Observables

8 Key excerpts on "Observables"

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  An Introduction to the Mathematical Structure of Quantum Mechanics

  A Short Course for Mathematicians

  • F Strocchi(Author)
  • 2005(Publication Date)
  • WSPC

    (Publisher)

  16 Mathematical description of a physical system 1.3 General mathematical description of a physical system In this section we argue that the structure of C*-algebra of Observables and states is the suitable language for the mathematical description of a physical system in general (including the atomic systems), with no reference to classical mechanics and its standard paradigms. l2 I. Observables. From an operational point of view, a physical system is defined by its physical properties, i.e. by the set 0 of the physical quantities (briefly called Observables) which can be measured on it and by the relations between them. Each observable has to be understood as characterized by a concrete physical apparatus yielding its measurements. l3 For any A E 0 and X E R, one can define the observable XA as the observable measured by rescaling the apparatus by A. By similar consider- ations one justifies the existence of elementary functions of an observable like the powers (with the standard elementary properties): if A E 0, A' may be interpreted as the observable associated with squaring the appara- tus scale (equivalently by squaring the results of measurements). Similarly, one defines the powers A, and their products A A = Am+n. It follows from this definition that A ' is the observable whose results of measure- ments always take the value 1, independently of the state on which the measurement is done. In the same way, one defines a polynomial of A as the observable ob- tained by taking as the new apparatus scale the given polynomial function of the scale for A. An element A E 0 is said to be positive if all the results of measurements of A are positive numbers. By the operational definition of elementary functions of A, this implies that (and it is actually equivalent to) A is of the form A = B2, B E 0.

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  The Mathematical Language of Quantum Theory

  From Uncertainty to Entanglement

  • Teiko Heinosaari, Mário Ziman(Authors)
  • 2011(Publication Date)
  • Cambridge University Press

    (Publisher)

  3 Observables The intrinsic randomness of measurement outcomes is a key feature of quantum theory. Any experiment produces a sequence of outcomes, each outcome occurring with a certain probability depending on the particular settings of the measur- ing and preparation devices. Quantum theory predicts only the probabilities of measurement outcomes, not individual occurrences of particular outcomes. The mathematical concept of an observable, which will be introduced and analyzed in this chapter, is used to capture the statistical essence of the measurement pro- cess. The monographs of Holevo [76], Busch et al. [34] and de Muynck [53] are recommended general references on this subject. Considering quantum theory as a framework for calculating measurement out- come probabilities, it has aspects which make it both a generalization and a restriction of the usual probability theory. It is a generalization in the sense that Observables are described by positive operator-valued measures. These are more general than probability measures, in much the same way as matrices are more general than numbers. However, quantum theory imposes inherent restrictions on the probability distributions that we can find in quantum measurements. In this sense, it can also be seen as a restriction of the usual probability theory. 3.1 Observables as positive operator-valued measures In Chapter 2 we introduced effects as statistical events. A prototypical example of a relevant event is ‘a measurement M gives an outcome x ’. This event is associated with an effect E , and the probability for the event to happen is given by the trace formula tr   E  , where  is the density operator representing the input state of the measured system. The whole measurement process can be thought of as a collection of events, each associated with an effect. In performing a measurement we want to observe which event is realized. The mathematical description of the possible events is called an 105

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  Symmetry, Structure, and Spacetime

  • Dean Rickles(Author)
  • 2007(Publication Date)
  • Elsevier Science

    (Publisher)

  ∞ ones) comprise the gauge freedom in the theory. The diffeomorphisms underlie the general covariance of general relativity, and this tells us that the models that are thus related will be indistinguishable as far as quantities not connected to the pointwise location of the fields are concerned. Therefore, the hole argument fails to get a grip: general relativity isn’t meant to predict values for these quantities, nor for any quantities localized to points of the (bare) manifold regardless of how well we can specify them in connected regions of spacetime. The kinds of quantity that do class as Observables in general relativity are relational (or highly non-local), much as Einstein argued, though not quite so narrowly defined. (I agree with this classification, but below I tame the view of many physicists that this implies relationalism about spacetime.) Let us say a little more about these Observables, and give some examples that connect with measurements.

  6.1 DEFINING Observables

  It is Observables that give us our connection to the world in the context of physics; they are the things we measure and whose values we predict. They form the qualitative character of a world in the sense that two worlds that are duplicates in terms of the Observables they contain, and in terms of their assigned values, are qualitatively indistinguishable. I think that the resolution of the hole argument can only come about once we have a proper grasp on what the theory of general relativity is about and what it is supposed to predict. This might sound rather obvious, but this involves the problem of defining the Observables of general relativity— a notorious problem that I discuss more fully in the next chapter. In this section I simply skirt over many ‘deep’ issues, and highlight the relevance of the definition of Observables to the hole problem (including a natural resolution) and, in a preliminary way, to the problem of time.

  In general relativity, the physically relevant quantities are taken to be the diffeomorphism invariant quantities, or Bergmann Observables. These are any quantities that spit out the same values under ‘draggings’ of the dynamical fields (the building blocks of the Observables) by diffeomorphisms. If we accept the view that general relativity is a gauge theory, with the gauge part given by the diffeomorphisms, then the Observables will be gauge invariant quantities, or Dirac Observables. This is usually taken to mean that any quantities that are related by a gauge-transformation describe the same physical state; one and the same physical possibility is being multiply represented, and the excess representational machinery is deemed to be mere surplus structure. But this is not necessary. As far as the physics goes, I think that such differences should not make a difference; the Observables grasp on to the qualitative character of the world, and any quantities that fail to distinguish between qualitative duplicates should be ruled out on the grounds that they would not be measurable or predictable in the context of the theory. This, I submit, is what is desired from gauge invariance; when physicists say physical equivalence, they simply mean to imply qualitative, or empirical equivalence. Gauge invariance of the Observables then implies that those differences that are non-qualitative do not get counted in, say, statistical algorithms or in the initial-value problem of the theory (or it does so only ‘up to a gauge transformation’). But this does not imply that there are not such differences, or that the ontological elements that are allowing for the qualitatively indistinguishable worlds (e.g. the manifold and its symmetries) do not exist. This cannot be read off the physics, but the Observables don’t care about such things.192 These quantities, as we have seen, avoid the hole argument, for that argument only targeted those quantities that were changed by a diffeomorphism, quantities localized at points of the spacetime manifold (e.g. the scalar curvature at the point

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  Quantum Mechanics

  • Gennaro Auletta, Mauro Fortunato, Giorgio Parisi(Authors)
  • 2009(Publication Date)
  • Cambridge University Press

    (Publisher)

  (2.104b) This means that there is a common basis in which both Observables are perfectly deter-mined. In other words, for each state | o k of the basis, the observable ˆ O , if measured, gives with certainty the eigenvalue o k as outcome and the observable ˆ O , if measured, gives with certainty the eigenvalue o k as outcome. Q.E.D As an immediate consequence of Cor. 2.1 we may state that non-commuting observ-ables cannot be simultaneously measurable with arbitrary precision. In general, therefore, given the set of Observables of a physical system S , it will be possible to divide them into separate subsets of reciprocally commuting Observables. These are called complete sets and represent the maximum number of properties of S that can be jointly known (see also Subsec. 2.2.7 ). We see here that, while in classical mechanics it is possible to know jointly all the properties of a system, in quantum mechanics by a complete description we mean the knowledge of all the Observables in certain complete (but not necessarily disjoint) sets. As we shall see in the following, the non-commutability between quantum mechanical Observables has extraordinary implications in the very foundations of the theory and in the corresponding interpretation of its physical reality (see e.g. Subsec. 2.3.3 ). 68 Quantum Observables and states 2.2 Wave function and basic Observables In this section we shall apply (and develop) the formalism of the previous section to concrete quantum Observables. After having introduced the concept of wave function (Subsec. 2.2.1 ), we will discuss the difficult problem of normalization (Subsec. 2.2.2 ). In Subsec. 2.2.3 we introduce the position operator whereas in Subsec. 2.2.4 we introduce the momentum observable. In Subsec. 2.2.5 we analyze the relationship between position and momentum representations. In Subsec. 2.2.6 the energy observable is shortly intro-duced – further developments can be found in Ch.

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  Quantum Mechanics in Hilbert Space

  • Eduard Prugovecki(Author)
  • 1982(Publication Date)
  • Academic Press

    (Publisher)

  We note that if Yo is a normalized vector representing a state at an instant to , then for any complex number a, I a I = 1, aYo is another normalized vector representing the same state. These This result can be deduced also from Radon-Nikodym's theorem (see Halmos [1950]) once it is observed that p(x) equals the Radon-Nikodym derivative dp,,(x)/dx. 270 IV. The Axiomatic Structure of Quantum Mechanics remarks justify the frequently made statement that “a state of a quantum system is (at any given instant) a ray in the Hilbert space.” In analyzing the concept of Observables we have demanded until now that an observable play the dual role of being related to a symbol of the formalism (a self-adjoint operator, in case of quantum mechanics) and at the same time be anchored directly in the experiment by being related to some empirical procedure (or procedures) for measuring it. We shall call Observables satisfying both these conditions fundamental Observables, to distinguish them as a subfamily of the wider family of Observables in general, which will be introduced next. In order to realize the necessity of enlarging the concept of Observables, consider the concept of a function of Observables in classical mechanics. Take, for instance, a one-particle system. If we know the position r and momentum p of the particle at some instant, then we can compute the values of other Observables-such as angular momentum r x p, kinetic energy” p2/2m, potential energy V(r)-since all these other Observables are functions of r and p. More generally, if we take any real-valued function F(r, p) in the six variables of r, p, we can think of it as an observable, because an indirect measurement of F(r, p) can be carried out by measuring the position and momentum of the particle and then computing F(r, p). The above argument indicates that we can generalize the concept of an observable in quantum mechanics by introducing functions of one or more already given fundamental Observables.

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  Einstein, Tagore and the Nature of Reality

  • Partha Ghose(Author)
  • 2016(Publication Date)
  • Routledge

    (Publisher)

  7 Physical reality and the unObservables of physical nature C. S. Unnikrishnan Introduction and scope The fundamental physical theories that interpret and explain the behaviour of matter in nature are dependent on several unObservables and insensibles in their construction. Entities like fields, wave functions and even space and time are all unObservables, except as manifestations of material existence and behaviour. There is thus an obvious difference of degree and meaning between the reality associated with these unobservable theoretical entities and that of perceptible matter. The success of the physical theory is often taken as evidence for the physical reality of such unObservables. While a rigorous natural philosophy will not be able to support or approve their reality with the same vigour and conviction as it might defend the reality of matter, there does not seem to be a way of avoiding such unObservables if we have to construct theories. Though there is compatibility and consistency between Observables and unObservables in most of classical physics, apparent conflicts and dissonance arise when microscopic physics is to be understood with a satisfactory theory. There are even observational consequences highlighting such conflicts when cosmology and the dynamics of the universe are included into the larger physical framework. In this chapter, I examine the nature of physical reality in the context of unavoidable unObservables in physics 1, 2 and discuss some examples. For the purpose of this discussion, I will work with a definition of an unobservable (in physics) as a quantitative entity, created and mathematically representable in relation to a theory of sensible matter and its behaviour, but whose ontological status cannot be established nor demonstrated directly or by deduction employing methods usually used for material entities

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  Progress in Theoretical Biology

  Volume 4

  • Robert J. Rosen, Fred M. Snell, Robert J. Rosen, Fred M. Snell(Authors)
  • 2013(Publication Date)
  • Academic Press

    (Publisher)

  Biological Observables Sorin Comorosan* Department of Research, J. M. Richards Laboratory, Grosse Pointe Park, Michigan I. States, Observables, and the Measurement Process in Quantum Theory and Biology 161 A. Basic Ideas 161 B. Operators and Observables 163 C. The Postulates of the Quantum Theory . 164 D. The Measurement Process in Biology . . . . 167 II. Biological Macromolecules as Measuring Systems 168 A. Enzyme-Substrate Interactions 168 B. Growth Factors-Cellular Receptors Interactions 182 C. Drug-Receptor Interactions 188 D. Physical Problems 193 III. Concluding Remarks 202 References 203 I. States, Observables, and the Measurement Process in Quantum Theory and Biology A. BASIC IDEAS In quantum mechanics the description of the microphysical events is always performed in the "common language, supplemented with termi- nology of classic physics" (Bohr, 1958). This aspect derives from the fact that we are obliged to discuss the quantum phenomena through those physical characteristics that are directly accessible to human observation. That is, in reality the quantum mechanics operates with the basic notions of classical physics. The attempt to introduce absolutely specific concepts, like the hidden Observables (Böhm, 1962), is far from a settled question. This situation is referred to as the Bohr correspondence principle and * Present address: Laboratorul de Biochimie, Fundeni Clinical Hospital, Bucharest, Romania. 161 162 SORIN COMOROSAN implies that any quantum theory should approach in the Classical limit" the results of classical mechanics. In this context it is interesting to point out that the correspondence principle indicates as reference system only classical physics. All other domains, like for example the biophysical one, are not taken into account. Accordingly, a generalization that would consider as reference system the biological domain might reveal quite new aspects.

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  Quantum Theory of High-Energy Ion-Atom Collisions

  • Dzevad Belkic(Author)
  • 2008(Publication Date)
  • CRC Press

    (Publisher)

  1 Basic notions and main Observables in scattering problems 1.1 Observables and elementary processes In this chapter, we shall devote our attention to the main Observables in scattering problems in light of the critical criteria for comparisons between theoretical and experimental data. Hereafter, any physical quantity which can be experimentally measured will be called an observable. We emphasize immediately that we shall not study collision experiments from the viewpoint of their realistic performance, which could certainly be very sophisticated, especially for modern measurements. Our primary concern will be focused on the exposition of the main concepts of scattering experiments regarding the boundary conditions and the corresponding requirements imposed by the theory. Here, we primarily think of the requirements that are a direct con-sequence of the limitations of the theory itself. Namely, scattering theory is based on the first principles of physics without reliance on any free or ad-justable parameters. Therefore, such a theory is complete. However, the price paid for this completeness is an obvious limitation to relatively clear-cut situ-ations dealing with elementary processes of the type of scattering of one given particle on a single particle from the target. These are the so-called binary collisions. Of course, this is an idealization, since in any realistic collision experiment, we encounter a beam of incident particles (projectiles) that scat-ter on the target which is itself comprized of many particles representing the elementary centers of scattering 1 . However, comparisons with experiments represent the ultimate test for the final validation of any theory in physics. In other words, no matter how apparently attractive and compelling a given theory might appear, from both the physical and mathematical standpoints, it would still be considered as inadequate if it fails to describe properly the available experimental data.

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