasons, since it was important in grasping the concept of uncertainty. Recent EPR-like experiments, however, indicate new ways of teaching the notions of uncertainty and complementarity. These allow us to achieve a clear understanding without difficult mathematical obstacles and promote also a well-founded discussion of philosophical aspects. The core of the considerations presented here are systems consisting of two-state objects, for example the spin states of photons or electrons.
The discussion of particle-wave dualism still plays an important part in teaching quantum theroy. This surely is for historical reasons, since it was important in grasping the concept of uncertainty. Recent EPR-like experiments, however, indicate new ways of teaching the notions of uncertainty and complementarity. These allow us to achieve a clear understanding without difficult mathematical obstacles and promote also a well-founded discussion of philosophical aspects. The core of the considerations presented here are systems consisting of two-state objects, for example the spin states of photons or electrons.
Complementarity, e.g., the wave–particle duality of nature, lies at the heart of quantum mechanics. It distinguishes the world of quantum phenomena from the reality of classical physics. In the 1920s, quantum theory as we know it today was still new, and examples used to illustrate complementarity emphasized the position (particle-like) and momentum (wave-like) attributes of a quantum mechanical object, be it a photon or a massive particle. This is the historical reason why complementarity is often superficially identified with the so-called wave-particle duality of matter . Complementarity, however, is a more general concept. We say that two observables are complementary if precise knowledge of one of them implies that all possible outcomes of measuring the other one are equally probable. We may illustrate this by two extreme examples. The first example consists of the position and momentum (along one direction) of a particle: if, say, the position is predetermined then the result of a momentum measurement cannot be predicted, all momentum values are equally probable (in a large range). The second extreme involves two orthogonal spin components of a spin- 1/2 particle: if, say, the vertical spin component has a definite value ( up or down ) then upon measuring a horizontal component both values ( left or right , for instance) are found, each with a probability of 50%. Thus, in the microcosmos complete knowledge in the sense of classical physics is not available. The classic example of the merger of wave and particle behavior is provided by Young's double-slit experiment.
Niels Bohr, as is well known, introduced the notion of complementarity into physics, as a fundamental principle of quantum mechanics. It holds that objects have complementary properties that cannot be measured accurately at the same time. For example, the particle and wave aspects of physical objects are such complementary phenomena. Both concepts are borrowed from classical mechanics, where it is impossible to be both, a particle and a wave at the same time. Particle and Wave represent the complementarity of the Discrete and the Continuous. Humans reason by means of concepts (meanings) and language, as well as, by means of logical or arithmetic symbolism. Meanings are continua, whereas logic and arithmetic are based on relations of identity and difference.
To resolve the challenges (paradoxes) attending the genesis of quantum theory, Niels Bohr in 1927 proposed a generalization of the conceptual framework of classical physics, termed complementarity, which posited pairs of classical concepts that could be manifested only in mutually exclusive experimental arrangements with both needed to embrace the full range of physical experience. In this context particle–wave complementarity has encountered two major obstacles; (1) No formulation of it has commanded a general consensus; and (2) empirical evidence abounds that contradicts the claim of mutual exclusivity of particle and wave properties of quantum objects in a single experiment. These problems are rooted in the unrestricted scope of this claim. It is shown that a restricted version of particle–wave complementarity, here called “which value–interference” complementarity, is consistent with observations, is naturally contained in the mathematical formalism of quantum theory and, in fact, holds in any representation of quantum theory defined by the eigenvectors of a complete set of commuting observables, such as the configuration space position (kinematic) variables or the complementary linear momentum (dynamic) variables. Thus, the which value–interference form of particle–wave complementarity exists within kinematic–dynamic complementarity, a “layered” complementarity, so to speak. It is hoped that a consistent and valid exposition of particle–wave complementarity will be helpful to those who find the framework of complementarity useful and appropriate for some purpose.
The wave-particle duality of photons and electrons was thoroughly discussed at a conference held at the Solvay Institute in October 1927, where Albert Einstein and Niels Bohr were the most prominent participants. From the beginning Einstein manifested great hesitation in accepting the renunciation of causal description of physical phenomena introduced by quantum theory. He clearly demonstrated his basic attitude by discussing the example of an atomic object (electron or photon) impinging upon a screen with a single slit (see figure 4). To the left of the screen the object is supposed to have a precisely defined momentum p and hence it is represented as a plane, harmonic wave. When passing through the slit, the plane wave is scattered into a cylindrical wave with an opening angle determined by the wavelength and the slit aperture. Einstein’s problem consisted of the fact that as soon as the object is detected at some point A on a photographic plate placed to the right of the screen, the object-wavecannot be detected in any other part of space whatever, which is directly contradictory to any wave phenomenon in classical physics. But the object-wave cannot be detected in any other part of space whatever, which is directly contradictory to any wave phenomenon is classical physics. But the object-wave just cannot be a classical wave phenomenon because this would imply the possibility of knowing the dynamical state of the object far more precisely than is compatible with the indeterminacy relations.
Opening Remarks.- Imagery, Probability and the Roots of Werner Heisenberg's Uncertainty Principle Paper.- Against An Exposition of Bell's Theorem.- Wave-Particle Duality: A Case Study.- Sein Kann Was Nicht Sein Darf, Or the Prehistory of the EPR, 1909-1935: Einstein's Early Worries About the Quantum Mechanics of Composite Systems.- to Two-Particle Interferometry.- A New View on the Uncertainty Principle.- Measurement Problems in Quantum Field Theory in the 1930's.- On the Collapse of the Wave Function.- Old and New Ideas in the Theory of Quantum Measurement.- Toward a Relativistic Theory of Statevector Reduction.- The Breakdown of the Superposition Principle.- Mathematical and Philosophical Questions in the Theory of Open and Macroscopic Quantum Systems.- Induced Nonlinear Time Evolution of Open Quantum Objects.- Quantum Logic: A Summary of Some Issues.- Remarks on the Measurement of Discrete Physical Quantities.- Some Comments and Reflections.- Invited Lecturers.
Bohr’s complementarity principle was introduced as a qualitative statement about quantum systems, which have properties that are equally real, but mutually exclusive. This principle, together with the uncertainty principle, is at the origin of Quantum Mechanics (QM), following the development of the theory since then. Another intriguing aspect of QM is entanglement, which is a type of correlation that is only possible within the mathematical formalism of QM. Its central importance in the Quantum foundations, as well as its important role in the fields of quantum information and quantum computation, has made the theory of entanglement achieve great progress in recent decades. In this dissertation we discuss the aspects and relations among complementarity relations, uncertainty, and entanglement. Such aspects are also investigated in relativistic scenarios, since the interest on how entanglement behaves in relativistic scenarios has grown more and more. First, by exploring the properties of the density operator, we obtain complementarity relations, using coherence measures well known in the literature. Next, we show that it is possible to obtain complete complementarity relations, i.e., relations that also take into account entanglement. For this, we explore the purity of a multipartite quantum system. In addition, we discuss the relation between complementarity and uncertainty of an observable, since it is possible to decompose uncertainty into its classical and quantum parts. We also obtain complementarity relations for uncertainty. Finally, we study the complete complementary relations and its Lorentz invariance. Although it is known that entanglement entropy does not remain invariant under Lorentz transformations, and neither does the measures that quantifies complementarity, we show that theses measures when taken together, in a complete complementarity relation, are Lorentz invariant.