Is There a Link Between Quantum Mechanics and Consciousness?
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Yes, More Decoherence: a Reply to Critics
Yes, More Decoherence: A Reply to Critics Elise M. Crull∗ Submitted 11 July 2017; revised 24 August 2017 1 Introduction A few years ago I published an article in this journal titled \Less interpretation and more decoherence in quantum gravity and inflationary cosmology" (Crull, 2015) that generated replies from three pairs of authors: Vassallo and Esfeld (2015), Okon and Sudarsky (2016) and Fortin and Lombardi (2017). As a philosopher of physics it is my chief aim to engage physicists and philosophers alike in deeper conversation regarding scientific theories and their implications. In as much as my earlier paper provoked a suite of responses and thereby brought into sharper relief numerous misconceptions regarding decoherence, I welcome the occasion provided by the editors of this journal to continue the discussion. In what follows, I respond to my critics in some detail (wherein the devil is often found). I must be clear at the outset, however, that due to the nature of these criticisms, much of what I say below can be categorized as one or both of the following: (a) a repetition of points made in the original paper, and (b) a reiteration of formal and dynamical aspects of quantum decoherence considered uncontroversial by experts working on theoretical and experimental applications of this process.1 I begin with a few paragraphs describing what my 2015 paper both was, and was not, about. I then briefly address Vassallo's and Esfeld's (hereafter VE) relatively short response to me, dedicating the bulk of my reply to the lengthy critique of Okon and Sudarsky (hereafter OS). -
Unit 1 Old Quantum Theory
UNIT 1 OLD QUANTUM THEORY Structure Introduction Objectives li;,:overy of Sub-atomic Particles Earlier Atom Models Light as clectromagnetic Wave Failures of Classical Physics Black Body Radiation '1 Heat Capacity Variation Photoelectric Effect Atomic Spectra Planck's Quantum Theory, Black Body ~diation. and Heat Capacity Variation Einstein's Theory of Photoelectric Effect Bohr Atom Model Calculation of Radius of Orbits Energy of an Electron in an Orbit Atomic Spectra and Bohr's Theory Critical Analysis of Bohr's Theory Refinements in the Atomic Spectra The61-y Summary Terminal Questions Answers 1.1 INTRODUCTION The ideas of classical mechanics developed by Galileo, Kepler and Newton, when applied to atomic and molecular systems were found to be inadequate. Need was felt for a theory to describe, correlate and predict the behaviour of the sub-atomic particles. The quantum theory, proposed by Max Planck and applied by Einstein and Bohr to explain different aspects of behaviour of matter, is an important milestone in the formulation of the modern concept of atom. In this unit, we will study how black body radiation, heat capacity variation, photoelectric effect and atomic spectra of hydrogen can be explained on the basis of theories proposed by Max Planck, Einstein and Bohr. They based their theories on the postulate that all interactions between matter and radiation occur in terms of definite packets of energy, known as quanta. Their ideas, when extended further, led to the evolution of wave mechanics, which shows the dual nature of matter -
Quantum Theory of the Hydrogen Atom
Quantum Theory of the Hydrogen Atom Chemistry 35 Fall 2000 Balmer and the Hydrogen Spectrum n 1885: Johann Balmer, a Swiss schoolteacher, empirically deduced a formula which predicted the wavelengths of emission for Hydrogen: l (in Å) = 3645.6 x n2 for n = 3, 4, 5, 6 n2 -4 •Predicts the wavelengths of the 4 visible emission lines from Hydrogen (which are called the Balmer Series) •Implies that there is some underlying order in the atom that results in this deceptively simple equation. 2 1 The Bohr Atom n 1913: Niels Bohr uses quantum theory to explain the origin of the line spectrum of hydrogen 1. The electron in a hydrogen atom can exist only in discrete orbits 2. The orbits are circular paths about the nucleus at varying radii 3. Each orbit corresponds to a particular energy 4. Orbit energies increase with increasing radii 5. The lowest energy orbit is called the ground state 6. After absorbing energy, the e- jumps to a higher energy orbit (an excited state) 7. When the e- drops down to a lower energy orbit, the energy lost can be given off as a quantum of light 8. The energy of the photon emitted is equal to the difference in energies of the two orbits involved 3 Mohr Bohr n Mathematically, Bohr equated the two forces acting on the orbiting electron: coulombic attraction = centrifugal accelleration 2 2 2 -(Z/4peo)(e /r ) = m(v /r) n Rearranging and making the wild assumption: mvr = n(h/2p) n e- angular momentum can only have certain quantified values in whole multiples of h/2p 4 2 Hydrogen Energy Levels n Based on this model, Bohr arrived at a simple equation to calculate the electron energy levels in hydrogen: 2 En = -RH(1/n ) for n = 1, 2, 3, 4, . -
Vibrational Quantum Number
Fundamentals in Biophotonics Quantum nature of atoms, molecules – matter Aleksandra Radenovic [email protected] EPFL – Ecole Polytechnique Federale de Lausanne Bioengineering Institute IBI 26. 03. 2018. Quantum numbers •The four quantum numbers-are discrete sets of integers or half- integers. –n: Principal quantum number-The first describes the electron shell, or energy level, of an atom –ℓ : Orbital angular momentum quantum number-as the angular quantum number or orbital quantum number) describes the subshell, and gives the magnitude of the orbital angular momentum through the relation Ll2 ( 1) –mℓ:Magnetic (azimuthal) quantum number (refers, to the direction of the angular momentum vector. The magnetic quantum number m does not affect the electron's energy, but it does affect the probability cloud)- magnetic quantum number determines the energy shift of an atomic orbital due to an external magnetic field-Zeeman effect -s spin- intrinsic angular momentum Spin "up" and "down" allows two electrons for each set of spatial quantum numbers. The restrictions for the quantum numbers: – n = 1, 2, 3, 4, . – ℓ = 0, 1, 2, 3, . , n − 1 – mℓ = − ℓ, − ℓ + 1, . , 0, 1, . , ℓ − 1, ℓ – –Equivalently: n > 0 The energy levels are: ℓ < n |m | ≤ ℓ ℓ E E 0 n n2 Stern-Gerlach experiment If the particles were classical spinning objects, one would expect the distribution of their spin angular momentum vectors to be random and continuous. Each particle would be deflected by a different amount, producing some density distribution on the detector screen. Instead, the particles passing through the Stern–Gerlach apparatus are deflected either up or down by a specific amount. -
Dirac Notation Frank Rioux
Elements of Dirac Notation Frank Rioux In the early days of quantum theory, P. A. M. (Paul Adrian Maurice) Dirac created a powerful and concise formalism for it which is now referred to as Dirac notation or bra-ket (bracket ) notation. Two major mathematical traditions emerged in quantum mechanics: Heisenberg’s matrix mechanics and Schrödinger’s wave mechanics. These distinctly different computational approaches to quantum theory are formally equivalent, each with its particular strengths in certain applications. Heisenberg’s variation, as its name suggests, is based matrix and vector algebra, while Schrödinger’s approach requires integral and differential calculus. Dirac’s notation can be used in a first step in which the quantum mechanical calculation is described or set up. After this is done, one chooses either matrix or wave mechanics to complete the calculation, depending on which method is computationally the most expedient. Kets, Bras, and Bra-Ket Pairs In Dirac’s notation what is known is put in a ket, . So, for example, p expresses the fact that a particle has momentum p. It could also be more explicit: p = 2 , the particle has momentum equal to 2; x = 1.23 , the particle has position 1.23. Ψ represents a system in the state Q and is therefore called the state vector. The ket can also be interpreted as the initial state in some transition or event. The bra represents the final state or the language in which you wish to express the content of the ket . For example,x =Ψ.25 is the probability amplitude that a particle in state Q will be found at position x = .25. -
Dirac Equation - Wikipedia
Dirac equation - Wikipedia https://en.wikipedia.org/wiki/Dirac_equation Dirac equation From Wikipedia, the free encyclopedia In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it 1 describes all spin-2 massive particles such as electrons and quarks for which parity is a symmetry. It is consistent with both the principles of quantum mechanics and the theory of special relativity,[1] and was the first theory to account fully for special relativity in the context of quantum mechanics. It was validated by accounting for the fine details of the hydrogen spectrum in a completely rigorous way. The equation also implied the existence of a new form of matter, antimatter, previously unsuspected and unobserved and which was experimentally confirmed several years later. It also provided a theoretical justification for the introduction of several component wave functions in Pauli's phenomenological theory of spin; the wave functions in the Dirac theory are vectors of four complex numbers (known as bispinors), two of which resemble the Pauli wavefunction in the non-relativistic limit, in contrast to the Schrödinger equation which described wave functions of only one complex value. Moreover, in the limit of zero mass, the Dirac equation reduces to the Weyl equation. Although Dirac did not at first fully appreciate the importance of his results, the entailed explanation of spin as a consequence of the union of quantum mechanics and relativity—and the eventual discovery of the positron—represents one of the great triumphs of theoretical physics. -
The Quantum Mechanical Model of the Atom
The Quantum Mechanical Model of the Atom Quantum Numbers In order to describe the probable location of electrons, they are assigned four numbers called quantum numbers. The quantum numbers of an electron are kind of like the electron’s “address”. No two electrons can be described by the exact same four quantum numbers. This is called The Pauli Exclusion Principle. • Principle quantum number: The principle quantum number describes which orbit the electron is in and therefore how much energy the electron has. - it is symbolized by the letter n. - positive whole numbers are assigned (not including 0): n=1, n=2, n=3 , etc - the higher the number, the further the orbit from the nucleus - the higher the number, the more energy the electron has (this is sort of like Bohr’s energy levels) - the orbits (energy levels) are also called shells • Angular momentum (azimuthal) quantum number: The azimuthal quantum number describes the sublevels (subshells) that occur in each of the levels (shells) described above. - it is symbolized by the letter l - positive whole number values including 0 are assigned: l = 0, l = 1, l = 2, etc. - each number represents the shape of a subshell: l = 0, represents an s subshell l = 1, represents a p subshell l = 2, represents a d subshell l = 3, represents an f subshell - the higher the number, the more complex the shape of the subshell. The picture below shows the shape of the s and p subshells: (notice the electron clouds) • Magnetic quantum number: All of the subshells described above (except s) have more than one orientation. -
It Is a Great Pleasure to Write This Letter on Behalf of Dr. Maria
Rigorous Quantum-Classical Path Integral Formulation of Real-Time Dynamics Nancy Makri Departments of Chemistry and Physics University of Illinois The path integral formulation of time-dependent quantum mechanics provides the ideal framework for rigorous quantum-classical or quantum-semiclassical treatments, as the spatially localized, trajectory-like nature of the quantum paths circumvents the need for mean-field-type assumptions. However, the number of system paths grows exponentially with the number of propagation steps. In addition, each path of the quantum system generally gives rise to a distinct classical solvent trajectory. This exponential proliferation of trajectories with propagation time is the quantum-classical manifestation of nonlocality. A rigorous real-time quantum-classical path integral (QCPI) methodology has been developed, which converges to the stationary phase limit of the full path integral with respect to the degrees of freedom comprising the system’s environment. The starting point is the identification of two components in the effects induced on a quantum system by a polyatomic environment. The first, “classical decoherence mechanism” is associated with phonon absorption and induced emission and is dominant at high temperature. Within the QCPI framework, the memory associated with classical decoherence is removable. A second, nonlocal in time, “quantum decoherence process”, which is associated with spontaneous phonon emission, becomes important at low temperatures and is responsible for detailed balance. The QCPI methodology takes advantage of the memory-free nature of system-independent solvent trajectories to account for all classical decoherence effects on the dynamics of the quantum system in an inexpensive fashion. Inclusion of the residual quantum decoherence is accomplished via phase factors in the path integral expression, which is amenable to large time steps and iterative decompositions. -
4 Nuclear Magnetic Resonance
Chapter 4, page 1 4 Nuclear Magnetic Resonance Pieter Zeeman observed in 1896 the splitting of optical spectral lines in the field of an electromagnet. Since then, the splitting of energy levels proportional to an external magnetic field has been called the "Zeeman effect". The "Zeeman resonance effect" causes magnetic resonances which are classified under radio frequency spectroscopy (rf spectroscopy). In these resonances, the transitions between two branches of a single energy level split in an external magnetic field are measured in the megahertz and gigahertz range. In 1944, Jevgeni Konstantinovitch Savoiski discovered electron paramagnetic resonance. Shortly thereafter in 1945, nuclear magnetic resonance was demonstrated almost simultaneously in Boston by Edward Mills Purcell and in Stanford by Felix Bloch. Nuclear magnetic resonance was sometimes called nuclear induction or paramagnetic nuclear resonance. It is generally abbreviated to NMR. So as not to scare prospective patients in medicine, reference to the "nuclear" character of NMR is dropped and the magnetic resonance based imaging systems (scanner) found in hospitals are simply referred to as "magnetic resonance imaging" (MRI). 4.1 The Nuclear Resonance Effect Many atomic nuclei have spin, characterized by the nuclear spin quantum number I. The absolute value of the spin angular momentum is L =+h II(1). (4.01) The component in the direction of an applied field is Lz = Iz h ≡ m h. (4.02) The external field is usually defined along the z-direction. The magnetic quantum number is symbolized by Iz or m and can have 2I +1 values: Iz ≡ m = −I, −I+1, ..., I−1, I. -
Less Decoherence and More Coherence in Quantum Gravity, Inflationary Cosmology and Elsewhere
Less Decoherence and More Coherence in Quantum Gravity, Inflationary Cosmology and Elsewhere Elias Okon1 and Daniel Sudarsky2 1Instituto de Investigaciones Filosóficas, Universidad Nacional Autónoma de México, Mexico City, Mexico. 2Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Mexico City, Mexico. Abstract: In [1] it is argued that, in order to confront outstanding problems in cosmol- ogy and quantum gravity, interpretational aspects of quantum theory can by bypassed because decoherence is able to resolve them. As a result, [1] concludes that our focus on conceptual and interpretational issues, while dealing with such matters in [2], is avoidable and even pernicious. Here we will defend our position by showing in detail why decoherence does not help in the resolution of foundational questions in quantum mechanics, such as the measurement problem or the emergence of classicality. 1 Introduction Since its inception, more than 90 years ago, quantum theory has been a source of heated debates in relation to interpretational and conceptual matters. The prominent exchanges between Einstein and Bohr are excellent examples in that regard. An avoid- ance of such issues is often justified by the enormous success the theory has enjoyed in applications, ranging from particle physics to condensed matter. However, as people like John S. Bell showed [3], such a pragmatic attitude is not always acceptable. In [2] we argue that the standard interpretation of quantum mechanics is inadequate in cosmological contexts because it crucially depends on the existence of observers external to the studied system (or on an artificial quantum/classical cut). We claim that if the system in question is the whole universe, such observers are nowhere to be found, so we conclude that, in order to legitimately apply quantum mechanics in such contexts, an observer independent interpretation of the theory is required. -
Chapter 5 the Dirac Formalism and Hilbert Spaces
Chapter 5 The Dirac Formalism and Hilbert Spaces In the last chapter we introduced quantum mechanics using wave functions defined in position space. We identified the Fourier transform of the wave function in position space as a wave function in the wave vector or momen tum space. Expectation values of operators that represent observables of the system can be computed using either representation of the wavefunc tion. Obviously, the physics must be independent whether represented in position or wave number space. P.A.M. Dirac was the first to introduce a representation-free notation for the quantum mechanical state of the system and operators representing physical observables. He realized that quantum mechanical expectation values could be rewritten. For example the expected value of the Hamiltonian can be expressed as ψ∗ (x) Hop ψ (x) dx = ψ Hop ψ , (5.1) h | | i Z = ψ ϕ , (5.2) h | i with ϕ = Hop ψ . (5.3) | i | i Here, ψ and ϕ are vectors in a Hilbert-Space, which is yet to be defined. For exampl| i e, c|omi plex functions of one variable, ψ(x), that are square inte 241 242 CHAPTER 5. THE DIRAC FORMALISM AND HILBERT SPACES grable, i.e. ψ∗ (x) ψ (x) dx < , (5.4) ∞ Z 2 formt heHilbert-Spaceofsquareintegrablefunctionsdenoteda s L. In Dirac notation this is ψ∗ (x) ψ (x) dx = ψ ψ . (5.5) h | i Z Orthogonality relations can be rewritten as ψ∗ (x) ψ (x) dx = ψ ψ = δmn. (5.6) m n h m| ni Z As see aboveexpressions look likeabrackethecalledthe vector ψn aket vector and ψ a bra-vector. -
A Numerical Study of Quantum Decoherence 1 Introduction
Commun. Comput. Phys. Vol. 12, No. 1, pp. 85-108 doi: 10.4208/cicp.011010.010611a July 2012 A Numerical Study of Quantum Decoherence 1 2, Riccardo Adami and Claudia Negulescu ∗ 1 Dipartimento di Matematica e Applicazioni, Universit`adi Milano Bicocca, via R. Cozzi, 53 20125 Milano, Italy. 2 CMI/LATP (UMR 6632), Universit´ede Provence, 39, rue Joliot Curie, 13453 Marseille Cedex 13, France. Received 1 October 2010; Accepted (in revised version) 1 June 2011 Communicated by Kazuo Aoki Available online 19 January 2012 Abstract. The present paper provides a numerical investigation of the decoherence ef- fect induced on a quantum heavy particle by the scattering with a light one. The time dependent two-particle Schr¨odinger equation is solved by means of a time-splitting method. The damping undergone by the non-diagonal terms of the heavy particle density matrix is estimated numerically, as well as the error in the Joos-Zeh approxi- mation formula. AMS subject classifications: 35Q40, 35Q41, 65M06, 65Z05, 70F05 Key words: Quantum mechanics, Schr¨odinger equation, heavy-light particle scattering, interfer- ence, decoherence, numerical discretization, numerical analysis, splitting scheme. 1 Introduction Quantum decoherence is nowadays considered as the key concept in the description of the transition from the quantum to the classical world (see e.g. [6, 7, 9, 19, 20, 22, 24]). As it is well-known, the axioms of quantum mechanics allow superposition states, namely, normalized sums of admissible wave functions are once more admissible wave functions. It is then possible to construct non-localized states that lack a classical inter- pretation, for instance, by summing two states localized far apart from each other.