Einstein's Physics

Total Page:16

File Type:pdf, Size:1020Kb

Einstein's Physics Einstein’s Physics Atoms, Quanta, and Relativity Derived, explained, and appraised Ta-Pei Cheng University of Missouri - St. Louis Portland State University To be published by Oxford University Press 2013 Contents I ATOMIC NATURE OF MATTER 1 1 Molecular size from classical fluid 3 1.1 Two relations of molecular size and the Avogadro number . 4 1.2 The relation for the effective viscosity . 6 1.2.1 The equation of motion for a viscous fluid . 7 1.2.2 Viscosity and heat loss in a fluid . 8 1.2.3 Volume fraction in terms of molecular dimensions . 11 1.3 The relation for the diffusion coefficient . 11 1.3.1 Osmoticforce....................... 12 1.3.2 Frictional drag force — the Stokes law . 13 1.4 SuppMat: Basics of fluid mechanics . 15 1.4.1 The equation of continuity . 15 1.4.2 The Euler equation for an ideal fluid . 16 1.5 SuppMat: Calculating the effective viscosity . 18 1.5.1 The induced velocity field v′ ............... 18 1.5.2 The induced pressure field p′ .............. 20 1.5.3 Heat dissipation in a fluid with suspended particles . 20 1.6 SuppMat: The Stokes formula for viscous force . 24 2 The Brownian motion 27 2.1 Diffusion and Brownian motion . 29 2.1.1 Einstein’s statistical derivation of the diffusion equation 29 2.1.2 The solution of the diffusion equation and the mean- squaredisplacement . 32 2.2 Fluctuations of a particle system . 32 2.2.1 Randomwalk....................... 33 2.2.2 Brownian motion as a random walk . 34 2.3 The Einstein-Smoluchowski relation . 34 2.3.1 Fluctuation and dissipation . 36 iii iv CONTENTS 2.3.2 Mean-square displacement and molecular dimensions . 36 2.4 Perrin’s experimental verification . 37 II QUANTUM THEORY 41 3 Blackbody radiation: From Kirchhoff to Planck 43 3.1 Radiation as a collection of oscillators . 45 3.2 Thermodynamics of blackbody radiation . 47 3.2.1 Radiation energy density is an universal function . 47 3.2.2 The Stefan-Boltzmann law . 49 3.2.3 Wien’s displacement law . 50 3.2.4 Planck’s distribution proposed . 54 3.3 Planck’s investigation of cavity oscillator entropy . 54 3.3.1 Relating the oscillator energy to the radiation density 55 3.3.2 The mean entropy of an oscillator . 55 3.4 Planck’s statistical analysis . 57 3.4.1 Calculating the complexion of Planck’s distribution . 57 3.4.2 The Planck’s constant and the Boltzmann’s constant . 61 3.4.3 Planck’s energy quantization proposal — a summary . 62 3.5 SuppMat: Radiation oscillator energy and frequency . 63 3.5.1 Ratio of the oscillator energy and frequency is an adi- abaticinvariant. .. .. .. .. .. .. 64 3.5.2 The thermodynamic derivation of the relation between radiation pressure and energy density . 67 4 Einstein’s proposal of light quanta 69 4.1 TheRayleigh-Jeanslaw . 71 4.1.1 Einstein’s derivation of the Rayleigh-Jeans law . 71 4.1.2 The history of the Rayleigh-Jeans law and "Planck’s fortunatefailure". 73 4.1.3 An excursion to Rayleigh’s calculation of the density ofwavestates....................... 74 4.2 Radiation entropy and complexion á la Einstein . 76 4.2.1 The entropy and complexion of radiation in the Wien limit............................ 77 4.2.2 The entropy and complexion of an ideal gas . 79 4.2.3 Radiation as a gas of light quanta . 80 4.2.4 Photons as quanta of radiation . 81 4.3 Thephotoelectriceffect . 81 CONTENTS v 4.4 SuppMat: The equipartition theorem . 83 5 Quantum theory of specific heat 85 5.1 The quantum postulate: Einstein vs. Planck.......... 86 5.1.1 Einstein’s derivation of Planck’s distribution . 87 5.2 Specific heat and the equipartition theorem . 88 5.2.1 The study of heat capacity in the pre-quantum era . 88 5.2.2 Einstein’s quantum insight . 90 5.3 TheEinsteinsolid ........................ 91 5.4 TheDebyesolidandphonons . 94 6 Waves, particles, and quantum jumps 99 6.1 Wave-particleduality. 101 6.1.1 Fluctuation theory (Einstein 1904) . 101 6.1.2 Energy fluctuation of radiation (Einstein 1909a) . 102 6.2 Bohr’satom............................105 6.2.1 Spectroscopy: Balmer and Rydberg . 106 6.2.2 Atomic structure: Thomson and Rutherford . 107 6.2.3 Bohr’s quantum model and the hydrogen spectrum . 107 6.3 Einstein’s A and B coefficients . 111 6.3.1 Probability introduced in quantum dynamics . 112 6.3.2 Stimulated emission and the idea of laser . 114 6.4 Looking ahead to quantum field theory . 115 6.4.1 Oscillators in matrix mechanics . 116 6.4.2 Quantum jumps: from emission and absorption of ra- diation to creation and annihilation of particles . 119 6.4.3 Resolving the riddle of wave-particle duality in radia- tionfluctuation. 122 6.5 SuppMat: Fluctuations of a wave system . 124 7 Bose-Einstein statistics and condensation 127 7.1 PhotonandtheComptoneffect . 129 7.2 Towards Bose-Einstein statistics . 130 7.2.1 Boltzmann statistics . 131 7.2.2 Bose’s counting of photon states . 133 7.2.3 Einstein’s elaboration of Bose’s counting . 136 7.3 Quantum mechanics and identical particles . 139 7.4 Bose-Einstein condensation . 142 7.4.1 Condensate occupancy calculated . 143 7.4.2 The condensation temperature . 144 vi CONTENTS 7.4.3 Laboratory observation of BE condensation . 146 7.5 SuppMat: Radiation pressure due to photons . 148 7.6 SuppMat: Planck’s analysis and BE statistics . 149 7.7 SuppMat: The role of indistinguishability in BEC . 150 8 Local reality and the Einstein-Bohr debate 153 8.1 QM basics — superposition and probability . 154 8.2 The Copenhagen interpretation . 154 8.3 EPR paradox : entanglement and nonlocality . 156 8.3.1 The post EPR era and Bell’s inequality . 159 8.3.2 Local reality vs. QM — the experimental outcome . 162 8.4 SuppMat: QM calculation of spin correlations . 164 III SPECIAL RELATIVITY 169 9 Prelude to special relativity 171 9.1 Relativity as a coordinate symmetry . 172 9.2 Maxwell’sequations . 174 9.2.1 The electromagnetic wave equation . 175 9.2.2 Aether as the medium for EM wave propagation . 176 9.3 Developments prior to special relativity . 177 9.3.1 Stellar aberration and Fizeau’s experiment . 177 9.3.2 Lorentz’s corresponding states and local time . 180 9.3.3 The Michelson-Morley experiment . 184 9.3.4 Length contraction and the Lorentz transformation . 187 9.3.5 Poincaré and special relativity . 188 9.4 Reconstructing Einstein’s motivation . 188 9.4.1 The magnet and conductor thought experiment . 189 9.4.2 From ‘no absolute time’ to the complete theory in five weeks ...........................191 9.4.3 Influence of prior investigators in physics and philosophy193 9.5 SuppMat: Lorentz transformation à la Lorentz . 194 9.5.1 Noncovariance under Galilean transformation . 195 9.5.2 Lorentz’s local time and noncovariance at O v2/c2 . 196 9.5.3 Maxwell’s equations are Lorentz covariant . 198 10 The new kinematics and E = mc2 201 10.1 Thenewkinematics . 203 10.1.1 Einstein’s two postulates . 203 CONTENTS vii 10.1.2 The new conception of time and the derivation of the Lorentz transformation . 204 10.1.3 Relativity of simultaneity, time dilation and length contraction ........................207 10.2 The new velocity addition rule . 212 10.2.1 The invariant space-time interval . 212 10.2.2 Adding velocities but keeping light speed constant . 212 10.3 Maxwell’s equations are covariant . 213 10.3.1 The Lorentz transformation of electromagnetic fields . 214 10.3.2 The Lorentz transformation of radiation energy . 216 10.4 TheLorentzforcelaw . 216 10.5 The E = mc2 relation ......................217 10.5.1 Work-energy theorem in relativity . 218 10.5.2 The E = mc2 paper three months later . 219 10.6 SuppMat: Relativistic wave motion . 220 10.6.1 The Fresnel formula from velocity addition rule . 220 10.6.2 The Doppler effect and aberration of light . 221 10.6.3 Derivation of the radiation energy transformation . 222 10.7 SuppMat: Relativistic momentum and force . 224 11 Geometric formulation of relativity 225 11.1 Minkowskispacetime. 227 11.1.1 Rotation in 3D space - a review . 227 11.1.2 The Lorentz transformation as a rotation in 4D space- time ............................228 11.2 Tensorsinaflatspacetime. 229 11.2.1 Tensor contraction and the metric . 230 11.2.2 Minkowski spacetime is pseudo-Euclidean . 232 11.2.3 Relativistic velocity, momentum, and energy . 233 11.2.4 The electromagnetic field tensor . 234 11.2.5 The energy-momentum-stress tensor for a field system 235 11.3 The spacetime diagram . 238 11.3.1 Basic features and invariant regions . 239 11.3.2 Lorentz transformation in the spacetime diagram . 241 11.4 The geometric formulation — a summary . 244 IV GENERAL RELATIVITY 245 12 Towards a general theory of relativity 247 viii CONTENTS 12.1 Einstein’s motivations for GR . 248 12.2 Theequivalenceprinciple . 249 12.2.1 The inertia mass vs. the gravitational mass . 249 12.2.2 ‘Myhappiestthought’ . 252 12.3 Implications of the equivalence principle . 253 12.3.1 Bendingofalightray . 254 12.3.2 Gravitational redshift . 255 12.3.3 Gravitational time dilation . 258 12.3.4 Gravity-induced index of refraction in free space . 259 12.3.5 Light ray deflection calculated . 260 12.3.6 From EP to ‘gravity as the structure of spacetime’ . 262 12.4 Elements of Riemannian geometry . 262 12.4.1 Gaussian coordinates and the metric tensor . 263 12.4.2 Geodesic equation . 265 12.4.3 Flatnesstheorem .
Recommended publications
  • Einstein Solid 1 / 36 the Results 2
    Specific Heats of Ideal Gases 1 Assume that a pure, ideal gas is made of tiny particles that bounce into each other and the walls of their cubic container of side `. Show the average pressure P exerted by this gas is 1 N 35 P = mv 2 3 V total SO 30 2 7 7 (J/K-mole) R = NAkB Use the ideal gas law (PV = NkB T = V 2 2 CO2 C H O CH nRT )and the conservation of energy 25 2 4 Cl2 (∆Eint = CV ∆T ) to calculate the specific 5 5 20 2 R = 2 NAkB heat of an ideal gas and show the following. H2 N2 O2 CO 3 3 15 CV = NAkB = R 3 3 R = NAkB 2 2 He Ar Ne Kr 2 2 10 Is this right? 5 3 N - number of particles V = ` 0 Molecule kB - Boltzmann constant m - atomic mass NA - Avogadro's number vtotal - atom's speed Jerry Gilfoyle Einstein Solid 1 / 36 The Results 2 1 N 2 2 N 7 P = mv = hEkini 2 NAkB 3 V total 3 V 35 30 SO2 3 (J/K-mole) V 5 CO2 C N k hEkini = NkB T H O CH 2 A B 2 25 2 4 Cl2 20 H N O CO 3 3 2 2 2 3 CV = NAkB = R 2 NAkB 2 2 15 He Ar Ne Kr 10 5 0 Molecule Jerry Gilfoyle Einstein Solid 2 / 36 Quantum mechanically 2 E qm = `(` + 1) ~ rot 2I where l is the angular momen- tum quantum number.
    [Show full text]
  • Otto Sackur's Pioneering Exploits in the Quantum Theory Of
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Catalogo dei prodotti della ricerca Chapter 3 Putting the Quantum to Work: Otto Sackur’s Pioneering Exploits in the Quantum Theory of Gases Massimiliano Badino and Bretislav Friedrich After its appearance in the context of radiation theory, the quantum hypothesis rapidly diffused into other fields. By 1910, the crisis of classical traditions of physics and chemistry—while taking the quantum into account—became increas- ingly evident. The First Solvay Conference in 1911 pushed quantum theory to the fore, and many leading physicists responded by embracing the quantum hypoth- esis as a way to solve outstanding problems in the theory of matter. Until about 1910, quantum physics had drawn much of its inspiration from two sources. The first was the complex formal machinery connected with Max Planck’s theory of radiation and, above all, its close relationship with probabilis- tic arguments and statistical mechanics. The fledgling 1900–1901 version of this theory hinged on the application of Ludwig Boltzmann’s 1877 combinatorial pro- cedure to determine the state of maximum probability for a set of oscillators. In his 1906 book on heat radiation, Planck made the connection with gas theory even tighter. To illustrate the use of the procedure Boltzmann originally developed for an ideal gas, Planck showed how to extend the analysis of the phase space, com- monplace among practitioners of statistical mechanics, to electromagnetic oscil- lators (Planck 1906, 140–148). In doing so, Planck identified a crucial difference between the phase space of the gas molecules and that of oscillators used in quan- tum theory.
    [Show full text]
  • Intermediate Statistics in Thermoelectric Properties of Solids
    Intermediate statistics in thermoelectric properties of solids André A. Marinho1, Francisco A. Brito1,2 1 Departamento de Física, Universidade Federal de Campina Grande, 58109-970 Campina Grande, Paraíba, Brazil and 2 Departamento de Física, Universidade Federal da Paraíba, Caixa Postal 5008, 58051-970 João Pessoa, Paraíba, Brazil (Dated: July 23, 2019) Abstract We study the thermodynamics of a crystalline solid by applying intermediate statistics manifested by q-deformation. We based part of our study on both Einstein and Debye models, exploring primarily de- formed thermal and electrical conductivities as a function of the deformed Debye specific heat. The results revealed that the q-deformation acts in two different ways but not necessarily as independent mechanisms. It acts as a factor of disorder or impurity, modifying the characteristics of a crystalline structure, which are phenomena described by q-bosons, and also as a manifestation of intermediate statistics, the B-anyons (or B-type systems). For the latter case, we have identified the Schottky effect, normally associated with high-Tc superconductors in the presence of rare-earth-ion impurities, and also the increasing of the specific heat of the solids beyond the Dulong-Petit limit at high temperature, usually related to anharmonicity of interatomic interactions. Alternatively, since in the q-bosons the statistics are in principle maintained the effect of the deformation acts more slowly due to a small change in the crystal lattice. On the other hand, B-anyons that belong to modified statistics are more sensitive to the deformation. PACS numbers: 02.20-Uw, 05.30-d, 75.20-g arXiv:1907.09055v1 [cond-mat.stat-mech] 21 Jul 2019 1 I.
    [Show full text]
  • Einstein's Physics
    Einstein’s Physics Albert Einstein at Barnes Foundation in Merion PA, c.1947. Photography by Laura Delano Condax; gift to the author from Vanna Condax. Einstein’s Physics Atoms, Quanta, and Relativity Derived, Explained, and Appraised TA-PEI CHENG University of Missouri–St. Louis Portland State University 3 3 Great Clarendon Street, Oxford, OX2 6DP, United Kingdom Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide. Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries © Ta-Pei Cheng 2013 The moral rights of the author have been asserted Impression: 1 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, by licence or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this work in any other form and you must impose this same condition on any acquirer British Library Cataloguing in Publication Data Data available ISBN 978–0–19–966991–2 Printed and bound by CPI Group (UK) Ltd, Croydon, CR0 4YY Links to third party websites are provided by Oxford in good faith and for information only. Oxford disclaims any responsibility for the materials contained in any third party website referenced in this work.
    [Show full text]
  • MOLAR HEAT of SOLIDS the Dulong–Petit Law, a Thermodynamic
    MOLAR HEAT OF SOLIDS The Dulong–Petit law, a thermodynamic law proposed in 1819 by French physicists Dulong and Petit, states the classical expression for the molar specific heat of certain crystals. The two scientists conducted experiments on three dimensional solid crystals to determine the heat capacities of a variety of these solids. They discovered that all investigated solids had a heat capacity of approximately 25 J mol-1 K-1 room temperature. The result from their experiment was explained as follows. According to the Equipartition Theorem, each degree of freedom has an average energy of 1 = 2 where is the Boltzmann constant and is the absolute temperature. We can model the atoms of a solid as attached to neighboring atoms by springs. These springs extend into three-dimensional space. Each direction has 2 degrees of freedom: one kinetic and one potential. Thus every atom inside the solid was considered as a 3 dimensional oscillator with six degrees of freedom ( = 6) The more energy that is added to the solid the more these springs vibrate. Figure 1: Model of interaction of atoms of a solid Now the energy of each atom is = = 3 . The energy of N atoms is 6 = 3 2 = 3 where n is the number of moles. To change the temperature by ΔT via heating, one must transfer Q=3nRΔT to the crystal, thus the molar heat is JJ CR=3 ≈⋅ 3 8.31 ≈ 24.93 molK molK Similarly, the molar heat capacity of an atomic or molecular ideal gas is proportional to its number of degrees of freedom, : = 2 This explanation for Petit and Dulong's experiment was not sufficient when it was discovered that heat capacity decreased and going to zero as a function of T3 (or, for metals, T) as temperature approached absolute zero.
    [Show full text]
  • First Principles Study of the Vibrational and Thermal Properties of Sn-Based Type II Clathrates, Csxsn136 (0 ≤ X ≤ 24) and Rb24ga24sn112
    Article First Principles Study of the Vibrational and Thermal Properties of Sn-Based Type II Clathrates, CsxSn136 (0 ≤ x ≤ 24) and Rb24Ga24Sn112 Dong Xue * and Charles W. Myles Department of Physics and Astronomy, Texas Tech University, Lubbock, TX 79409-1051, USA; [email protected] * Correspondence: [email protected]; Tel.: +1-806-834-4563 Received: 12 May 2019; Accepted: 11 June 2019; Published: 14 June 2019 Abstract: After performing first-principles calculations of structural and vibrational properties of the semiconducting clathrates Rb24Ga24Sn112 along with binary CsxSn136 (0 ≤ x ≤ 24), we obtained equilibrium geometries and harmonic phonon modes. For the filled clathrate Rb24Ga24Sn112, the phonon dispersion relation predicts an upshift of the low-lying rattling modes (~25 cm−1) for the Rb (“rattler”) compared to Cs vibration in CsxSn136. It is also found that the large isotropic atomic displacement parameter (Uiso) exists when Rb occupies the “over-sized” cage (28 atom cage) rather than the 20 atom counterpart. These guest modes are expected to contribute significantly to minimizing the lattice’s thermal conductivity (κL). Our calculation of the vibrational contribution to the specific heat and our evaluation on κL are quantitatively presented and discussed. Specifically, the heat capacity diagram regarding CV/T3 vs. T exhibits the Einstein-peak-like hump that is mainly attributable to the guest oscillator in a 28 atom cage, with a characteristic temperature 36.82 K for Rb24Ga24Sn112. Our calculated rattling modes are around 25 cm−1 for the Rb trapped in a 28 atom cage, and 65.4 cm−1 for the Rb encapsulated in a 20 atom cage.
    [Show full text]
  • Equipartition of Energy
    Equipartition of Energy The number of degrees of freedom can be defined as the minimum number of independent coordinates, which can specify the configuration of the system completely. (A degree of freedom of a system is a formal description of a parameter that contributes to the state of a physical system.) The position of a rigid body in space is defined by three components of translation and three components of rotation, which means that it has six degrees of freedom. The degree of freedom of a system can be viewed as the minimum number of coordinates required to specify a configuration. Applying this definition, we have: • For a single particle in a plane two coordinates define its location so it has two degrees of freedom; • A single particle in space requires three coordinates so it has three degrees of freedom; • Two particles in space have a combined six degrees of freedom; • If two particles in space are constrained to maintain a constant distance from each other, such as in the case of a diatomic molecule, then the six coordinates must satisfy a single constraint equation defined by the distance formula. This reduces the degree of freedom of the system to five, because the distance formula can be used to solve for the remaining coordinate once the other five are specified. The equipartition theorem relates the temperature of a system with its average energies. The original idea of equipartition was that, in thermal equilibrium, energy is shared equally among all of its various forms; for example, the average kinetic energy per degree of freedom in the translational motion of a molecule should equal that of its rotational motions.
    [Show full text]
  • Fastest Frozen Temperature for a Thermodynamic System
    Fastest Frozen Temperature for a Thermodynamic System X. Y. Zhou, Z. Q. Yang, X. R. Tang, X. Wang1 and Q. H. Liu1, 2, ∗ 1School for Theoretical Physics, School of Physics and Electronics, Hunan University, Changsha, 410082, China 2Synergetic Innovation Center for Quantum Effects and Applications (SICQEA), Hunan Normal University,Changsha 410081, China For a thermodynamic system obeying both the equipartition theorem in high temperature and the third law in low temperature, the curve showing relationship between the specific heat and the temperature has two common behaviors: it terminates at zero when the temperature is zero Kelvin and converges to a constant value of specific heat as temperature is higher and higher. Since it is always possible to find the characteristic temperature TC to mark the excited temperature as the specific heat almost reaches the equipartition value, it is also reasonable to find a temperature in low temperature interval, complementary to TC . The present study reports a possibly universal existence of the such a temperature #, defined by that at which the specific heat falls fastest along with decrease of the temperature. For the Debye model of solids, below the temperature # the Debye's law manifest itself. PACS numbers: I. INTRODUCTION A thermodynamic system usually obeys two laws in opposite limits of temperature; in high temperature the equipar- tition theorem works and in low temperature the third law of thermodynamics holds. From the shape of a curve showing relationship between the specific heat and the temperature, the behaviors in these two limits are qualitatively clear: in high temperature it approaches to a constant whereas it goes over to zero during the temperature is lowering to the zero Kelvin.
    [Show full text]
  • Exercise 18.4 the Ideal Gas Law Derived
    9/13/2015 Ch 18 HW Ch 18 HW Due: 11:59pm on Monday, September 14, 2015 To understand how points are awarded, read the Grading Policy for this assignment. Exercise 18.4 ∘ A 3.00­L tank contains air at 3.00 atm and 20.0 C. The tank is sealed and cooled until the pressure is 1.00 atm. Part A What is the temperature then in degrees Celsius? Assume that the volume of the tank is constant. ANSWER: ∘ T = ­175 C Correct Part B If the temperature is kept at the value found in part A and the gas is compressed, what is the volume when the pressure again becomes 3.00 atm? ANSWER: V = 1.00 L Correct The Ideal Gas Law Derived The ideal gas law, discovered experimentally, is an equation of state that relates the observable state variables of the gas­­pressure, temperature, and density (or quantity per volume): pV = NkBT (or pV = nRT ), where N is the number of atoms, n is the number of moles, and R and kB are ideal gas constants such that R = NA kB, where NA is Avogadro's number. In this problem, you should use Boltzmann's constant instead of the gas constant R. Remarkably, the pressure does not depend on the mass of the gas particles. Why don't heavier gas particles generate more pressure? This puzzle was explained by making a key assumption about the connection between the microscopic world and the macroscopic temperature T . This assumption is called the Equipartition Theorem. The Equipartition Theorem states that the average energy associated with each degree of freedom in a system at T 1 k T k = 1.38 × 10−23 J/K absolute temperature is 2 B , where B is Boltzmann's constant.
    [Show full text]
  • Lecture 8 - Dec 2, 2013 Thermodynamics of Earth and Cosmos; Overview of the Course
    Physics 5D - Lecture 8 - Dec 2, 2013 Thermodynamics of Earth and Cosmos; Overview of the Course Reminder: the Final Exam is next Wednesday December 11, 4:00-7:00 pm, here in Thimann Lec 3. You can bring with you two sheets of paper with any formulas or other notes you like. A PracticeFinalExam is at http://physics.ucsc.edu/~joel/Phys5D The Physics Department has elected to use the new eCommons Evaluations System to collect end of the quarter instructor evaluations from students in our courses. All students in our courses will have access to the evaluation tool in eCommons, whether or not the instructor used an eCommons site for course work. Students will receive an email from the department when the evaluation survey is available. The email will provide information regarding the evaluation as well as a link to the evaluation in eCommons. Students can click the link, login to eCommons and find the evaluation to take and submit. Alternately, students can login to ecommons.ucsc.edu and click the Evaluation System tool to see current available evaluations. Monday, December 2, 13 20-8 Heat Death of the Universe If we look at the universe as a whole, it seems inevitable that, as more and more energy is converted to unavailable forms, the total entropy S will increase and the ability to do work anywhere will gradually decrease. This is called the “heat death of the universe”. It was a popular theme in the late 19th century, after physicists had discovered the 2nd Law of Thermodynamics. Lord Kelvin calculated that the sun could only shine for about 20 million years, based on the idea that its energy came from gravitational contraction.
    [Show full text]
  • Hostatmech.Pdf
    A computational introduction to quantum statistics using harmonically trapped particles Martin Ligare∗ Department of Physics & Astronomy, Bucknell University, Lewisburg, PA 17837 (Dated: March 15, 2016) Abstract In a 1997 paper Moore and Schroeder argued that the development of student understanding of thermal physics could be enhanced by computational exercises that highlight the link between the statistical definition of entropy and the second law of thermodynamics [Am. J. Phys. 65, 26 (1997)]. I introduce examples of similar computational exercises for systems in which the quantum statistics of identical particles plays an important role. I treat isolated systems of small numbers of particles confined in a common harmonic potential, and use a computer to enumerate all possible occupation-number configurations and multiplicities. The examples illustrate the effect of quantum statistics on the sharing of energy between weakly interacting subsystems, as well as the distribution of energy within subsystems. The examples also highlight the onset of Bose-Einstein condensation in small systems. PACS numbers: 1 I. INTRODUCTION In a 1997 paper in this journal Moore and Schroeder argued that the development of student understanding of thermal physics could be enhanced by computational exercises that highlight the link between the statistical definition of entropy and the second law of thermodynamics.1 The first key to their approach was the use of a simple model, the Ein- stein solid, for which it is straightforward to develop an exact formula for the number of microstates of an isolated system. The second key was the use of a computer, rather than analytical approximations, to evaluate these formulas.
    [Show full text]
  • Einstein Solids Interacting Solids Multiplicity of Solids PHYS 4311: Thermodynamics & Statistical Mechanics
    Justin Einstein Solids Interacting Solids Multiplicity of Solids PHYS 4311: Thermodynamics & Statistical Mechanics Alejandro Garcia, Justin Gaumer, Chirag Gokani, Kristian Gonzalez, Nicholas Hamlin, Leonard Humphrey, Cullen Hutchison, Krishna Kalluri, Arjun Khurana Justin Introduction • Debye model(collective frequency motion) • Dulong–Petit law (behavior at high temperatures) predicted by Debye model •Polyatomic molecules 3R • dependence predicted by Debye model, too. • Einstein model not accurate at low temperatures Chirag Debye Model: Particle in a Box Chirag Quantum Harmonic Oscillator m<<1 S. eq. Chirag Discrete Evenly-Spaced Quanta of Energy Kristian What about in 3D? A single particle can "oscillate" each of the 3 cartesian directions: x, y, z. If we were just to look at a single atom subject to the harmonic oscillator potential in each cartesian direction, its total energy is the sum of energies corresponding to oscillations in the x direction, y direction, and z direction. Kristian Einstein Solid Model: Collection of Many Isotropic/Identical Harmonic Oscillators If our solid consists of N "oscillators" then there are N/3 atoms (each atom is subject to the harmonic oscillator potential in the 3 independent cartesian directions). Krishna What is the Multiplicity of an Einstein Solid with N oscillators? Our system has exactly N oscillators and q quanta of energy. This is the macrostate. There would be a certain number of microstates resulting from individual oscillators having different amounts of energy, but the whole solid having total q quanta of energy. Note that each microstate is equally probable! Krishna What is the Multiplicity of an Einstein Solid with N oscillators? Recall that energy comes in discrete quanta (units of hf) or "packets"! Each of the N oscillators will have an energy measure in integer units of hf! Question: Let an Einstein Solid of N oscillators have a fixed value of exactly q quanta of energy.
    [Show full text]