DOCSLIB.ORG

Lecture 1: Old Quantum Theory

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Lecture 1: Old Quantum Theory Lecture 1: Old Quantum Theory September 3, 2019 1 Early Atom Models 1. Thompson Model ∆p F ∆t ZZ e2=4π" R2 · (2R=v) ZZ e2 θ ∼ ∼ ∼ 1 0 ∼ 1 p p mv RE 4π"0 e2 e02 = = 1:44fm Mev 4π"0 5 Z −5 - For R ∼ 1:0Å = 10 fm, α particle Z = 2, θ ∼ E × 10 (very small) - The effect of electrons can be neglected (because the mass of electron is very small) 2. Rutherford Model (a) Coulomb scattering formula 1 a θ ZZ e02 Coulomb Scattering Formula: b = cot with a = 1 scattering factor 2 2 E dv ZZ e02 m = 1 ^r dt r2 dφ mr2 = L = mvb dt The trajectory is then ZZ e02 dv = 1 ^rdφ mvb Integrate both sides Z f θ dv = vf − vi = 2v sin i 2 Z f Z π−θ θ θ θ ^rdφ = (i cos φ + j sin φ) dφ = i sin θ+j (1 + cos θ) = 2 cos i sin + j cos i 0 2 2 2 θ θ Note the direction of vf − vi is (π − θ) =2, i.e, i sin 2 + j cos 2 , the R f same with i ^rdφ. Thus we have θ θ ZZ e02 2v sin = 2 cos · 1 2 2 mvb ZZ e02 θ a θ b = 1 cot = cot mv2 2 2 2 with Coulomb scattering factor ZZ e02 a = 1 : E Clearly, θ ! 0 for b ! 1, and θ = π for b = 0. 2 (b) Rutherform Scattering Formula i. Question: Given N incident particles, how many scattering in the solid angle dΩ = 2π sin θdθ per unit area, per target particle? ii. This gives the differential cross section (DCS) 1 dN 0 σ (θ) = c Nnd dΩ where n is the number density and d the sample depth (ndA gives the number of target particles with A the area) iii. Particles in b ! b + db, scattering between angle θ ! θ + dθ. Assume the target particles scatter the incident particle inde- pendently. dN 0 2πbjdbj a2 cot (θ=2) = · (ndA) = 2π dθ · nd N A 8 sin2 (θ=2) Thus 1 dN 0 a2 cot (θ=2) 1 a2 σc (θ) = = 2π dθ = Nnd dΩ 8 sin2 (θ=2) 2π sin θdθ 16 sin4 (θ=2) iv. Written explicitly 02 2 ZZ1e 1 σc (θ) = · 4E sin4 (θ=2) v. This DCS can be verified experimentally. It has the dimension of area (Note e02 = 1:44 fm Mev). If this formula holds, then dN 0 sin4 (θ=2), dN 0 · E2, and dN 0=Z2 should be constants given other paramaters fixed. These were verified in 1913. (c) Rutherford Planet model fails for the following reasons. A charged particle like the electron circulating in orbit would be expected to radiate light, with the same frequency as the orbital motion. The frequencies of these orbital motions could be anything. Worse, as the electron lost energy to radiation it would spiral down into the atomic nucleus. Atoms cannot remain stable. 2 Planck Quantum Concept 1. Rayleigh-Jeans formula 3 (a) Question: The energy density E (ν; T ) in the frequency range ν ! ν + dν at temperature T ? (b) According to classical physics, the radiation field is the sum of Fourier normal modes proportional to exp (iq · r) with q = 2πn=L where n = (n1; n2; n3) and L is the length of a cubic box. n1;2;3 are intergers. 3 (c) # of modes within dq = dqxdqydqz is dn = (L=2π) dq. # of modes V 2 V 2 with q ! q + dq is then 8π3 4πq dq = 2π2 q dq. Since q = 2π/λ, ν = c/λ, thus q = 2πν=c. (d) The number of normal modes in the range ν ! ν + dν is then V V 2πν 2 2π 8πν2V N (ν) dν = 2 × q2dq = 2 × dν = dν 2π2 2π2 c c c3 where the first factor 2 stands for two polorization directions. (e) At temperture T , according to classical statistical mechanics, the average energy of the normal modes (viewed as harmonic oscillators) is given by R 1 0 exp (−βE) EdE −1 E = R 1 = β = kBT 0 exp (−βE) dE (f) The energy density is then N (v) dν · E 8πν2 E (ν; T ) dν = = · k T dν L3 c3 B which is Rayleigh-Jeans (RJ) formula (g) The R-J formula fails in two aspects: Deviates badly for large ν, called ultraviolet catastrophe. Also the integration R E (ν; T ) dν di- verges if it holds for any ν. 4 2. Planck Formula (a) 1900, proceedings of German Physical Society, Planck noted the data can be fitted very well by (a guess work) 8πv2 hν E (ν; T ) dν = 3 · dν c exp (hν=kBT ) − 1 with fitting constants −23 −34 −27 kB ' 1:4 × 10 J/K and h ' 6:6 × 10 J·s = 6:63 × 10 erg s (b) Planck’s quantization assumption: the radiation was the same as if it were in equilibrium with a large number of charged oscillators with different frequencies, the energy of any oscillator of frequency ν being an integer multiple of hν. Later, Lorenz gave a derivation in 1910, simply assuming discrete energy: 1 1 ! P exp (−βnhν) · (nhν) @ X E = 0 = − ln exp (−βnhν) P1 exp (−βnhν) @β 0 0 @ 1 hν exp (−βhν) hν = − ln = = @β 1 − exp (−βhν) 1 − exp (−βhν) exp (βhν) − 1 (c) Hence 8πν2 8πv2 hν E (ν; T ) dν = 3 · Edν = 3 · dν c c exp (hν=kBT ) − 1 Z 1 5 4 4 8π kB E (ν; T ) dv = αBT with αB = 3 3 0 15h c (d) Application of Planck’s work: Calculate NA = R=kB, the size of atom, the electron charge ... 3 Einstein Photon Assumption 1. Photo-Electronic Effect Some basic facts: 5 (a) About 10−9s, electronic current becomes saturated immediately (b) Threshold frequency, below which no photo-electron comes out Classically, the energy of lights are proportional to the intensity, which cannot understand the above facts. Einstein assumes light quanta: E = hν and the kinetic energy of photo-electron is given by K = hν − W 2. Compton Scattering (1922-23) Accordingly, the momentum of the photon is given by p = E=c = h/λ m2c4 = E2 − p2c2 = 0 For X-ray scattering by electron, assuming the light scattering backwards with frequency ν0, then the electron forwards with momentum hν=c + hν0=c, and energy conservation gives s hν + hν0 2 hν + m c2 = hν0 + m2c4 + c2 e e c Thus 2 0 2 2 0 0 mec ν 2h (ν − ν ) mec = 4h νν ! ν = 2 2hν + mec Written in the wave-length 2h λ0 = c/ν0 = λ + = λ + 2 × 2:425 × 10−10 cm mec −10 h=mec = 2:425 × 10 cm is known as Comptom wavelength of electron. If scattering at an angle θ, the factor 2 before h=mec is replaced by 1 − cos θ. Later after verification of this relation, chemist Lewis gave the name photon. 4 Bohr Model 1. Bohr was a visitor to Rutherford’s group in 1913. He proposed in the first place that the energies of atoms are quantized, in the sense that atoms exist in only a discrete set of states. • Discrete stable orbital: v2 Ze02 m = r r2 6 • Angular momentum quantization: (~ is a new parameter) n mvr = n (n = 1; 2; 3 ::: ) ! r = ~ ~ mv • Substitute into the first Eq., we get Ze02 vn = n~ 2 2 2 n~ n ~ 2 a1 ~ rn = = 02 = n with a1 = 02 mvn mZe Z me • Energy 02 2 04 1 2 Ze 1 Z mee En = mvn − = − 2 2 2 rn 2 n ~ • Transition frequency E − E 1 1 Z2m e04 ν = m n = − e h n2 m2 2~2h For transition n ! n + 1 and n ! 1, 2 Z2m e02 Z2m e04 ν ' · e = e n3 2~2h n3~2h • Correspondence principle: If an electron in an atom is moving on an orbit with period T, classically the electromagnetic radiation will repeat itself every orbital period. If the coupling to the electromag- netic field is weak, so that the orbit doesn’t decay very much in one cycle, the radiation will be emitted in a pattern which repeats every period, so that the Fourier transform will have frequencies which are only multiples of 1/T. This is the classical radiation law: the fre- quencies emitted are integer multiples of 1/T. For n ! 1, ν should be the frequency of the orbital 02 2 04 vn Ze =n~ Z mee f = = 2 2 02 = 3 2 2πrn 2πn ~ =meZe n ~ (2π~) Letting f = ν, we thus obtain as expected ~ = h=2π • Results Ze02 Z e02 Z e02 1 vn = = c = αc with α = ' n~ n ~c n ~c 137 n2 2 n2 2 r = ~ = a with a = ~ = 0:529 × 10−8cm n mZe02 Z 1 1 me02 2 04 2 1 Z mee 1 Z En = − = − × 27:2 eV 2 n2 ~2 2 n2 7 2. Atomic spectrum of Hydrogen 2 04 1 ν 1 1 Z mee 1 1 ν~ = = = − = − RH λ c n2 m2 2~2hc n2 m2 2 04 1 Z mee −E1 RH = = hc 2~2 hc For Hydrogen, E1 = −13:6 eV, 13:6 × 1:6 × 10−19(J) R = ' 109677:58 cm−1 H 6:63 × 10−34(J s) × 3 × 1010(cm/s) Actually, for hydrogen atom, me should be replaced by reduced mass memH 1837 µH = ' me + mH 1838 If we consider Deuterium, then the reduced mass is memD 3674 µD = ' me + mD 3675 Thus we have slight isotrope effect for Hydrogen spectrum.
Load more

Recommended publications
  • 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
    [Show full text]
  • Arxiv:1206.1084V3 [Quant-Ph] 3 May 2019
    Overview of Bohmian Mechanics Xavier Oriolsa and Jordi Mompartb∗ aDepartament d'Enginyeria Electr`onica, Universitat Aut`onomade Barcelona, 08193, Bellaterra, SPAIN bDepartament de F´ısica, Universitat Aut`onomade Barcelona, 08193 Bellaterra, SPAIN This chapter provides a fully comprehensive overview of the Bohmian formulation of quantum phenomena. It starts with a historical review of the difficulties found by Louis de Broglie, David Bohm and John Bell to convince the scientific community about the validity and utility of Bohmian mechanics. Then, a formal explanation of Bohmian mechanics for non-relativistic single-particle quantum systems is presented. The generalization to many-particle systems, where correlations play an important role, is also explained. After that, the measurement process in Bohmian mechanics is discussed. It is emphasized that Bohmian mechanics exactly reproduces the mean value and temporal and spatial correlations obtained from the standard, i.e., `orthodox', formulation. The ontological characteristics of the Bohmian theory provide a description of measurements in a natural way, without the need of introducing stochastic operators for the wavefunction collapse. Several solved problems are presented at the end of the chapter giving additional mathematical support to some particular issues. A detailed description of computational algorithms to obtain Bohmian trajectories from the numerical solution of the Schr¨odingeror the Hamilton{Jacobi equations are presented in an appendix. The motivation of this chapter is twofold.
    [Show full text]
  • Walther Nernst, Albert Einstein, Otto Stern, Adriaan Fokker
    Walther Nernst, Albert Einstein, Otto Stern, Adriaan Fokker, and the Rotational Specific Heat of Hydrogen Clayton Gearhart St. John’s University (Minnesota) Max-Planck-Institut für Wissenschaftsgeschichte Berlin July 2007 1 Rotational Specific Heat of Hydrogen: Widely investigated in the old quantum theory Nernst Lorentz Eucken Einstein Ehrenfest Bohr Planck Reiche Kemble Tolman Schrödinger Van Vleck Some of the more prominent physicists and physical chemists who worked on the specific heat of hydrogen through the mid-1920s. See “The Rotational Specific Heat of Molecular Hydrogen in the Old Quantum Theory” http://faculty.csbsju.edu/cgearhart/pubs/sel_pubs.htm (slide show) 2 Rigid Rotator (Rotating dumbbell) •The rigid rotator was among the earliest problems taken up in the old (pre-1925) quantum theory. • Applications: Molecular spectra, and rotational contri- bution to the specific heat of molecular hydrogen • The problem should have been simple Molecular • relatively uncomplicated theory spectra • only one adjustable parameter (moment of inertia) • Nevertheless, no satisfactory theoretical description of the specific heat of hydrogen emerged in the old quantum theory Our story begins with 3 Nernst’s Heat Theorem Walther Nernst 1864 – 1941 • physical chemist • studied with Boltzmann • 1889: lecturer, then professor at Göttingen • 1905: professor at Berlin Nernst formulated his heat theorem (Third Law) in 1906, shortly after appointment as professor in Berlin. 4 Nernst’s Heat Theorem and Quantum Theory • Initially, had nothing to do with quantum theory. • Understand the equilibrium point of chemical reactions. • Nernst’s theorem had implications for specific heats at low temperatures. • 1906–1910: Nernst and his students undertook extensive measurements of specific heats of solids at low (down to liquid hydrogen) temperatures.
    [Show full text]
  • Feynman Quantization
    3 FEYNMAN QUANTIZATION An introduction to path-integral techniques Introduction. By Richard Feynman (–), who—after a distinguished undergraduate career at MIT—had come in as a graduate student to Princeton, was deeply involved in a collaborative effort with John Wheeler (his thesis advisor) to shake the foundations of field theory. Though motivated by problems fundamental to quantum field theory, as it was then conceived, their work was entirely classical,1 and it advanced ideas so radicalas to resist all then-existing quantization techniques:2 new insight into the quantization process itself appeared to be called for. So it was that (at a beer party) Feynman asked Herbert Jehle (formerly a student of Schr¨odinger in Berlin, now a visitor at Princeton) whether he had ever encountered a quantum mechanical application of the “Principle of Least Action.” Jehle directed Feynman’s attention to an obscure paper by P. A. M. Dirac3 and to a brief passage in §32 of Dirac’s Principles of Quantum Mechanics 1 John Archibald Wheeler & Richard Phillips Feynman, “Interaction with the absorber as the mechanism of radiation,” Reviews of Modern Physics 17, 157 (1945); “Classical electrodynamics in terms of direct interparticle action,” Reviews of Modern Physics 21, 425 (1949). Those were (respectively) Part III and Part II of a projected series of papers, the other parts of which were never published. 2 See page 128 in J. Gleick, Genius: The Life & Science of Richard Feynman () for a popular account of the historical circumstances. 3 “The Lagrangian in quantum mechanics,” Physicalische Zeitschrift der Sowjetunion 3, 64 (1933). The paper is reprinted in J.
    [Show full text]
  • The Concept of the Photon—Revisited
    The concept of the photon—revisited Ashok Muthukrishnan,1 Marlan O. Scully,1,2 and M. Suhail Zubairy1,3 1Institute for Quantum Studies and Department of Physics, Texas A&M University, College Station, TX 77843 2Departments of Chemistry and Aerospace and Mechanical Engineering, Princeton University, Princeton, NJ 08544 3Department of Electronics, Quaid-i-Azam University, Islamabad, Pakistan The photon concept is one of the most debated issues in the history of physical science. Some thirty years ago, we published an article in Physics Today entitled “The Concept of the Photon,”1 in which we described the “photon” as a classical electromagnetic field plus the fluctuations associated with the vacuum. However, subsequent developments required us to envision the photon as an intrinsically quantum mechanical entity, whose basic physics is much deeper than can be explained by the simple ‘classical wave plus vacuum fluctuations’ picture. These ideas and the extensions of our conceptual understanding are discussed in detail in our recent quantum optics book.2 In this article we revisit the photon concept based on examples from these sources and more. © 2003 Optical Society of America OCIS codes: 270.0270, 260.0260. he “photon” is a quintessentially twentieth-century con- on are vacuum fluctuations (as in our earlier article1), and as- Tcept, intimately tied to the birth of quantum mechanics pects of many-particle correlations (as in our recent book2). and quantum electrodynamics. However, the root of the idea Examples of the first are spontaneous emission, Lamb shift, may be said to be much older, as old as the historical debate and the scattering of atoms off the vacuum field at the en- on the nature of light itself – whether it is a wave or a particle trance to a micromaser.
    [Show full text]
  • Theory and Experiment in the Quantum-Relativity Revolution
    Theory and Experiment in the Quantum-Relativity Revolution expanded version of lecture presented at American Physical Society meeting, 2/14/10 (Abraham Pais History of Physics Prize for 2009) by Stephen G. Brush* Abstract Does new scientific knowledge come from theory (whose predictions are confirmed by experiment) or from experiment (whose results are explained by theory)? Either can happen, depending on whether theory is ahead of experiment or experiment is ahead of theory at a particular time. In the first case, new theoretical hypotheses are made and their predictions are tested by experiments. But even when the predictions are successful, we can’t be sure that some other hypothesis might not have produced the same prediction. In the second case, as in a detective story, there are already enough facts, but several theories have failed to explain them. When a new hypothesis plausibly explains all of the facts, it may be quickly accepted before any further experiments are done. In the quantum-relativity revolution there are examples of both situations. Because of the two-stage development of both relativity (“special,” then “general”) and quantum theory (“old,” then “quantum mechanics”) in the period 1905-1930, we can make a double comparison of acceptance by prediction and by explanation. A curious anti- symmetry is revealed and discussed. _____________ *Distinguished University Professor (Emeritus) of the History of Science, University of Maryland. Home address: 108 Meadowlark Terrace, Glen Mills, PA 19342. Comments welcome. 1 “Science walks forward on two feet, namely theory and experiment. ... Sometimes it is only one foot which is put forward first, sometimes the other, but continuous progress is only made by the use of both – by theorizing and then testing, or by finding new relations in the process of experimenting and then bringing the theoretical foot up and pushing it on beyond, and so on in unending alterations.” Robert A.
    [Show full text]
  • Quantum Mechanics I - Iii
    QUANTUM MECHANICS I - III PHYS 516 - 518 Jan 1 - Dec. 31, 2015 Prof. R. Gilmore 12-918 X-2779 [email protected] office hours: 2:00 ! \1" Course Schedule: (Winter Quarter) MWF 11:00 - 11:50, Disque 919 Objective: To provide the foundations for modern physics. Course Requirements and Obligations Course grading will be based on assigned homework problem sets and a midterm and final exam. Texts Two texts and one supplement will be used for this course. The first text has been chosen from among many admirable texts because it provides a more comprehensive treatment of quantum physics discovered since 1970 than other texts. The second text will be used primarily during the second quarter of this course (PHYS517). It provides hands-on experience for solving binding and scattering problems in one dimension and potentials involving periodic poten- tials, again in one dimension. The third text (optional) is strongly recommended for those who feel their undergratuate experience in this beautiful subject may be deficient in some way. It is out of print but a limited number of copies are often available through Amazon in the event our book store has sold out of their reprinted copies. 1 David H. McIntyre Quantum Mechanics NY: Pearson, 2012 ISBN-10: 0-321-76579-6 R. Gilmore Elementary Quantum Mechanics in One Dimension Baltimore, Johns Hopkins University Press, 2004 ISBN 0-8018-8015-7 R. H. Dicke and J. P. Wittke Introduction to Quantum Mechanics Reading, MA: Addison-Wesley, 1960 ISBN 0-? If it becomes a hardship to acquire this fine text, you can get about the same information from another, later, fine text: David J.
    [Show full text]
  • Chapter 2 Quantum Theory
    Chapter 2 - Quantum Theory At the end of this chapter – the class will: Have basic concepts of quantum physical phenomena and a rudimentary working knowledge of quantum physics Have some familiarity with quantum mechanics and its application to atomic theory Quantization of energy; energy levels Quantum states, quantum number Implication on band theory Chapter 2 Outline Basic concept of quantization Origin of quantum theory and key quantum phenomena Quantum mechanics Example and application to atomic theory Concept introduction The quantum car Imagine you drive a car. You turn on engine and it immediately moves at 10 m/hr. You step on the gas pedal and it does nothing. You step on it harder and suddenly, the car moves at 40 m/hr. You step on the brake. It does nothing until you flatten the brake with all your might, and it suddenly drops back to 10 m/hr. What’s going on? Continuous vs. Quantization Consider a billiard ball. It requires accuracy and precision. You have a cue stick. Assume for simplicity that there is no friction loss. How fast can you make the ball move using the cue stick? How much kinetic energy can you give to the ball? The Newtonian mechanics answer is: • any value, as much as energy as you can deliver. The ball can be made moving with 1.000 joule, or 3.1415926535 or 0.551 … joule. Supposed it is moving with 1-joule energy, you can give it an extra 0.24563166 joule by hitting it with the cue stick by that amount of energy.
    [Show full text]
  • Quantum Mechanics of the Electron Particle-Clock
    Quantum Mechanics of the electron particle-clock David Hestenes Department of Physics, Arizona State University, Tempe, Arizona 85287-1504∗ Understanding the electron clock and the role of complex numbers in quantum mechanics is grounded in the geometry of spacetime, and best expressed with Spacetime Algebra (STA). The efficiency of STA is demonstrated with coordinate-free applications to both relativistic and non- relativistic QM. Insight into the structure of Dirac theory is provided by a new comprehensive analysis of local observables and conservation laws. This provides the foundation for a comparative analysis of wave and particle models for the hydrogen atom, the workshop where Quantum Mechanics was designed, built and tested. PACS numbers: 10,03.65.-w Keywords: pilot waves, zitterbewegung, spacetime algebra I. INTRODUCTION The final Section argues for a realist interpretation of the Dirac wave function describing possible particle De Broglie always insisted that Quantum Mechanics is paths. a relativistic theory. Indeed, it began with de Broglie`s relativistic model for an electron clock. Ironically, when Schr¨odingerintroduced de Broglie`s idea into his wave II. SPACETIME ALGEBRA equation, he dispensed with both the clock and the rela- tivity! Spacetime Algebra (STA) plays an essential role in the This paper aims to revitalize de Broglie's idea of an formulation and analysis of electron theory in this paper. electron clock by giving it a central role in physical inter- Since thorough expositions of STA are available in many pretation of the Dirac wave function. In particular, we places [2{4], a brief description will suffice here, mainly aim for insight into structure of the wave function and to establish notations and define terms.
    [Show full text]
  • Quantization Conditions, 1900–1927
    Quantization Conditions, 1900–1927 Anthony Duncan and Michel Janssen March 9, 2020 1 Overview We trace the evolution of quantization conditions from Max Planck’s intro- duction of a new fundamental constant (h) in his treatment of blackbody radiation in 1900 to Werner Heisenberg’s interpretation of the commutation relations of modern quantum mechanics in terms of his uncertainty principle in 1927. In the most general sense, quantum conditions are relations between classical theory and quantum theory that enable us to construct a quantum theory from a classical theory. We can distinguish two stages in the use of such conditions. In the first stage, the idea was to take classical mechanics and modify it with an additional quantum structure. This was done by cut- ting up classical phase space. This idea first arose in the period 1900–1910 in the context of new theories for black-body radiation and specific heats that involved the statistics of large collections of simple harmonic oscillators. In this context, the structure added to classical phase space was used to select equiprobable states in phase space. With the arrival of Bohr’s model of the atom in 1913, the main focus in the development of quantum theory shifted from the statistics of large numbers of oscillators or modes of the electro- magnetic field to the detailed structure of individual atoms and molecules and the spectra they produce. In that context, the additional structure of phase space was used to select a discrete subset of classically possible motions. In the second stage, after the transition to modern quantum quan- tum mechanics in 1925–1926, quantum theory was completely divorced from its classical substratum and quantum conditions became a means of con- trolling the symbolic translation of classical relations into relations between quantum-theoretical quantities, represented by matrices or operators and no longer referring to orbits in classical phase space.1 More specifically, the development we trace in this essay can be summa- rized as follows.
    [Show full text]
  • A Conceptual Introduction to Nelson's Mechanics
    A Conceptual Introduction to Nelson’s Mechanics Guido Bacciagaluppi To cite this version: Guido Bacciagaluppi. A Conceptual Introduction to Nelson’s Mechanics. R. Buccheri, M. Saniga, A. Elitzur. Endophysics, time, quantum and the subjective, World Scientific, Singapore, pp.367-388, 2005. halshs-00996258 HAL Id: halshs-00996258 https://halshs.archives-ouvertes.fr/halshs-00996258 Submitted on 26 May 2014 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. A Conceptual Introduction to Nelson’s Mechanics∗ Guido Bacciagaluppi† May 27, 2014 Abstract Nelson’s programme for a stochastic mechanics aims to derive the wave function and the Schr¨odinger equation from natural conditions on a diffusion process in configuration space. If successful, this pro- gramme might have some advantages over the better-known determin- istic pilot-wave theory of de Broglie and Bohm. The essential points of Nelson’s strategy are reviewed, with particular emphasis on concep- tual issues relating to the role of time symmetry. The main problem in Nelson’s approach is the lack of strict equivalence between the cou- pled Madelung equations and the Schr¨odinger equation. After a brief discussion, the paper concludes with a possible suggestion for trying to overcome this problem.
    [Show full text]
  • Lectures on Quantum Mechanics for Mathematicians
    Lectures on Quantum Mechanics for mathematicians A.I. Komech 1 Faculty of Mathematics of Vienna University Institute for Information Transmission Problems of RAS, Moscow Department Mechanics and Mathematics of Moscow State University (Lomonosov) [email protected] Abstract The main goal of these lectures – introduction to Quantum Mechanics for mathematically-minded readers. The second goal is to discuss the mathematical interpretation of the main quantum postulates: transitions between quantum stationary orbits, wave-particle duality and probabilistic interpretation. We suggest a dynamical interpretation of these phenomena based on the new conjectures on attractors of nonlinear Hamiltonian partial differential equations. This conjecture is confirmed for a list of model Hamiltonian nonlinear PDEs by the results obtained since 1990 (we survey sketchy these results). However, for the Maxwell–Schr¨odinger equations this conjecture is still an open problem. We calculate the diffraction amplitude for the scattering of electron beams and Aharonov–Bohm shift via the Kirchhoff approximation. Keywords: Schr¨odinger equation; Maxwell equations; Maxwell–Schr¨odinger equations; semiclassical asymp- totics; Hamilton–Jacobi equation; energy; charge; momentum; angular momentum; quantum transitions; wave- particle duality; probabilistic interpretation; magnetic moment; normal Zeeman effect; spin; rotation group; Pauli equation; electron diffraction; Kirchhoff approximation; Aharonov–Bohm shift; attractors; Hamiltonian equations; nonlinear partial differential equations; Lie group; symmetry group. Contents 1 Introduction 3 arXiv:1907.05786v2 [math-ph] 15 Jul 2019 2 Planck’s law, Einstein’s photons and de Broglie wave-particle duality 4 2.1 Planck’s law and Einstein’s photon . ................ 4 2.2 De Broglie wave-particle duality . ................ 4 3 The Schr¨odinger quantum mechanics 5 3.1 CanonicalQuantisation .
    [Show full text]