The Principle of Relativity and the De Broglie Relation Abstract
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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. -
On the Linkage Between Planck's Quantum and Maxwell's Equations
The Finnish Society for Natural Philosophy 25 years, K.V. Laurikainen Honorary Symposium, Helsinki 11.-12.11.2013 On the Linkage between Planck's Quantum and Maxwell's Equations Tuomo Suntola Physics Foundations Society, www.physicsfoundations.org Abstract The concept of quantum is one of the key elements of the foundations in modern physics. The term quantum is primarily identified as a discrete amount of something. In the early 20th century, the concept of quantum was needed to explain Max Planck’s blackbody radiation law which suggested that the elec- tromagnetic radiation emitted by atomic oscillators at the surfaces of a blackbody cavity appears as dis- crete packets with the energy proportional to the frequency of the radiation in the packet. The derivation of Planck’s equation from Maxwell’s equations shows quantizing as a property of the emission/absorption process rather than an intrinsic property of radiation; the Planck constant becomes linked to primary electrical constants allowing a unified expression of the energies of mass objects and electromagnetic radiation thereby offering a novel insight into the wave nature of matter. From discrete atoms to a quantum of radiation Continuity or discontinuity of matter has been seen as a fundamental question since antiquity. Aristotle saw perfection in continuity and opposed Democritus’s ideas of indivisible atoms. First evidences of atoms and molecules were obtained in chemistry as multiple proportions of elements in chemical reac- tions by the end of the 18th and in the early 19th centuries. For physicist, the idea of atoms emerged for more than half a century later, most concretely in statistical thermodynamics and kinetic gas theory that converted the mole-based considerations of chemists into molecule-based considerations based on con- servation laws and probabilities. -
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. -
Guide for the Use of the International System of Units (SI)
Guide for the Use of the International System of Units (SI) m kg s cd SI mol K A NIST Special Publication 811 2008 Edition Ambler Thompson and Barry N. Taylor NIST Special Publication 811 2008 Edition Guide for the Use of the International System of Units (SI) Ambler Thompson Technology Services and Barry N. Taylor Physics Laboratory National Institute of Standards and Technology Gaithersburg, MD 20899 (Supersedes NIST Special Publication 811, 1995 Edition, April 1995) March 2008 U.S. Department of Commerce Carlos M. Gutierrez, Secretary National Institute of Standards and Technology James M. Turner, Acting Director National Institute of Standards and Technology Special Publication 811, 2008 Edition (Supersedes NIST Special Publication 811, April 1995 Edition) Natl. Inst. Stand. Technol. Spec. Publ. 811, 2008 Ed., 85 pages (March 2008; 2nd printing November 2008) CODEN: NSPUE3 Note on 2nd printing: This 2nd printing dated November 2008 of NIST SP811 corrects a number of minor typographical errors present in the 1st printing dated March 2008. Guide for the Use of the International System of Units (SI) Preface The International System of Units, universally abbreviated SI (from the French Le Système International d’Unités), is the modern metric system of measurement. Long the dominant measurement system used in science, the SI is becoming the dominant measurement system used in international commerce. The Omnibus Trade and Competitiveness Act of August 1988 [Public Law (PL) 100-418] changed the name of the National Bureau of Standards (NBS) to the National Institute of Standards and Technology (NIST) and gave to NIST the added task of helping U.S. -
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. -
Turbulence Examined in the Frequency-Wavenumber Domain*
Turbulence examined in the frequency-wavenumber domain∗ Andrew J. Morten Department of Physics University of Michigan Supported by funding from: National Science Foundation and Office of Naval Research ∗ Arbic et al. JPO 2012; Arbic et al. in review; Morten et al. papers in preparation Andrew J. Morten LOM 2013, Ann Arbor, MI Collaborators • University of Michigan Brian Arbic, Charlie Doering • MIT Glenn Flierl • University of Brest, and The University of Texas at Austin Robert Scott Andrew J. Morten LOM 2013, Ann Arbor, MI Outline of talk Part I{Motivation (research led by Brian Arbic) • Frequency-wavenumber analysis: {Idealized Quasi-geostrophic (QG) turbulence model. {High-resolution ocean general circulation model (HYCOM)∗: {AVISO gridded satellite altimeter data. ∗We used NLOM in Arbic et al. (2012) Part II-Research led by Andrew Morten • Derivation and interpretation of spectral transfers used above. • Frequency-domain analysis in two-dimensional turbulence. • Theoretical prediction for frequency spectra and spectral transfers due to the effects of \sweeping." {moving beyond a zeroth order approximation. Andrew J. Morten LOM 2013, Ann Arbor, MI Motivation: Intrinsic oceanic variability • Interested in quantifying the contributions of intrinsic nonlinearities in oceanic dynamics to oceanic frequency spectra. • Penduff et al. 2011: Interannual SSH variance in ocean models with interannual atmospheric forcing is comparable to variance in high resolution (eddying) ocean models with no interannual atmospheric forcing. • Might this eddy-driven low-frequency variability be connected to the well-known inverse cascade to low wavenumbers (e.g. Fjortoft 1953)? • A separate motivation is simply that transfers in mixed ! − k space provide a useful diagnostic. Andrew J. -
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. -
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. -
Variable Planck's Constant
Variable Planck’s Constant: Treated As A Dynamical Field And Path Integral Rand Dannenberg Ventura College, Physics and Astronomy Department, Ventura CA [email protected] [email protected] January 28, 2021 Abstract. The constant ħ is elevated to a dynamical field, coupling to other fields, and itself, through the Lagrangian density derivative terms. The spatial and temporal dependence of ħ falls directly out of the field equations themselves. Three solutions are found: a free field with a tadpole term; a standing-wave non-propagating mode; a non-oscillating non-propagating mode. The first two could be quantized. The third corresponds to a zero-momentum classical field that naturally decays spatially to a constant with no ad-hoc terms added to the Lagrangian. An attempt is made to calibrate the constants in the third solution based on experimental data. The three fields are referred to as actons. It is tentatively concluded that the acton origin coincides with a massive body, or point of infinite density, though is not mass dependent. An expression for the positional dependence of Planck’s constant is derived from a field theory in this work that matches in functional form that of one derived from considerations of Local Position Invariance violation in GR in another paper by this author. Astrophysical and Cosmological interpretations are provided. A derivation is shown for how the integrand in the path integral exponent becomes Lc/ħ(r), where Lc is the classical action. The path that makes stationary the integral in the exponent is termed the “dominant” path, and deviates from the classical path systematically due to the position dependence of ħ. -
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. -
Variable Planck's Constant
Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 29 January 2021 doi:10.20944/preprints202101.0612.v1 Variable Planck’s Constant: Treated As A Dynamical Field And Path Integral Rand Dannenberg Ventura College, Physics and Astronomy Department, Ventura CA [email protected] Abstract. The constant ħ is elevated to a dynamical field, coupling to other fields, and itself, through the Lagrangian density derivative terms. The spatial and temporal dependence of ħ falls directly out of the field equations themselves. Three solutions are found: a free field with a tadpole term; a standing-wave non-propagating mode; a non-oscillating non-propagating mode. The first two could be quantized. The third corresponds to a zero-momentum classical field that naturally decays spatially to a constant with no ad-hoc terms added to the Lagrangian. An attempt is made to calibrate the constants in the third solution based on experimental data. The three fields are referred to as actons. It is tentatively concluded that the acton origin coincides with a massive body, or point of infinite density, though is not mass dependent. An expression for the positional dependence of Planck’s constant is derived from a field theory in this work that matches in functional form that of one derived from considerations of Local Position Invariance violation in GR in another paper by this author. Astrophysical and Cosmological interpretations are provided. A derivation is shown for how the integrand in the path integral exponent becomes Lc/ħ(r), where Lc is the classical action. The path that makes stationary the integral in the exponent is termed the “dominant” path, and deviates from the classical path systematically due to the position dependence of ħ. -
ATMS 310 Rossby Waves Properties of Waves in the Atmosphere Waves
ATMS 310 Rossby Waves Properties of Waves in the Atmosphere Waves – Oscillations in field variables that propagate in space and time. There are several aspects of waves that we can use to characterize their nature: 1) Period – The amount of time it takes to complete one oscillation of the wave\ 2) Wavelength (λ) – Distance between two peaks of troughs 3) Amplitude – The distance between the peak and the trough of the wave 4) Phase – Where the wave is in a cycle of amplitude change For a 1-D wave moving in the x-direction, the phase is defined by: φ ),( = −υtkxtx − α (1) 2π where φ is the phase, k is the wave number = , υ = frequency of oscillation (s-1), and λ α = constant determined by the initial conditions. If the observer is moving at the phase υ speed of the wave ( c ≡ ), then the phase of the wave is constant. k For simplification purposes, we will only deal with linear sinusoidal wave motions. Dispersive vs. Non-dispersive Waves When describing the velocity of waves, a distinction must be made between the group velocity and the phase speed. The group velocity is the velocity at which the observable disturbance (energy of the wave) moves with time. The phase speed of the wave (as given above) is how fast the constant phase portion of the wave moves. A dispersive wave is one in which the pattern of the wave changes with time. In dispersive waves, the group velocity is usually different than the phase speed. A non- dispersive wave is one in which the patterns of the wave do not change with time as the wave propagates (“rigid” wave).