DOCSLIB.ORG

Quantum Aspects of Light and Matter Notes on Quantum Mechanics

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Quantum Aspects of Light and Matter Notes on Quantum Mechanics Quantum aspects of light and matter Notes on Quantum Mechanics http://quantum.bu.edu/notes/QuantumMechanics/QuantumAspectsOfLightAndMatter.pdf Last updated Tuesday, September 20, 2005 8:07:07-05:00 Copyright © 2004 Dan Dill ([email protected]) Department of Chemistry, Boston University, Boston MA 02215 Near the end of his life Albert Einstein wrote, “All the fifty years of conscious brooding have brought me no closer to the answer to the question: What are light quanta? Of course today every rascal thinks he knows the answer, but he is deluding himself.” We are today in the same state of “learned ignorance” with respect to light as was Einstein. Arthur Zajonc, "Light Reconsidered," Optics & Photonic News, October 2003 So, chemistry is about atoms, but atoms are too small to "see" directly. The reason is that our eyes are sensitive to the visible range of light, corresponding to wavelengths from 380 nm (deep violet) to 740 nm (dark red), but even the shortest of these wavelengths is much, much larger than atoms, which are about 1 Þ = 10-8 cm = 0.1 nm across. What to do? Our only option is to explore the structure of atoms indirectly, using a fortunate feature of atoms that they do absorb and emit light at precisely defined wavelengths. It turns out that atoms of each element in the periodic table have a characteristic pattern of interaction with light, and this pattern, called a spectrum, serves as a fingerprint to identify the atoms of the different elements. The simplest example is the spectrum of atomic hydrogen. Here is the absorption spectrum of hydrogen atoms in the visible wavelength range. Absorption spectrum of gaseous hydrogen atoms at high temperature. Wavelength increases from 380 nm on the left to 740 nm on the right The dark lines mark those wavelengths of the light that hydrogen atoms absorb. In the visible region of the spectrum there are only four such absorption lines for hydrogen atoms. Other atoms typically absorb light at many, many more different wavelengths. Absorption spectra are typically seen in light emitted from stars as it passes through clouds of atoms. In the laboratory it is more typical to see emission spectra. Here is a representation of the emission spectrum of hydrogen atoms in the visible wavelength range. Emission spectrum of gaseous hydrogen atoms at high temperature. Wavelength increases from 380 nm on the left to 740 nm on the right A nice display of the emission spectrum of the Sun and the atomic absorption spectra of hydrogen, helium, mercury and uranium is at http://quantum.bu.edu/images/atomSpectra.jpeg. To keep things as simple as possible, let's set as our goal trying to understand the hydrogen spectrum and what this understanding can teach us about the atom. 2 Quantum aspects of light and matter à Properties of waves The very first step in using the interaction of light with atoms to sort out what is going on inside atoms is to understand what light itself is. In simplest terms, light is paired electric and magnetic field that oscillate in unison perpendicular to the direction of travel of the light. The magnetic field has a negligible effect on atoms compared to the electric field and so we'll focus our attention on the oscillations of the electrical component of light. (Because of the simultaneous presence of oscillating electric and magnetic fields, light is also called electromagnetic radiation. Sometime, too, the term light is used to mean just the visible range of light. We will use light to mean electromagnetic radiation, independently of whether it is visible to our eyes.) To do this, we begin by describing the mathematics of waves. A wave is the oscillatory variation of some property in time and in space. If we focus on a particular point in space, the value of the changing property—the height of the wave—is seen to oscillate about some average value. If instead we focus on a particular part of the wave, such a one of its crests—a point of maximum amplitude—the wave is seen to move through space. Examples of waves are Wave type Changing quantity water height of water sound density of air chemical concentration light electric and magnetic fields A review of sines and cosines: SOH-CAH-TOA The trigonometric functions sine, cosine, and tangent are used in working with waves. Hopefully these are familiar to you from high school, but here is a brief review of these functions and their values. The sine of the angle q (the Greek letter "theta"), sin q , subtended by a right triangle with hypotenuse H and side O opposite the angle q is O H. The cosine of the angle q, cos(q), subtended by a right triangle with hypotenuse H and side A adjacent to the angle q is A H. The tangent of the angle q, tan(q), subtended by a right triangle side O oppositeH L the angle q and side A adjacent to the angle q is O A. ê ê Show that a consequence of these definitions is that the tangent if an angle is the ratio of itsê sine and cosine, tan q = sin q cos q . The so-called "American Indian chief" mnemonic SOH-CAH-TOA incorporates these definitions: SOH means Sine is Opposite overH Hypotenuse;L H L ê CAHH L means Cosine is Adjacent over Hypotenuse; and TOA means Tangent is Opposite over Adjacent. For angles greater than 90 °, the following identities can be used to express the values of the trigonometric functions in terms of those for angles in the range 0 §q§90 °. sin a ≤ b = sin a cos b ≤ cos a sin b cos a ≤ b = cos a cos b ¡ sin a sin b H L H L H L H L H L H L H L H L H L H L Copyright © 2004 Dan Dill ([email protected]). All rights reserved Quantum aspects of light and matter 3 For example, at 45 ° the sides and hypotenuse of a right triangle are in the ratio 1 2 = 0.71. This means sin 45 ° = cos 45 ° = 1 2 . Using the identities, we can evaluate the cosine of 135° to be è!!! cos 135 ° ë è!!! H L = Hcos 90L ° +ë45 ° = cos 90 ° cos 45 ° - sin 90 ° sin 45 ° H =L 0 μ 1 2 - 1 μ 1 2 H L =-1 2 H L H L H L H L è!!! è!!! ë ë Use the identities to show that sin -45 ° =-1 2. è!!! ë The two trigonometric identities are so useful that it is well worthwhile to memorize them. è!!! H L ë Usually it is easiest to determine the values of sine and cosine of an angle with reference to plots of the sine and cosine and a few key values. Here are the plots, with key values marked. sin q , cos q 1 H L H L 0.5 q p p 3 p 2p ÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ 2 2 -0.5 -1 Plot of the sine (black curve and points) and cosine (gray curve and points). The points on the curve are for angles 0, p 6 = 30 °, p 4 = 45 °, p 3 = 60 °, p 2 = 90 °, etc. Here is a table of the values of sine and cosine for key angles q. ê ê ê ê q degrees q radians sin q cos q 0001 30 p 612 3 2 H L H L H L H L 45 p 4121 2 è!!! 60 p ê 3 3ê 21ë2 è!!! è!!! 90 p ê 210ë ë è!!! Values of sine and cosine for key anglesê q. Key decimalë valuesê are 0, 0.5, 1 2 = 0.71, 3 2 = 0.87, and 1. ê The decimal values for the key angles are 0, 0.5, 1 2 = 0.71,è!!!! 3 è2!!!!= 0.87, and 1. It is easy to ë ë memorize the values of sine and cosine—which you should do—since their values in the first quadrant, 0 §q§p 2, determine the values in the other three quadrants in way that is easy to see è!!! è!!! using a sketch of the two functions. Once these valuesë are memorized,ë determining the value of the sine or cosine of an angle is not hard at all. ê Copyright © 2004 Dan Dill ([email protected]). All rights reserved 4 Quantum aspects of light and matter What is sin 405 ° ? Answer: 1 2 = 0.71. What is tan -45 ° ? Answer: -1. è!!! H L ë What is sin -30 ° ? Answer: -1 2. H L What is cos -180 ° ? Answer: -1. H L ê Representing waveH propertiesL mathematically Waves can be expressed mathematically in terms of sine or cosine curves. The value of the curve at a particular time and location represents the amplitude of the wave at that time and location. For example, to represent a sound wave with a sine curve, the value of the sine curve at a given time at each point in space corresponds to the pressure, relative to ambient pressure, at that time and location. What does a sine curve represent for a water wave? What does a sine curve represent for the electric field part of a light (electromagnetic) wave? The most striking feature of waves is that they move through space (standing waves, which oscillate in place, are a special case). To keep things as simple as possible, let's assume we have a wave only along one direction, which we call x. If we look at a wave at a particular instant in time, it is a sine curve, say.
Load more

Recommended publications
  • Schrödinger Equation)
    Lecture 37 (Schrödinger Equation) Physics 2310-01 Spring 2020 Douglas Fields Reduced Mass • OK, so the Bohr model of the atom gives energy levels: • But, this has one problem – it was developed assuming the acceleration of the electron was given as an object revolving around a fixed point. • In fact, the proton is also free to move. • The acceleration of the electron must then take this into account. • Since we know from Newton’s third law that: • If we want to relate the real acceleration of the electron to the force on the electron, we have to take into account the motion of the proton too. Reduced Mass • So, the relative acceleration of the electron to the proton is just: • Then, the force relation becomes: • And the energy levels become: Reduced Mass • The reduced mass is close to the electron mass, but the 0.0054% difference is measurable in hydrogen and important in the energy levels of muonium (a hydrogen atom with a muon instead of an electron) since the muon mass is 200 times heavier than the electron. • Or, in general: Hydrogen-like atoms • For single electron atoms with more than one proton in the nucleus, we can use the Bohr energy levels with a minor change: e4 → Z2e4. • For instance, for He+ , Uncertainty Revisited • Let’s go back to the wave function for a travelling plane wave: • Notice that we derived an uncertainty relationship between k and x that ended being an uncertainty relation between p and x (since p=ћk): Uncertainty Revisited • Well it turns out that the same relation holds for ω and t, and therefore for E and t: • We see this playing an important role in the lifetime of excited states.
    [Show full text]
  • Uniting the Wave and the Particle in Quantum Mechanics
    Uniting the wave and the particle in quantum mechanics Peter Holland1 (final version published in Quantum Stud.: Math. Found., 5th October 2019) Abstract We present a unified field theory of wave and particle in quantum mechanics. This emerges from an investigation of three weaknesses in the de Broglie-Bohm theory: its reliance on the quantum probability formula to justify the particle guidance equation; its insouciance regarding the absence of reciprocal action of the particle on the guiding wavefunction; and its lack of a unified model to represent its inseparable components. Following the author’s previous work, these problems are examined within an analytical framework by requiring that the wave-particle composite exhibits no observable differences with a quantum system. This scheme is implemented by appealing to symmetries (global gauge and spacetime translations) and imposing equality of the corresponding conserved Noether densities (matter, energy and momentum) with their Schrödinger counterparts. In conjunction with the condition of time reversal covariance this implies the de Broglie-Bohm law for the particle where the quantum potential mediates the wave-particle interaction (we also show how the time reversal assumption may be replaced by a statistical condition). The method clarifies the nature of the composite’s mass, and its energy and momentum conservation laws. Our principal result is the unification of the Schrödinger equation and the de Broglie-Bohm law in a single inhomogeneous equation whose solution amalgamates the wavefunction and a singular soliton model of the particle in a unified spacetime field. The wavefunction suffers no reaction from the particle since it is the homogeneous part of the unified field to whose source the particle contributes via the quantum potential.
    [Show full text]
  • 1 the Principle of Wave–Particle Duality: an Overview
    3 1 The Principle of Wave–Particle Duality: An Overview 1.1 Introduction In the year 1900, physics entered a period of deep crisis as a number of peculiar phenomena, for which no classical explanation was possible, began to appear one after the other, starting with the famous problem of blackbody radiation. By 1923, when the “dust had settled,” it became apparent that these peculiarities had a common explanation. They revealed a novel fundamental principle of nature that wascompletelyatoddswiththeframeworkofclassicalphysics:thecelebrated principle of wave–particle duality, which can be phrased as follows. The principle of wave–particle duality: All physical entities have a dual character; they are waves and particles at the same time. Everything we used to regard as being exclusively a wave has, at the same time, a corpuscular character, while everything we thought of as strictly a particle behaves also as a wave. The relations between these two classically irreconcilable points of view—particle versus wave—are , h, E = hf p = (1.1) or, equivalently, E h f = ,= . (1.2) h p In expressions (1.1) we start off with what we traditionally considered to be solely a wave—an electromagnetic (EM) wave, for example—and we associate its wave characteristics f and (frequency and wavelength) with the corpuscular charac- teristics E and p (energy and momentum) of the corresponding particle. Conversely, in expressions (1.2), we begin with what we once regarded as purely a particle—say, an electron—and we associate its corpuscular characteristics E and p with the wave characteristics f and of the corresponding wave.
    [Show full text]
  • Path Integral Quantum Monte Carlo Study of Coupling and Proximity Effects in Superfluid Helium-4
    Path Integral Quantum Monte Carlo Study of Coupling and Proximity Effects in Superfluid Helium-4 A Thesis Presented by Max T. Graves to The Faculty of the Graduate College of The University of Vermont In Partial Fullfillment of the Requirements for the Degree of Master of Science Specializing in Materials Science October, 2014 Accepted by the Faculty of the Graduate College, The University of Vermont, in partial fulfillment of the requirements for the degree of Master of Science in Materials Science. Thesis Examination Committee: Advisor Adrian Del Maestro, Ph.D. Valeri Kotov, Ph.D. Frederic Sansoz, Ph.D. Chairperson Chris Danforth, Ph.D. Dean, Graduate College Cynthia J. Forehand, Ph.D. Date: August 29, 2014 Abstract When bulk helium-4 is cooled below T = 2.18 K, it undergoes a phase transition to a su- perfluid, characterized by a complex wave function with a macroscopic phase and exhibits inviscid, quantized flow. The macroscopic phase coherence can be probed in a container filled with helium-4, by reducing one or more of its dimensions until they are smaller than the coherence length, the spatial distance over which order propagates. As this dimensional reduction occurs, enhanced thermal and quantum fluctuations push the transition to the su- perfluid state to lower temperatures. However, this trend can be countered via the proximity effect, where a bulk 3-dimensional (3d) superfluid is coupled to a low (2d) dimensional su- perfluid via a weak link producing superfluid correlations in the film at temperatures above the Kosterlitz-Thouless temperature. Recent experiments probing the coupling between 3d and 2d superfluid helium-4 have uncovered an anomalously large proximity effect, leading to an enhanced superfluid density that cannot be explained using the correlation length alone.
    [Show full text]
  • Matter-Wave Diffraction of Quantum Magical Helium Clusters Oleg Kornilov * and J
    s e r u t a e f Matter-wave diffraction of quantum magical helium clusters Oleg Kornilov * and J. Peter Toennies , Max-Planck-Institut für Dynamik und Selbstorganisation, Bunsenstraße 10 • D-37073 Göttingen • Germany. * Present address: Chemical Sciences Division, Lawrence Berkeley National Lab., University of California • Berkeley, CA 94720 • USA. DOI: 10.1051/epn:2007003 ack in 1868 the element helium was discovered in and named Finally, helium clusters are the only clusters which are defi - Bafter the sun (Gr.“helios”), an extremely hot environment with nitely liquid. This property led to the development of a new temperatures of millions of degrees. Ironically today helium has experimental technique of inserting foreign molecules into large become one of the prime objects of condensed matter research at helium clusters, which serve as ultra-cold nano-containers. The temperatures down to 10 -6 Kelvin. With its two very tightly bound virtually unhindered rotations of the trapped molecules as indi - electrons in a 1S shell it is virtually inaccessible to lasers and does cated by their sharp clearly resolved spectral lines in the infrared not have any chemistry. It is the only substance which remains liq - has been shown to be a new microscopic manifestation of super - uid down to 0 Kelvin. Moreover, owing to its light weight and very fluidity [3], thus linking cluster properties back to the bulk weak van der Waals interaction, helium is the only liquid to exhib - behavior of liquid helium. it one of the most striking macroscopic quantum effects: Today one of the aims of modern helium cluster research is to superfluidity.
    [Show full text]
  • Lecture 3: Wave Properties of Matter
    3 WAVE PROPERTIES OF MATTER 1 3 Wave properties of matter In Lecture 1 we mapped out some of the principal interactions between photons and matter (elec- trons). At the quantum level, individual photons and electrons are described as particles in the various relationships used to describe phenomena such as the photo{electric effect and Compton scattering. We are building up a map of quantum interactions (Table 1). Under certain circum- particle wave electron Thomson's experiment Photo{electric effect Compton effect low{energy photon photo{electric effect interference diffraction high{energy photon Compton effect Pair production Table 1: A simple map of quantum behaviour. stances, low energy photons can act as particles or as waves. Importantly, such photons never display both wave{like and particle{like properties in the same experiment (Two-slit experiment). The blank spaces in Table 1 provide the directions we must investigate in order to complete our simple quantum map: can high energy photons, X{ and γ{rays, act as waves, and can electrons { classically viewed as matter particles { also act in a wave{like manner? • X{ray diffraction { high energy photons can behave as waves. • De Broglie suggests a relationship between momentum (mass) and wavelength { a wave{like description of matter? • Electron scattering and transmission electron diffraction photography confirm wave phenom- ena in electrons. • A wave mechanics primer. • The uncertainty principle derived using wave physics. • Bohr's complementarity principle and an introduction to the two slit experiment. • The Copenhagen interpretation of quantum theory. • A wave description of a particle in a box leads to quantised energy states.
    [Show full text]
  • The Principle of Relativity and the De Broglie Relation Abstract
    The principle of relativity and the De Broglie relation Julio G¨u´emez∗ Department of Applied Physics, Faculty of Sciences, University of Cantabria, Av. de Los Castros, 39005 Santander, Spain Manuel Fiolhaisy Physics Department and CFisUC, University of Coimbra, 3004-516 Coimbra, Portugal Luis A. Fern´andezz Department of Mathematics, Statistics and Computation, Faculty of Sciences, University of Cantabria, Av. de Los Castros, 39005 Santander, Spain (Dated: February 5, 2016) Abstract The De Broglie relation is revisited in connection with an ab initio relativistic description of particles and waves, which was the same treatment that historically led to that famous relation. In the same context of the Minkowski four-vectors formalism, we also discuss the phase and the group velocity of a matter wave, explicitly showing that, under a Lorentz transformation, both transform as ordinary velocities. We show that such a transformation rule is a necessary condition for the covariance of the De Broglie relation, and stress the pedagogical value of the Einstein-Minkowski- Lorentz relativistic context in the presentation of the De Broglie relation. 1 I. INTRODUCTION The motivation for this paper is to emphasize the advantage of discussing the De Broglie relation in the framework of special relativity, formulated with Minkowski's four-vectors, as actually was done by De Broglie himself.1 The De Broglie relation, usually written as h λ = ; (1) p which introduces the concept of a wavelength λ, for a \material wave" associated with a massive particle (such as an electron), with linear momentum p and h being Planck's constant. Equation (1) is typically presented in the first or the second semester of a physics major, in a curricular unit of introduction to modern physics.
    [Show full text]
  • High Energy Physics Quantum Information Science Awards Abstracts
    High Energy Physics Quantum Information Science Awards Abstracts Towards Directional Detection of WIMP Dark Matter using Spectroscopy of Quantum Defects in Diamond Ronald Walsworth, David Phillips, and Alexander Sushkov Challenges and Opportunities in Noise‐Aware Implementations of Quantum Field Theories on Near‐Term Quantum Computing Hardware Raphael Pooser, Patrick Dreher, and Lex Kemper Quantum Sensors for Wide Band Axion Dark Matter Detection Peter S Barry, Andrew Sonnenschein, Clarence Chang, Jiansong Gao, Steve Kuhlmann, Noah Kurinsky, and Joel Ullom The Dark Matter Radio‐: A Quantum‐Enhanced Dark Matter Search Kent Irwin and Peter Graham Quantum Sensors for Light-field Dark Matter Searches Kent Irwin, Peter Graham, Alexander Sushkov, Dmitry Budke, and Derek Kimball The Geometry and Flow of Quantum Information: From Quantum Gravity to Quantum Technology Raphael Bousso1, Ehud Altman1, Ning Bao1, Patrick Hayden, Christopher Monroe, Yasunori Nomura1, Xiao‐Liang Qi, Monika Schleier‐Smith, Brian Swingle3, Norman Yao1, and Michael Zaletel Algebraic Approach Towards Quantum Information in Quantum Field Theory and Holography Daniel Harlow, Aram Harrow and Hong Liu Interplay of Quantum Information, Thermodynamics, and Gravity in the Early Universe Nishant Agarwal, Adolfo del Campo, Archana Kamal, and Sarah Shandera Quantum Computing for Neutrino‐nucleus Dynamics Joseph Carlson, Rajan Gupta, Andy C.N. Li, Gabriel Perdue, and Alessandro Roggero Quantum‐Enhanced Metrology with Trapped Ions for Fundamental Physics Salman Habib, Kaifeng Cui1,
    [Show full text]
  • Exploring the Universe with Matter Waves
    NEWS & VIEWS RESEARCH maintains a steady functionality despite the motifs. Such a hierarchy of motor controllers Unknown, 1400–038 Lisbon, Portugal. animal’s continuously increasing body size. has long been thought to be a key principle e-mails: adrien.jouary@neuro. Next, the researchers investigated underlying behaviour in most animals, includ- fchampalimaud.org; christian.machens@ the dynamics of radial-muscle contrac- ing humans6. However, recording the activity neuro.fchampalimaud.org tion and relaxation around tens of thou- of every muscle in a human is currently impos- 1. Reiter, S. et al. Nature 562, 361–366 (2018). sands of chromato phores. They discovered sible. The simple readout provided by the skin- 2. Mather, J. A. & Dickel, L. Curr. Opin. Behav. Sci. 16, co-variations in muscle movements at many display system of cuttlefish could well lead us 131–137 (2017). spatial scales, indicating that chromatophores to a greater understanding of motor control. ■ 3. Hanlon, R. T. & Messenger, J. B. Phil. Trans. R. Soc. B are regulated by modules of motor neurons that 320, 437–487 (1988). 4. Messenger, J. B. Biol. Rev. 76, 473–528 (2001). function in synchrony, and that operate on skin Adrien Jouary and Christian K. Machens 5. Churchland, M. M. et al. Nature 487, 51–56 (2012). patches of different sizes. The smallest modules are in the Champalimaud Neuroscience 6. Lashley, K. S. in Cerebral Mechanisms in Behavior consisted of fewer than ten adjacent chromato- Programme, Champalimaud Centre for the (ed. Jefffries, L. A.) 112–136 (Wiley, 1951). phores of the same colour. By contrast, larger modules, when contracted in synchrony, dis- played more-complex shapes, such as rings, QUANTUM PHYSICS rectangles or disjointed structures resembling eye spots.
    [Show full text]
  • The Dynamics of Wave-Particle Duality
    Journal of Applied Mathematics and Physics, 2018, 6, 1840-1859 http://www.scirp.org/journal/jamp ISSN Online: 2327-4379 ISSN Print: 2327-4352 The Dynamics of Wave-Particle Duality Adriano Orefice*, Raffaele Giovanelli, Domenico Ditto Department of Agricultural and Environmental Sciences (DISAA), University of Milan, Milano, Italy How to cite this paper: Orefice, A., Gi- Abstract ovanelli, R. and Ditto, D. (2018) The Dy- namics of Wave-Particle Duality. Journal of Both classical and wave-mechanical monochromatic waves may be treated in Applied Mathematics and Physics, 6, terms of exact ray-trajectories (encoded in the structure itself of Helm- 1840-1859. holtz-like equations) whose mutual coupling is the one and only cause of any https://doi.org/10.4236/jamp.2018.69157 diffraction and interference process. In the case of Wave Mechanics, de Brog- Received: August 22, 2018 lie’s merging of Maupertuis’s and Fermat’s principles (see Section 3) pro- Accepted: September 15, 2018 vides, without resorting to the probability-based guidance-laws and flow-lines Published: September 18, 2018 of the Bohmian theory, the simple law addressing particles along the Helm- Copyright © 2018 by authors and holtz rays of the relevant matter waves. The purpose of the present research Scientific Research Publishing Inc. was to derive the exact Hamiltonian ray-trajectory systems concerning, re- This work is licensed under the Creative spectively, classical electromagnetic waves, non-relativistic matter waves and Commons Attribution International License (CC BY 4.0). relativistic matter waves. We faced then, as a typical example, the numerical http://creativecommons.org/licenses/by/4.0/ solution of non-relativistic wave-mechanical equation systems in a number of Open Access numerical applications, showing that each particle turns out to “dances a wave-mechanical dance” around its classical trajectory, to which it reduces when the ray-coupling is neglected.
    [Show full text]
  • What Might the Matter Wave Be Telling Us of the Nature of Matter?
    What might the matter wave be telling us of the nature of matter? Daniel Shanahan PO Box 301, Cleveland, Queensland, 4163, Australia August 21, 2019 Abstract Various attempts at a thoroughly wave-theoretic explanation of mat- ter have taken as their fundamental ingredient the de Broglie or matter wave. But that wave is superluminal whereas it is implicit in the Lorentz transformation that influences propagate ultimately at the velocity c of light. It is shown that if the de Broglie wave is understood, not as a wave in its own right, but as the relativistically induced modulation of an underlying standing wave comprising counter-propagating influences of velocity c, the energy, momentum, mass and inertia of a massive particle can be explained from the manner in which the modulated wave struc- ture must adapt to a change of inertial frame. With those properties of the particle explained entirely from wave structure, nothing remains to be apportioned to anything discrete or “solid”within the wave. Considera- tion may thus be given to the possibility of wave-theoretic explanations of particle trajectories, and to a deeper understanding of the Klein-Gordon, Schrödinger and Dirac equations, all of which were conceived as equations for the de Broglie wave. Keywords de Broglie wave Planck-Einstein relation wave-particle duality inertia pilot wave theory· Dirac bispinor Lorentz· transforma- tion · · · · 1 Introduction It might be thought that the de Broglie wave can say very little regarding the nature of solid matter. As this “matter wave”is usually understood, it seems to make no sense at all.
    [Show full text]
  • An Introduction to Quantum Computing and Quantum Random Number Generators
    An introduction to Quantum Computing and Quantum Random Number Generators Giuseppe Vallone email: [email protected] INFN and The Future of Scientific Computing - 4 May 2018 Summary 1 Introduction 2 Quantum Computing Circuit model Alternatives to circuit model Implementations 3 Quantum Random Number Generators 4 Conclusions Pag. 2 Intro Quantum Computing QRNG Conclusions Summary 1 Introduction 2 Quantum Computing Circuit model Alternatives to circuit model Implementations 3 Quantum Random Number Generators 4 Conclusions Pag. 3 Intro Quantum Computing QRNG Conclusions What is Quantum Information? Information Theory Quantum Mechanics , Merging two big XXth century revolutions: information theory (Shannon, Turing) and Quantum Mechanics. Pag. 4 Intro Quantum Computing QRNG Conclusions Examples of applications Quantum computer Quantum cryptography Quantum metrology Pag. 5 Intro Quantum Computing QRNG Conclusions Examples of applications Quantum sensing Quantum imaging Quantum random number Quantum simulation generation Pag. 6 Intro Quantum Computing QRNG Conclusions But... ...be aware of fake! Pag. 7 Intro Quantum Computing QRNG Conclusions Summary 1 Introduction 2 Quantum Computing Circuit model Alternatives to circuit model Implementations 3 Quantum Random Number Generators 4 Conclusions Pag. 8 Intro Quantum Computing QRNG Conclusions Summary 1 Introduction 2 Quantum Computing Circuit model Alternatives to circuit model Implementations 3 Quantum Random Number Generators 4 Conclusions Pag. 9 Intro Quantum Computing QRNG Conclusions One-slide
    [Show full text]