Born Rule

7 Key excerpts on "Born Rule"

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  A Student's Guide to the Schrödinger Equation

  • Daniel A. Fleisch(Author)
  • 2020(Publication Date)
  • Cambridge University Press

    (Publisher)

  Specifically, the wavefunction may be used to determine the possible results of measurements of quantum observables and to calculate the probabilities of each of those results. The Born Rule plays an extremely important role in quantum mechanics, since it explains the meaning of the solutions to the Schr¨ odinger equation in a way that matches experimental results. But the Born Rule is silent about other critical aspects of quantum mechanics, and unlike the almost immediate and widespread acceptance of the Born Rule, those other aspects have been the subject of continuing debate for nearly a century. That debate has not led to a set of universally agreed-upon principles. The most widely accepted (and widely disputed) explanation of quantum mechanics is called the “Copenhagen interpretation,” since it was developed in large part at the Niels Bohr Institute in Copenhagen. In spite of the ambivalence many quantum theorists express toward the Copenhagen interpretation, it’s worth your time to understand its basic tenets. With that understanding, you’ll be able to appreciate the features and drawbacks of the Copenhagen interpre-tation, as well as the advantages and difficulties of alternative interpretations. 4.1 The Born Rule and Copenhagen Interpretation 97 So exactly what are those tenets? That’s not easy to say, since there seem to be almost as many versions of the Copenhagen interpretation as there are bicycles in Copenhagen. But the principles usually attributed to the Copenhagen interpretation include the completeness of the information in the quantum state, the smooth time evolution of quantum states, wavefunction collapse, the relationship of operator eigenvalues to measurement results, the uncertainty principle, the Born Rule, the correspondence principle between classical and quantum physics, and the complementary wave and particle aspects of matter.

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  Do We Really Understand Quantum Mechanics?

  • Franck Laloë(Author)
  • 2012(Publication Date)
  • Cambridge University Press

    (Publisher)

  The latter then appears as an attractor. The reasoning can be seen as a derivation of the Born Rule, which no longer appears as an independent postulate, but as a consequence of the dynamics. See [420] for more recent numerical simulations along the same line. 236 Various interpretations they remain in regions of space where it does not vanish, which ensures that neither the guiding formula (10.17) nor the quantum potential (10.18) become indetermi- nate. Another important consequence of this postulate is to ensure compatibility with relativity [421], since arbitrary distributions would introduce the possibility of superluminal signaling (Appendix H). Since the Born Rule is a consequence of the quantum equilibrium, one can then consider that this rule is not an independent postulate of quantum mechanics, but merely a consequence of relativity. One can then restore determinism, and assume that the results of measurements merely reveal the initial pre-existing value of the positions, chosen among all pos- sible values in the initial probability distribution (we come back in more detail to this point in §10.6.1.c). This assumption solves many difficulties, for instance those related to understanding why quantum systems can manifest both wave and particle properties in interference experiments. The system always contains two objects, a wave and a particle; the wave may produce interference effects and guide the parti- cle in a way that forces its trajectory to reproduce the interference pattern – nothing especially mysterious conceptually (below we study Bohmian trajectories in more detail).

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  Interpreting Quantum Theory

  A Therapeutic Approach

  • S. Friederich(Author)
  • 2014(Publication Date)
  • Palgrave Macmillan

    (Publisher)

  In addition, it seems not to do justice to quantum theoretical practice, where claims about the values of observables are often considered (and Born Rule probabilities computed for them) even where these values are not ‘measured’ or otherwise determined experimentally. 40 This observation makes the challenge of clarifying the appropriate range of applicability of the Born Rule even more pressing. The response to this challenge found in the literature which seems to go best with the Rule Perspective is essentially that of Healey’s pragmatist interpretation of quantum theory. Appealing to environment-induced decoherence, it holds that Born Rule probabilities apply precisely to those NQMCs which are in terms of observables for which taking into account the system’s interaction with its environment renders the density operator assigned to the system (at least approximately) diagonal. In Healey’s own words: Born-rule probabilities are well-defined only over claims licensed by quantum theory. According to the quantum theory, interaction of a system with its environment typically induces decoherence in such a way as (approximately) to select a preferred basis of states in the system’s Hilbert space. Quantum theory will fully license claims about the real value only of a dynamical variable represented by an operator that is diagonal in a preferred basis: it will grant a slightly less complete license to claims about approximately diagonal observables. All these dynamical variables can consistently be assigned simultaneous real values distributed in accordance with the Born probabilities. So there is no need to formulate the Born Rule so that its probabilities concern only measurement outcomes. (Healey [2012a], p

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  Phenomenology and QBism

  New Approaches to Quantum Mechanics

  • Philipp Berghofer, Harald A. Wiltsche, Philipp Berghofer, Harald A. Wiltsche(Authors)
  • 2023(Publication Date)
  • Routledge

    (Publisher)

  However those beliefs are updated, QBists adopt a subjectivist interpretation of probability and the question now is how to understand the probabilities yielded by quantum mechanics, which are (apparently) objective. These are given by the Born Rule, which relates such probabilities to the square of the amplitude of the relevant wave-function.

  According to Earman, QBists could give a straightforward answer to this question in the form of a proof that demonstrates that quantum probabilities are just ‘objectified’ forms of subjective probabilities (Earman 2019 ).5 Unfortunately, he alleges, QBists decline to take advantage of this and, partly as a result, create ‘faux difficulties’ for themselves and ‘fail to convey some of the strengths of their stance’ (2019, p. 404). However, Earman has been accused of fundamentally misunderstanding QBism and, indeed, of begging the question (Fuchs & Stacey 2020 , p. 1). In particular, appealing to features of the quantum mechanical formalism to ground the Born Rule and relate subjective probabilities to the apparently objective quantum forms puts the cart before the horse, as it ‘… ultimately ends up re-objectifying what had been initially supposed to be subjective probabilities’ (ibid., p. 3).

  What Earman failed to appreciate is that according to QBism these features of the theory should not be regarded as representational, in the sense of describing how potential events are related, with the Born Rule arising as a consequence; rather the latter should be understood as primitive and one of the aims of the programme is then to recover the formal structure of the theory from that basis (Fuchs & Stacey 2020 , p. 3). In other words, according to Fuchs, ‘[t]he interpretation should come first, with the mathematical structure of the theory derivative from it’ (in Crease & Sares 2020

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  Something Deeply Hidden

  Quantum Worlds and the Emergence of Spacetime

  • Sean Carroll(Author)
  • 2019(Publication Date)
  • Oneworld Publications

    (Publisher)

  act, not about how often things happen. And “what we should believe” isn’t something that really has a place in the postulates of a physical theory; it should be implied by them.

  Moreover, as we will see, there is neither any room for an extra postulate, nor any need for one. Given the basic structure of quantum mechanics, the Born Rule is natural and automatic. Since we tend to see Born Rule–like behavior in nature, this should give us confidence that we’re on the right track. A framework in which an important result can be derived from more fundamental postulates should, all else being equal, be preferred to one where it needs to be separately assumed.

  If we successfully address this question, we will have made significant headway toward showing the world we would expect to see if Many-Worlds were true is the world we actually do see. That is, a world that is closely approximated by classical physics, except for quantum measurement events, during which the probability of obtaining any particular outcome is given by the Born Rule.

  The issue of probabilities is often phrased as trying to derive why probabilities are given by amplitudes squared. But that’s not really the hard part. Squaring amplitudes in order to get probabilities is a very natural thing to do; there weren’t any worries that it might have been the wave function to the fifth power or anything like that. We learned that back in Chapter Five, when we used qubits to explain that the wave function can be thought of as a vector. That vector is like the hypotenuse of a right triangle, and the individual amplitudes are like the shorter sides of that triangle. The length of the vectors equals one, and by Pythagoras’s theorem that’s the sum of the squares of all the amplitudes. So “amplitudes squared” naturally look like probabilities: they’re positive numbers that add up to one.

  The deeper issue is why there is anything unpredictable about Everettian quantum mechanics at all, and if so, why there is any specific rule for attaching probabilities. In Many-Worlds, if you know the wave function at one moment in time, you can figure out precisely what it’s going to be at any other time, just by solving the Schrödinger equation. There’s nothing chancy about it. So how in the world is such a picture supposed to recover the reality of our observations, where the decay of a nucleus or the measurement of a spin seems irreducibly random?

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  Physics 1942 – 1962

  Including Presentation Speeches and Laureates' Biographies

  • Sam Stuart(Author)
  • 2013(Publication Date)
  • Elsevier

    (Publisher)

  This theory was called quantum mechanics. The following year Born got a new result of fundamental significance. Schrödinger had just then found a new formulation for quantum mechanics. Schrödinger's work expanded the earlier ideas of De Broglie which imply that atomic phenomena are connected with a wave undulation. However, Schrödinger had not solved the problem of how it is possible to make state-ments about the positions and velocities of particles if one knows the wave corresponding to the particle. Born provided the solution to the problem. He found that the waves determine the probability of the measuring results. For this reason, according to Born, quantum mechanics gives only a statistical description. This can be illustrated by a simple example. When you shoot at a target it is possible in principle - according to the older conception - to aim the shot from the start so that it is certain to hit the target in the middle. Quantum mechanics teaches us to the contrary - that in principle we cannot predict where a single 254 P H Y S I C S 1 9 5 4 shot will hit the target. But we can achieve this much, that from a large num-ber of shots the average point of impact will lie in the middle of the target. In contradiction to the deterministic predictions of the older mechanics, quantum mechanics accordingly poses laws which are of a statistical char-acter, and as regards single phenomena will only determine the probabilities that one or another of various possibilities will occur. For material bodies of ordinary dimensions the uncertainty of the predictions of quantum mechan-ics is practically of no significance. But in atomic phenomena, on the other hand, it is fundamental. Such a radical break with older ideas could not of course prevail without opposition. But Born's conception is now generally accepted by physicists, with a few exceptions.

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  The Transactional Interpretation of Quantum Mechanics

  A Relativistic Treatment

  • Ruth E. Kastner(Author)
  • 2022(Publication Date)
  • Cambridge University Press

    (Publisher)

  3 The Original TI Fundamentals 3.1 Background In his famous Lectures, Richard Feynman said: Now in the further development of science, we want more than just a formula. First we have an observation, then we have numbers that we measure, then we have a law which summarizes all the numbers. But the real glory of science is that we can find a way of thinking such that the law is evident. (Feynman et al., 1964) The Transactional Interpretation of quantum mechanics (TIQM) is precisely that “way of thinking such that the law is evident.” In this case, the law in question is the Born Rule for the probabilities of outcomes of measurements. In this chapter, we introduce the basics of TIQM (or more concisely, just “TI”). TI was first proposed by John G. Cramer in a series of papers in the 1980s (Cramer, 1980, 1983, 1986, 1988). The 1986 paper presented the key ideas and showed how the interpretation gives rise to a physical basis for the Born Rule which prescribes that the probability of an event is given by the square of the wave function corresponding to that event. TI was originally inspired by the Wheeler–Feynman (WF) time- symmetric theory of classical electrodynamics (Wheeler and Feynman, 1945, 1949). The WF theory is also called the “absorber” theory or the “direct-action” theory, because it abolished the idea of the electromagnetic field as a separate mechanical system and proposed instead that radiation is a direct interaction between emitters and absorbers, without any independently existing field as an intermediary. The interaction is a time-symmetric process, in which a charge emits a field in the form of half-retarded, half-advanced solutions to the wave equation, and the response of absorbers combines with that primary field to create a radiative process that transfers energy from an emitter to an absorber. This process is symbolized in TI by a “handshake.” Let’s first review the WF proposal, and then we’ll see how TI generalizes the idea to the quantum domain.

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