The Extended Bloch Representation of Quantum Mechanics. Explaining Superposition, Interference and Entanglement Preprint, 2015 - PDF (with Diederik Aerts)

The extended Bloch representation of quantum mechanics was recently derived to offer a (hidden-measurement) solution to the measurement problem. In this article we use it to investigate the geometry of superposition and entangled states, explaining the interference effects, and the entanglement correlations, in terms of the different orientations that a state-vector can take within the generalized Bloch sphere. We also introduce a tensorial determination of the generators of SU(N), particularly suitable to describe multipartite systems, from the viewpoint of the sub-entities. We then use it to show that non-product states admit a general description in which the sub-entities can always remain in well-defined states, even when they are entangled. Therefore, the completed version of quantum mechanics provided by the extended Bloch representation, in which the density operators are also representative of pure states, allows to solve not only the well-known measurement problem, but also the lesser-known entanglement problem. This because we no longer need to give up the general physical principle saying that a composite entity exists, and therefore is in a pure state, if and only if its components also exist, and therefore are in well-defined pure states.

Solving the Entanglement Paradox Preprint, 2015 - PDF (with Diederik Aerts)

Entangled states are in conflict with the general physical principle saying that a composite entity exists if and only if its components also exist, and therefore are in pure states. To solve this paradox one has to complete the standard formulation of quantum mechanics, by adding more pure states. We show that this can be done, in a consistent way, by using the extended Bloch representation of quantum mechanics, recently introduced to solve the measurement problem, which therefore can also be exploited to restore the full intelligibility of entangled states.

Do spins have directions? Preprint, 2015 - PDF (with Diederik Aerts)

The standard Bloch sphere representation has been recently generalized to describe not only systems of arbitrary dimension, but also their measurements, in what has been called the \emph{extended Bloch representation of quantum mechanics}. This model, which offers a solution to the longstanding measurement problem, is based on the \emph{hidden-measurement interpretation of quantum mechanics}, according to which the Born rule results from our lack of knowledge of the measurement interaction that each time is actualized between the measuring apparatus and the measured entity. In this article we present the extended Bloch model and use it to investigate, more specifically, the nature of the quantum spin entities and of their relation to our three-dimensional Euclidean theater. Our analysis shows that spin eigenstates cannot generally be associated with directions in the Euclidean space, but only with generalized directions in the Blochean space, which apart from the special case of spin one-half entities, is a space of higher dimensionality. Accordingly, spin entities have to be considered as genuine non-spatial entities. We also show, however, that specific vectors can be identified in the Blochean theater that are isomorphic to the Euclidean space directions, and therefore representative of them, and that spin eigenstates always have a predetermined orientation with respect to them. We use the details of our results to put forward a new view of realism, that we call multiplex realism, providing a specific framework with which to interpret the human observations and understanding of the component parts of the world. Elements of reality can be represented in different theaters, one being our customary Euclidean space, and another one the quantum realm, revealed to us through our sophisticated experiments, whose elements of reality, in the quantum jargon, are the eigenvalues and eigenstates. Our understanding of the component parts of the world can then be guided by looking for the possible connections, in the form of partial morphisms, between the different representations, which is precisely what we do in this article with regard to spin entities.

Many-Measurements or Many-Worlds? A Dialogue To appear in: Foundations of Science, 2014 - PDF (with Diederik Aerts)

Many advocates of the Everettian interpretation consider that their approach is the only one taking quantum mechanics really seriously, and that by doing so a fantastic scenario for our reality can be deduced, made of infinitely many and continuously branching parallel worlds. In this article, written in the form of a dialogue, we suggest that quantum mechanics can be taken even more seriously, if the 'many-worlds' view is replaced by a 'many-measurements' view. In this way, not only the Born rule can be derived, thus solving the measurement problem, but a one-world 'non-spatial' reality can also be deduced, providing an even more fantastic scenario than that of the multiverse.

The Extended Bloch Representation of Quantum Mechanics and the Hidden-Measurement Solution to the Measurement Problem Annals of Physics 351, Pages 975–1025 (2014) - PDF (open access) (with Diederik Aerts)

A generalized Poincare-Bloch sphere, in which the states of a quantum entity of arbitrary dimension are geometrically represented, is investigated and further extended, to also incorporate the measurements. This extended representation constitutes a general solution to the measurement problem, inasmuch it allows to derive the Born rule as an average over hidden-variables, describing not the state of the quantum entity, but its interaction with the measuring system. According to this modelization, a quantum measurement is to be understood, in general, as a tripartite process, formed by an initial deterministic decoherence-like process, a subsequent indeterministic collapse-like process, and a final deterministic purification-like process. We also show that quantum probabilities can be generally interpreted as the probabilities of a first-order non-classical theory, describing situations of maximal lack of knowledge regarding the process of actualization of potential interactions, during a measurement.

Solving the Hard Problem of Bertrand's Paradox J. Math. Phys. 55, 083503 (2014) - PDF (with Diederik Aerts)

Bertrand's paradox is a famous problem of probability theory, pointing to a possible inconsistency in Laplace's principle of insufficient reason. In this article we show that Bertrand's paradox contains two different problems: an "easy" problem and a "hard" problem. The easy problem can be solved by formulating Bertrand's question in sufficiently precise terms, so allowing for a non ambiguous modelization of the entity subjected to the randomization. We then show that once the easy problem is settled, also the hard problem becomes solvable, provided Laplace's principle of insufficient reason is applied not to the outcomes of the experiment, but to the different possible "ways of selecting" an interaction between the entity under investigation and that producing the randomization. This consists in evaluating a huge average over all possible "ways of selecting" an interaction, which we call a 'universal average'. Following a strategy similar to that used in the definition of the Wiener measure, we calculate such universal average and therefore solve the hard problem of Bertrand's paradox. The link between Bertrand's problem of probability theory and the measurement problem of quantum mechanics is also briefly discussed.

The unreasonable success of quantum probability II: Quantum measurements as universal measurements Preprint, 2014 - PDF (to appear in: J. Math. Psychology) (with Diederik Aerts)

In the first part of this two-part article, we have introduced and analyzed a multidimensional model, called the rho-model, able to describe general quantum-like measurements with an arbitrary number of outcomes, and we have used it as a general theoretical framework to study the most general possible condition of lack of knowledge in a measurement, so defining what we have called a universal measurement. In this second part, we present the formal proof that universal measurements, which are averages over all possible forms of fluctuations, produce the same probabilities as measurements characterized by uniform fluctuations on the measurement situation. Since quantum probabilities can be proven to arise from the presence of such uniform fluctuations, we have proven that they can be interpreted as the probabilities of a first-order non-classical theory, describing situations in which the experimenter lacks complete knowledge about the nature of the interaction between the measuring apparatus and the entity under investigation. This same explanation can be applied - mutatis mutandis - to the case of cognitive measurements, made by human subjects on conceptual entities, or in decision processes, although it is not necessarily the case that the structure of the set of states would be in this case strictly Hilbertian. We also show that universal measurements correspond to maximally robust descriptions of indeterministic reproducible experiments, and since quantum measurements can also be shown to be maximally robust, this adds plausibility to their interpretation as universal measurements, and provides a further element of explanation for the great success of the quantum statistics in the description of a large class of phenomena.

The unreasonable success of quantum probability I: Quantum measurements as uniform fluctuations Journal Mathematical Psychology, March 31, 2015 - open access (with Diederik Aerts)

We introduce a model which allows to represent the probabilities associated with an arbitrary measurement situation as it appears in different domains of science - form cognitive science to physics - and use it to explain the emergence of quantum probabilities (the Born rule) as uniform fluctuations on this measurement situation. The model exploits the geometry of simplexes to represent the states both of the system and the measuring apparatus, in a way that the measurement probabilities can be derived as the Lebesgue measure of suitably defined convex subregions of the simplex under consideration. Although this Lebesgue-model is an abstract construct, it admits physical realizations. In this article we consider a very simple and evocative one, using a material point particle which is acted upon by special elastic membranes, which by breaking and collapsing are able to produce the different possible outcomes. This easy to visualize mechanical realization allows one to gain considerable insight into the possible hidden structure of a measurement process, be it from a measurement associated with a situation in cognitive science or in physics, or in any other domain. We also show that the Lebesgue-model can be further generalized into a model describing conditions of lack of knowledge generated by non-uniform fluctuations, which we call the rho-model. In this more general framework, which is more suitable to describe typical experiments in cognitive science, we define and motivate a notion of universal measurement, describing the most general possible condition of lack of knowledge in a measurement, emphasizing that the uniform fluctuations characterizing quantum measurements can also be understood as an average over all possible forms of non-uniform fluctuations which can be actualized in a measurement context. This means that the Born rule of quantum mechanics can be understood as a first order approximation of a more general non-uniform theory, thus explaining part of the great success of quantum probability in the description of different domains of reality. And more specifically, also providing a possible explanation for the success of quantum cognition, a research field in cognitive science employing the quantum formalism as a modeling tool. This is the first part of a two-part article. In the second part, the proof of the equivalence between universal measurements and uniform measurements, and its significance for quantum theory as a first order approximation, is given and further analyzed.

A remark on the role of indeterminism and non-locality in the violation of Bell's inequality Preprint - PDF Annals of Physics, Volume 342, March 2014, Pages 133–142

Some years ago Aerts et al. presented a macroscopic model in which the amount of non-locality and indeterminism could be continuously varied, and used it to show that by increasing non-locality one increases, as expected, the degree of violation of Bell's inequality (BI), whereas, more surprisingly, by increasing indeterminism one decreases the degree of the violation of BI. In this note we propose a different macroscopic model in which the amount of non-locality and indeterminism can also be parameterized, and therefore varied, and we find that, in accordance with the model of Aerts et al., an increase of non-locality produces a stronger violation of BI. However, differently from their model, we also find that, depending on the initial state in which the system is prepared, an increase of indeterminism can either strengthen or weaken the degree of violation of BI.

Quantum dice Preprint - PDF Annals of Physics, Volume 336, September 2013, Pages 56–75

In a letter to Born, Einstein wrote: "Quantum mechanics is certainly imposing. But an inner voice tells me that it is not yet the real thing. The theory says a lot, but does not really bring us any closer to the secret of the old one. I, at any rate, am convinced that He does not throw dice." In this paper we take seriously Einstein's famous metaphor, and show that we can gain considerable insight into quantum mechanics by doing something as simple as rolling dice. More precisely, we show how to perform measurements on a single die, to create typical quantum interference effects, and how to connect (entangle) two identical dice, to maximally violate Bell's inequality.

God may not play dice, but human observers surely do Preprint - PDF Foundations of Science: Volume 20, Issue 1 (2015), Page 77-105

We investigate indeterminism in physical observations. For this, we introduce a distinction between genuinely indeterministic (creation-1 and discovery-1) observational processes, and fully deterministic (creation-2 and discovery-2) observational processes, which we analyze by drawing a parallel between the localization properties of microscopic entities, like electrons, and the lateralization properties of macroscopic entities, like simple elastic bands. We show that by removing the randomness incorporated in certain of our observational processes, acquiring over them a better control, we also alter these processes in such a radical way that in the end they do not correspond anymore to the observation of the same property. We thus conclude that a certain amount of indeterminism must be accepted and welcomed in our physical observations, as we cannot get rid of it without also diminishing our discriminative power. We also provide in our analysis some elements of clarification regarding the non-spatial nature of microscopic entities, which we illustrate by using an analogy with the process of objectification of human concepts. Finally, the important notion of relational properties is properly defined, and the role played by indeterminism in their characterization clarified.

Quantum "fields" are not fields. Comment on "There are no particles, there are only fields," by Art Hobson Preprint - PDF Am. J. Phys. 81 , 707 (2013)

We comment on a recent paper by Hobson, explaining that quantum "fields" are no more fields than quantum "particles" are particles, so that the replacement of a particle ontology by an all-field ontology cannot solve the typical interpretational problems of quantum mechanics.

Quantum measurements are physical processes. Comment on "Consciousness and the double-slit interference pattern: Six experiments," By Dean Radin et al. Reprint - PDF Physics Essays, March issue, Vol. 26, No. 1 (2013)

The validity of the assertion that some recent double-slit interference experiments, conducted by Radin et al., would have tested the possible role of the experimenter's mind in the collapse of the quantum wave function, is questioned. It is emphasized that quantum mechanics doesn't need any psychophysical ingredient to explain the measurement processes, and therefore parapsychologists shouldn't resort to the latter to support the possibility of psychokinesis, but search for more convincing explanations.

Using simple elastic bands to explain quantum mechanics: a conceptual review of two of Aerts' machine-models Preprint - PDF Central European Journal of Physics, Volume 11, Issue 2, pp 147-161 (2013)

From the beginning of his research, the Belgian physicist Diederik Aerts has shown great creativity in inventing a number of concrete machine-models that have played an important role in the development of general mathematical and conceptual formalisms for the description of the physical reality. These models can also be used to demystify much of the strangeness in the behavior of quantum entities, by allowing to have a peek at what's going on - in structural terms - behind the "quantum scenes," during a measurement. In this author's view, the importance of these machine-models, and of the approaches they have originated, have been so far seriously underappreciated by the physics community, despite their success in clarifying many challenges of quantum physics. To fill this gap, and encourage a greater number of researchers to take cognizance of the important work of so-called Geneva-Brussels school, we describe and analyze in this paper two of Aerts' historical machine-models, whose operations are based on simple breakable elastic bands. The first one, called the spin quantum-machine, is able to replicate the quantum probabilities associated with the spin measurement of a spin-1/2 entity. The second one, called the \emph{connected vessels of water model} (of which we shall present here an alternative version based on elastics) is able to violate Bell's inequality, as coincidence measurements on entangled states can do.

The observer effect Preprint - PDF Foundations of Science, June issue, Volume 18, Issue 2, pp 213-243 (2013)

Founding our analysis on the Geneva-Brussels approach to the foundations of physics, we provide a clarification and classification of the key concept of observation. An entity can be observed with or without a scope. In the second case, the observation is a purely non-invasive discovery process; in the first case, it is a purely invasive process, which can involve either creation or destruction aspects. An entity can also be observed with or without a full control over the observational process. In the latter case, the observation can be described by a symmetry breaking mechanism, through which a specific deterministic observational process is selected among a number of potential ones, as explained in Aerts' hidden measurement approach. This is what is called a product test, or product observation, whose consequences are that outcomes can only be predicted in probabilistic terms, as it is the case in typical quantum measurements. We also show that observations can be about intrinsic (stable) properties of the observed entity, or about relational (ephemeral) properties between the observer and observed entities; also, they can be about intermediate properties, neither purely classical, nor purely quantum. Our analysis allows us to propose a general conceptual characterization of quantum measurements, as observational processes involving three aspects: (1) product observations, (2) pure creation aspects and (3) ephemeral relational properties. We also discuss the important concept of non-spatiality and emphasize some of the differences and similarities between quantum and classical/relativistic observations.

The delta-quantum machine, the k-model, and the non-ordinary spatiality of quantum entities Preprint - PDF Foudations of Science, March issue, Volume 18, Issue 1, pp 11-41 (2013)

The purpose of this article is threefold. Firstly, it aims to present, in an educational and non-technical fashion, the main ideas at the basis of Aerts' creation-discovery view and hidden measurement approach: a fundamental explanatory framework whose importance, in this author's view, has been seriously underappreciated by the physics community, despite its success in clarifying many conceptual challenges of quantum physics. Secondly, it aims to introduce a new quantum-machine - that we call the delta-quantum-machine - which is able to reproduce the transmission and reflection probabilities of a one-dimensional quantum scattering process by a Dirac delta-function potential. The machine is used not only to demonstrate the pertinence of the above mentioned explanatory framework, in the general description of physical systems, but also to illustrate (in the spirit of Aerts' epsilon-model) the origin of classical and quantum structures, by revealing the existence of processes which are neither classical nor quantum, but irreducibly intermediate. We do this by explicitly introducing what we call the k-model and by proving that its processes cannot be modelized by a classical or quantum scattering system. The third purpose of this work is to exploit the powerful metaphor provided by our quantum-machine, to investigate the intimate relation between the concept of potentiality and the notion of non-spatiality, that we characterize in precise terms, introducing for this the new concept of weak actuality.

Time-delay of classical and quantum scattering processes: a conceptual overview and a general definition Preprint - PDF Central European Journal of Physics, Vol. 10, No. 2, pp. 282–319 (2012)

We present a step by step introduction to the notion of time-delay inclassical and quantum mechanics, with the aim of clarifying its foundation at aconceptual level. In doing so, we motivate the introduction of the concepts of"fuzzy" and "free-flight" sojourn times, that we use to provide the mostgeneral possible definition for the quantum time-delay, valid for simple andmultichannel scattering systems, with or without conditions on the observationof the scattering particle, and for incoming wave packets whose energy can besmeared out or sharply peaked (fixed energy). We conclude our conceptualanalysis by presenting what we think is the right interpretation of theconcepts of sojourn and delay times in quantum mechanics, explaining why, inultimate analysis, they shouldn't be called "times".

From permanence to total availability: a quantum conceptual upgrade Preprint - PDF Foundations of Science, Vol. 17, Issue 3, pp. 223-244 (2012)

We consider the classical concept of time of permanence and observe that itsquantum equivalent is described by a bona fide self-adjoint operator. Itsinterpretation, by means of the spectral theorem, reveals that we have toabandon not only the idea that quantum entities would be characterizable interms of spatial trajectories but, more generally, that they would possess thevery attribute of spatiality. Consequently, a permanence time shouldn't beinterpreted as a "time" in quantum mechanics, but as a measure of the totalavailability of a quantum entity in participating to a creative process ofspatial localization.

Ephemeral properties and the illusion of microscopic particles Preprint - PDF Foundations of Science, Vol. 16, Issue 4, pp. 393-409 (2011)

Founding our analysis on the Geneva-Brussels approach to quantum mechanics, we use conventional macroscopic objects as guiding examples to clarify the content of two important results of the beginning of twentieth century: Einstein-Podolsky-Rosen's reality criterion and Heisenberg's uncertainty principle. We then use them in combination to show that our widespread belief in the existence of microscopic particles is only the result of a cognitive illusion, as microscopic particles are not particles, but are instead the ephemeral spatial and local manifestations of non-spatial and non-local entities.

Proprietà effimere e l'illusione delle particelle microscopiche Versione Italiana: AutoRicerca, Numero 2, Anno 2011

Fondando la nostra analisi sull’approccio alla meccanica quantistica della scuola di Ginevra-Brussel, ci lasceremo guidare da alcuni esempi di oggetti macroscopici convenzionali, alfine di chiarire il contenuto di due importanti risultati dell’inizio del ventesimo secolo: il criterio di realtà di Einstein-Podolsky-Rosen e il principio di indeterminazione di Heisenberg. Combinando questi due risultati, mostreremo che la credenza diffusa nell’esistenza delle particelle microscopiche è solo il risultato di un’illusione cognitiva, in quanto le particelle microscopiche non sono particelle, bensì la manifestazione spaziale e locale, di natura effimera, di entità non-spaziali e non-locali.

Comment on "The quantum mechanics of electric conduction in crystals," by R. J. Olsen and G. Vignale PDF - Am. J. Phys. 79, p. 546-549, 2011

In this note we use the notion of time-delay to explain the physical content of the transformation properties of the transmission and reflection amplitudes, as a result of a displacement of the potential. We then reconsider the derivation in the above mentioned paper of Olsen and Vignale, to obtain the condition for total reflection, in the limit of an infinite number of cells composing the finite-periodic potential. In doing so, we also obtain an expression of Hartman's effect, showing that the group velocity of the transmitted particle inside the chain can become arbitrary large, as the number of cells tends to infinity.

Comment on “Generalized composition law from 2x2 matrices," by R. Giust, J.-M. Vigoureux, and J. Lages Am. J. Phys. 78, p. 645-646, 2010

We point out that the result derived by R. Giust, J.-M. Vigoureux, and J. Lages was previously obtained by different authors and different methods, in a more compact form, and comment on the utility of the factorization formula in relation to Levinson's theorem.

Comment on “The role of mediation in collisions and related analogs,” by E. Bashkansky and N. Netzer PDF - Am. J. Phys. 75, p. 1166, 2007

We show that the result derived by Bashkansky and Netzer is more general and not limited to the specific choice of a geometric sequence chosen for the mediator masses.

A Simple semiclassical derivation of Hartman’s effect PDF - Eur. J. Phys., 21, 2000, pp. L1-L4

We present a very simple semiclassical derivation of Hartman's effect valid for potential barriers of general shape. The derivation also gives some insight into the tunnelling time phenomenon.

A remark on the high-energy limit of the one-dimensional scattering problem with position dependent mass Solid State Communications., 106, No. 5, 1998, pp. 249-251 (with M. Di Ventra)

We show that, contrary to what would be expected on the basis of the mass barrier model, the transmission probability for the one-dimensional scattering problem with position-dependent mass normally tends to unity as energy goes to infinity, provided the mass is a continuous function of position.

Comment on "Factorization of scattering matrices in one-dimensional Schrödinger-type equations" by T. Aktosun, M. Klaus, and C. van der Mee PDF - J. Math. Phys., 38, 1997, pp. 4882-4883

We point out a recent proof of the factorization formula using an adaptation of the variable phase method, and present a third alternative proof that uses integral equations instead of Schroedinger differential equations.

On the One-Dimensional Scattering by Time-Periodic Potentials: General Theory and Application to Specific Models PDF - Helv. Phys. Acta 70, 1997, pp. 751-779 (with D. Saraga)

A comprehensive introduction to the basic formalism of the one-dimensional scattering by time-periodic short-ranged potentials is presented. The fundamental objects of the theory (transmission and reflexion probabilities, sidebands and time delays) are defined, and a generalized Born expansion derived. Particular emphasis is given to the connection between the time-dependent approach and the quasi-stationary one. In particular, the independence of the scattering process of the choice of time-origin, in the limit of a monoenergetic wave packet, is clearly established. The generalized Born expansion is applied to two archetypical models: the square barrier with modulated height (the celebrated Büttiker-Landauer model) and the square barrier with oscillating position. For these two models, the full transmission probability is calculated up to the first non-vanishing correction in the time-dependent perturbation.

Dynamical capture in quantum mechanics PDF - J. Phys. A: Math. Gen., 30, 1997, pp. 1011 - 1015 (with Ph. A. Martin)

Using simple time-dependent methods, we study the phenomenon of dynamical capture in non relativistic quantum mechanics. We show that for time-dependent potentials asymptotically constant in time, the probability for an incoming particle to be trapped in the interaction region is in general non-zero. Capture in a stationary beam is also discussed.

Quelques aspects de la diffusion quantique: temps de retard, théorème de Levinson et potentiels dépendant du temps (PhD thesis) PDF - Thèse N° 1438 (1995) présentée au département de physique (EPFL), pour l’obtention du grade de docteur ès sciences

In this Thesis we study several aspects of potential scattering in non-relativistic quantum mechanics. In the first part, we study the concept of time delay. More precisely, after an elementary introduction to the theory (chapter I), we clarify in chapter II the role played by the localizing regions in the definition of global time delay. In chapter III, we generalize the concept to arbitrary conditions of observation for the scattered particle (conditional time delay). In chapters IV and V, we address the problem of the measure of time delay by physical clocks and, more specifically, by the Larmor clock which exploit the mechanism of precession of a spin in a magnetic field. In the second part, we are interested in the spectral property of time delay, its connection with Levinson's theorem, and the application of the latter to one-dimensional systems. We derive Levinson's theorem in chapter VI, using as a single ingredient the completeness of physical states. In chapter VII, we apply Levinson's result to determine the number of bound-states of a finite periodic potential, as a function of its period. For this, we use the the factorization property of the scattering matrix which we derive in chapter VIII in the more general situation of a particle with position-dependent mass. In the third part, we are concerned by time-dependent potentials. In chapter IX, we generalize the concept of time delay, and for the particular case of a periodic variation we derive a Levinson theorem. In chapter X, we consider potentials with very slow or very rapid variations in time. The low and high frequency limits are derived as well as their first corrections, and their physical significance discussed.

Dans cette thèse on aborde divers aspects de la diffusion par potentiel en mécanique quantique non-relativiste. Dans la première partie, nous étudions le concept de temps de retard. Plus précisément, après une introduction élémentaire à la théorie (chapitre I), nous clarifions au chapitre II le rôle joué par les régions de localisation dans la définition du temps de retard global. Au chapitre III, nous généralisons le concept à des conditions d'observation arbitraires pour la particule diffusée (temps de retard conditionnel). Aux chapitres IV et V, nous abordons le problème de la mesure du temps de retard par des horloges physiques et, plus particuliérement, par l'horloge de Larmor qui exploite le mécanisme de précession d'un spin dans un champ magnétique. Dans la deuxième partie nous nous intéressons à la propriété spectrale du temps de retard, à son lien avec le théorème de Levinson, et à l'application de ce dernier à des systèmes unidimensionnels. Nous dérivons le théorème de Levinson au chapitre VI, en utilisant comme seul ingrédient la complétude des états physiques. Au chapitre VII, nous appliquons le résultat de Levinson pour déterminer le nombre d'états liés d'un potentiel périodique fini, enfonction de sa période. Pour cela, nous utilisons la propriété de factorisation de la matrice de diffusion que nous dérivons au chapitre VIII dans le cadre plus général d'une particule avec masse dépendante de la position. Dans la troisième partie, nous sommes concernés par les potentiels dépendant du temps. Au chapitre IX, nous généralisons le concept de temps de retard, et pour le cas particulier d'une variation périodique nous dérivons un théorème de Levinson. Au chapitre X, nous considérons des potentiels qui varient très lentement ou très rapidement au cours du temps. Les limites basse et haute fréquence sont dérivées ainsi que les premières corrections, et leur interprétation physique est discutée.

Levinson's theorem for time-periodic potentials PDF - Europhys. Lett., 34 (9), 1995, pp. 639 - 643 (with Ph. A. Martin)

Levinson's theorem is generalized to quantum scattering with time-periodic potentials. The zero-energy limit of the phase shift in the elastic channel is linked to the number of bound (cyclic) states of the time-dependent potential by the same relation as in the static case. [PDF]

How many bound-states does a one-dimensional finite superlattice have? Superlattices and Microstructures, 20, No 2, 1996, pp. 149-153 (with M. Di Ventra)

We give a simple and general description of the bound-state structure of a one-dimensional finite superlattice as a function of its period d and some quantities characterizing the single unit cell.

Differential équations and factorization property for the one-dimensional Schrödinger equation with position-dependent mass Eur. J. Phys. 16, 1995, pp. 260 - 265 (with M. Di Ventra)

The variable phase method is applied to the one-dimensional Schrödinger equation with position-dependent (effective) mass, to derive first-order differential equations for the transmission and reflection amplitudes, and bound-state energies, which are particularly convenient for numerical computations. When the mass and potential have the same asymptotic at both ends of the real line, the method also allows to prove a factorization property of the scattering matrix.

Il metodo della fase variabile è applicato all’equazione di Schrödinger unidimensionale, con massa (efficace) dipendente dalla posizione, allo scopo di derivare equazioni differenziali del primo ordine per le ampiezze di trasmissione, riflessione e per le energie degli stati legati, che siano particolarmente adatte per calcoli numerici. Per una massa ed un potenziale aventi lo stesso andamento asintotico a destra e a sinistra, il metodo permette anche di dimostrare una proprietà di fattorizzazione della matrice di diffusione.

On the low- and high-frequency limit of quantum scattering by time-dependent potentials PDF - J. Phys. A, 28, 1995, pp. 2403-2427 (with Ph. A. Martin)

Using time-dependent methods, we study the scattering of a quantum mechanical particle by short-range potentials with very slow or very fast periodic variations in time. The low- and high-frequency limits are derived as well as their first non-vanishing corrections, and their physical significance discussed.

On the number of states bound by one-dimensional finite periodic potentials J. Math. Phys., 36, 1995, pp. 1753-1764 (with M. Di Ventra)

Bound-states and zero-energy resonances of one-dimensional finite periodic potentials are investigated, by means of Levinson's theorem. For finite range potentials supporting no bound states, a lower bound for the (reduced) time delay at threshold is derived.

Spin precession revisited PDF - Found. Phys., 24, 1994, pp. 1371-1378 (with Ph. A. Martin)

The passage of a spin 1/2 neutral particle through a region of uniform magnetic field and the corresponding precession mechanism is analyzed from the view point of scattering theory, with particular consideration on the role of the field boundaries.

Levinson's theorem, zero-energy resonances, and time delay in one-dimensional scattering systems J. Math. Phys., 35, 1994, pp. 2719-2733

The one-dimensional Levinson's theorem is derived and used to study zero-energy resonances in a double-potential system. The low energy behavior of time delay is also investigated. In particular, we show that the quantum mechanical time delay admits a classical lower bound, in the low energy limit, if the potential has no bound-state solutions.

Conditional time-delay in scattering theory Helv. Phys. Acta 66, 1993, pp. 361-377

We give a general and mathematically precise definition of the notion of conditional time delay in scattering theory i.e., a notion of time delay for a given condition of observation of the scattered particle. A formula, generalizing the Eisenbud-Wigner time delay formula, is derived. The basic concept entering in the definition of the conditional time delay is that of conditional sojourn time. Although conditional sojourn times cannot be uniquely defined in quantum mechanics because of the uncertainty principle, we show that conditional time delays admit a well defined probabilistic interpretation in the limit of infinitely extended spatial regions. Some comments are presented in relation with the tunneling time problem.

On the definition of time-delay in scattering theory Helv. Phys. Acta 65, 1992, pp. 1119-1126 (with Ph. A. Martin)

We show that the time delay of a scattering process, defined as the difference of interacting and free sojourn times for increasing spatial regions, can only exist for sequences of dilated balls. The transformation properties of the Eisenbud-Wigner formula under translations are discussed.

On the theory of the Larmor clock and time-delay PDF - J. Phys. A, 25, 1992, pp. 3627-3647 (with Ph. A. Martin)

Using the time dependent scattering theory we prove that, in any spatial dimension and for arbitrary spin, the reading of the Larmor clock agrees with the global (Eisenbud-Wigner) time delay in the limit of an infinitesimal magnetic field. We show that convergence is also achieved at fixed energy (without oscillating terms) in the limit where the spatial switching on of the field occurs on distances much larger than the de Broglie wave length of the particle. Finally, we investigate the functioning of the spin clock beyond the linear response regime.