The goal of equilibrium statistical mechanics is to calculate the diagonal elements of ˆρ eq so we can evaluate average observables < A >= Tr{Â ˆρ eq


 Noah Wright
 1 years ago
 Views:
Transcription
1 Chapter. The microcanonical ensemble The goal of equilibrium statistical mechanics is to calculate the diagonal elements of ˆρ eq so we can evaluate average observables < A >= Tr{Â ˆρ eq } = A that give us fundamental relations or equations of state. Just as thermodynamics has its potentials U A H G etc. so statistical mechanics has its ensembles which are useful depending on what macroscopic variables are specified. We first consider the microcanonical ensemble because it is the one directly defined in postulate II of statistical mechanics. In the microcanonical ensemble U is fixed (Postulate I) and other constraints that are fixed are the volume V and mole number n (for a simple system) or other extensive parameters (for more complicated systems).. Definition of the partition function The partition function of an ensemble describes how probability is partitioned among the available microstates compatible with the constraints imposed on the ensemble. In the case of the microcanonical ensemble the partitioning is equal in all microstates at the same energy: according to postulate II with p i = ρ (eq) ii = / W (U) for each microstate i at energy U. Using just this we can evaluate equations of state and fundamental relations.. Calculation of thermodynamic quantities from W(U) Example : Fundamental relation for lattice gas: entropyvolume part. Consider again the model system of a box with M=V/V 0 volume elements V 0 and particles of volume V 0 so each particle can fill one volume elements. The particles can randomly hop among unoccupied volume elements to randomly sample the full volume of the box. This is a simple model of an ideal gas. As shown in the last chapter M! W = (M )!! for identical particles and we can approximate this if M<< by W V! V 0 since M!/(M!) M in that case. Assuming the hopping samples all microstates so the system reaches equilibrium we compute the equilibrium entropy as proved in chapter 0 from postulate III as S = k B lnw S 0 + k B ln(v / V 0 ) where S 0 is independent of volume. This gives the volume dependence of the entropy of an ideal gas. ote that by taking the derivative ( S/ V) = k B /V = P/T we can immediately derive the ideal gas law PV = k B T = nrt. Example : Fundamental relation for a lattice gas: entropyenergy part. The above model does not give us the energy dependence since we did not explicitly consider the energy of the particles other than to assume there was enough energy for
2 them to randomly hop around. We now remedy this by considering the energy levels of particles in a box. The result will also demonstrate once more that Ω increases so ferociously fast that it is equal to W with incredibly high accuracy for more than a handful of particles. Let the total energy U be randomly distributed among particles in a box of volume L 3 = V. The energy is given by U = 3 p i m i= where i=3 are the xyz coordinates of particle # and so forth to i=333 are the xyz coordinates of particle #. In quantum mechanics the momentum of a free particle is given by p=h/λ where h is Planck s constant. Only certain waves Ψ(x) are allowed in the box such that Ψ(x) = 0 at the boundaries of the box as shown in the figure below. Fig.. Particle in a box wavefunction can only have wavelengths so that Ψ=0 at the boundaries. The state space (or quantum number space) with 3 axes contains a hypersphere of constant energy. In a large number of dimensions the states (black dots) in a layer at the surface of this sphere is essentially equal to the total number of states within that surface. The wavelengths λ=l/ L 3L/... can be inserted in the equation for total energy yielding U = 3 3 hn i h n m L = i n i= 8mL i = 3 i= the energy for a bunch of particles in a box. W(U) is the number of states at energy U. Looking at the figure again all the energy levels are dots in a 3dimensional cartesian space called the state space or action space or sometimes quantum number space. The surface of constant energy U is the surface of a hypersphere of dimension 3 in state space. The reason is that the above equation is of the form constant = x +y +... where the variables are the quantum numbers. The number of states within a thin shell of energy U at the surface of the sphere is W (U) where lim W (U) = Ω.
3 Ω is the total number of states inside the sphere which at a first glance would seem to be much larger than W(U) the states in the shell. In fact for a very high dimensional hypervolume a thin shell at the surface contains all the volume so in fact Ω is essentially equal to W(U) and we can just calculate the former to a good approximation when is large. If this is hard to believe consider an analogous example of a hypercube instead of a hypersphere. Its volume is L m where m is the number of dimensions. The change in volume with side length L is V/ L=mL m so ΔV=mL m ΔL is the volume of a shell of width ΔL at the surface of the cube. The ratio of that volume to the total volume is ΔV/V=mΔL/L. Let s take the example our intuition is built on m=3 and assume ΔL/L=0.00 just a 0.% surface layer. Then ΔV/V= <<V indeed. But now consider m=0 0 a typical number of particles in a statistical mechanical system. ow ΔV/V= =0 7. The little increment in volume is much greater than the original volume of the cube and contains essentially all the volume of the new slightly larger cube. It may be slightly large in side length but it is astronomically larger in volume. ow back to our hypersphere in the figure. Its volume which is essentially equal to W(U) the number of states just at the surface of the sphere is 3 W (U) = V = 3 / π hypersphere Γ(3/ + ) R3 = U U 0. The (/) 3 is there because all quantum numbers must be greater than zero so only the positive part of the sphere should be counted. The Gamma function Γ is related to the factorial function and R is the radius of the sphere which is given by R = n max = 8mL U h the largest quantum number if all energy is in a single mode. They key is that R~ U so U is raised to the 3/ power where is the number of particles 3 is because there are three modes per particle and the / is because of the energy of a free particle depends on the square of the quantum number. Thus 3 S(U) = k B lnw (U) = S k B lnu = S nrlnu where the constant S 0 is not the same as in the previous example. We used the volume equation from the previous example to obtain an equation of state (PV=nRT) and we can obtain another equation of state here: S = V n T = 3 nr U or U = 3 nrt. This equation relates the energy of an ideal gas to its temperature. 3n is the number of modes or degrees of freedom (3 velocities per particle time n moles of particles) whereas the factor of comes directly from the particleinabox energy function in case you ever wondered where that comes from. So for a harmonic oscillator n~u ( E = ω(n + / ) as you may recall) instead of n~u / and you might expect U=3nRT for 3 particles held together by springs into a solid crystal lattice. And indeed that is true for an ideal lattice at high temperature (in analogy to an ideal gas at high temperature). Unlike free particles the energy of oscillators does not have the factor of 3 /
4 /. The deep reason is that an oscillator has two degrees of freedom to store energy in each direction not just one: there s still the kinetic energy but there s also potential energy. Example 3: A system of uncoupled spins s z =±/ The Hamiltonian for this system in a magnetic field is given by H = s zj B + B j = where the extra term at the end is added so the energy equals zero when all the spins are pointing down. At energy U=0 no spin is excited. For each excited spin the energy increases by B so at energy U U/B atoms are excited. These U/B excitations are indistinguishable and can be distributed in sites:! Γ( + ) W (U) = U! U =! Γ + U Γ U. B + This is our usual formula for permutations; the right side is in terms of Gamma functions which are defined even when U/B is not an integer. Gamma functions basically interpolate the factorial function for noninteger values. This formula has a potential problem builtin: clearly when U starts out at 0 and then increases W initially increases. But for U=B (the maximum energy) W= again. In fact W reaches its maximum for U=B/. But if W(U) is not monotonic in U then S isn t either violating P3 of thermodynamics. Let s see how this works out. For large and temperature neither so low that U ~ O() nor so high that B U ~ O() we can use the Stirling expansion ln! ln yielding B S = lnω ln U k B ln U + U B U B lnu B + U B ln U ln U U B lnu B + U B ln U B ln ln U B + U B ln U U B ln U B U B ln U U B ln U B after canceling terms as much as possible. We can now calculate the temperature and obtain the equation of state U(T): T = S k B ln B U B U + e B/kT In this equations at T ~ 0 U 0; and as T U B /. So even at infinite temperature the energy can only go up to half the maximum value where W(U) is monotonic. The population cannot be inverted to have more spins point up than down.
5 At most half the spins can be made to point up by heating. This should come as no surprise: if the number of microstates is maximized by having only half the spins point up when energy is added then that s the state you will get (this is true even in the exact solution). ote that this does not mean that it is impossible to get all spins to point up. It is just not an equilibrium state at any temperature between 0 and. Such nonequilibrium states with more spins up (or atoms excited) than down are called inverted populations. In lasers such states are created by putting the system (like a laser crystal) far out of equilibrium. Such a state will then relax back to an equilibrium state releasing a pulse of energy as the spins (or atoms) drop from the excited to the ground state. The heat capacity of the above example system is c v = U T B e B/kT kt ( + e B/kT ) peaked at 4k BT/B so we can calculate thermodynamic quantities as input for thermodynamic manipulations. As we shall see in detail later (actually we saw it in the previous example!) in any real system the heat capacity must eventually approach c v = k B / where is the number of degrees of freedom. However a broad peak at 4k B T / B is a sign of two lowlying energy levels spaced by B. Levels at higher energy will eventually contribute to c v making sure it does not drop. Example 4: Let us check that T derived from S = k lnw B = T indeed agrees with the intuitive concept of temperature. Consider two baths within a closed system so U = U + U = const. du = 0 du = du. If we know for each bath then W i (U i ) dw i = W i U i du i W tot W ( ) ( W + dw ) W W + O(dW ) dw tot + dw dw + W dw W = W W + W U U du = 0 at equilibrium because the maximum number of states is already occupied. For this to be true for any infinitesimal energy flow du W W U = W W U lnw U V V = lnw U V V or S U V = = S T At equilibrium the temperatures are equal fitting out thermodynamic definition that temperature is equalized when heat flow is allowed. V = T
Chapter 3. Entropy, temperature, and the microcanonical partition function: how to calculate results with statistical mechanics.
Chapter 3. Entropy, temperature, and the microcanonical partition function: how to calculate results with statistical mechanics. The goal of equilibrium statistical mechanics is to calculate the density
More informationChapter 4: Going from microcanonical to canonical ensemble, from energy to temperature.
Chapter 4: Going from microcanonical to canonical ensemble, from energy to temperature. All calculations in statistical mechanics can be done in the microcanonical ensemble, where all copies of the system
More informationto satisfy the large number approximations, W W sys can be small.
Chapter 12. The canonical ensemble To discuss systems at constant T, we need to embed them with a diathermal wall in a heat bath. Note that only the system and bath need to be large for W tot and W bath
More informationLecture 8. The Second Law of Thermodynamics; Energy Exchange
Lecture 8 The Second Law of Thermodynamics; Energy Exchange The second law of thermodynamics Statistics of energy exchange General definition of temperature Why heat flows from hot to cold Reading for
More informationLecture 8. The Second Law of Thermodynamics; Energy Exchange
Lecture 8 The Second Law of Thermodynamics; Energy Exchange The second law of thermodynamics Statistics of energy exchange General definition of temperature Why heat flows from hot to cold Reading for
More informationChE 503 A. Z. Panagiotopoulos 1
ChE 503 A. Z. Panagiotopoulos 1 STATISTICAL MECHANICAL ENSEMLES 1 MICROSCOPIC AND MACROSCOPIC ARIALES The central question in Statistical Mechanics can be phrased as follows: If particles (atoms, molecules,
More information5.60 Thermodynamics & Kinetics Spring 2008
MIT OpenCourseWare http://ocw.mit.edu 5.60 Thermodynamics & Kinetics Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 5.60 Spring 2008 Lecture
More information2m + U( q i), (IV.26) i=1
I.D The Ideal Gas As discussed in chapter II, microstates of a gas of N particles correspond to points { p i, q i }, in the 6Ndimensional phase space. Ignoring the potential energy of interactions, the
More informationStatistical. mechanics
CHAPTER 15 Statistical Thermodynamics 1: The Concepts I. Introduction. A. Statistical mechanics is the bridge between microscopic and macroscopic world descriptions of nature. Statistical mechanics macroscopic
More informationIV. Classical Statistical Mechanics
IV. Classical Statistical Mechanics IV.A General Definitions Statistical Mechanics is a probabilistic approach to equilibrium macroscopic properties of large numbers of degrees of freedom. As discussed
More informationJoint Entrance Examination for Postgraduate Courses in Physics EUF
Joint Entrance Examination for Postgraduate Courses in Physics EUF Second Semester 013 Part 1 3 April 013 Instructions: DO NOT WRITE YOUR NAME ON THE TEST. It should be identified only by your candidate
More informationCHEMUA 652: Thermodynamics and Kinetics
1 CHEMUA 652: Thermodynamics and Kinetics Notes for Lecture 2 I. THE IDEAL GAS LAW In the last lecture, we discussed the MaxwellBoltzmann velocity and speed distribution functions for an ideal gas. Remember
More information1 Multiplicity of the ideal gas
Reading assignment. Schroeder, section.6. 1 Multiplicity of the ideal gas Our evaluation of the numbers of microstates corresponding to each macrostate of the twostate paramagnet and the Einstein model
More informationThermal and Statistical Physics Department Exam Last updated November 4, L π
Thermal and Statistical Physics Department Exam Last updated November 4, 013 1. a. Define the chemical potential µ. Show that two systems are in diffusive equilibrium if µ 1 =µ. You may start with F =
More information[S R (U 0 ɛ 1 ) S R (U 0 ɛ 2 ]. (0.1) k B
Canonical ensemble (Two derivations) Determine the probability that a system S in contact with a reservoir 1 R to be in one particular microstate s with energy ɛ s. (If there is degeneracy we are picking
More informationIntroduction to Quantum Mechanics (Prelude to Nuclear Shell Model) Heisenberg Uncertainty Principle In the microscopic world,
Introduction to Quantum Mechanics (Prelude to Nuclear Shell Model) Heisenberg Uncertainty Principle In the microscopic world, x p h π If you try to specify/measure the exact position of a particle you
More informationPhase space in classical physics
Pase space in classical pysics Quantum mecanically, we can actually COU te number of microstates consistent wit a given macrostate, specified (for example) by te total energy. In general, eac microstate
More information8 Wavefunctions  Schrödinger s Equation
8 Wavefunctions  Schrödinger s Equation So far we have considered only free particles  i.e. particles whose energy consists entirely of its kinetic energy. In general, however, a particle moves under
More informationThermal & Statistical Physics Study Questions for the Spring 2018 Department Exam December 6, 2017
Thermal & Statistical Physics Study Questions for the Spring 018 Department Exam December 6, 017 1. a. Define the chemical potential. Show that two systems are in diffusive equilibrium if 1. You may start
More informationStatistical Mechanics Notes. Ryan D. Reece
Statistical Mechanics Notes Ryan D. Reece August 11, 2006 Contents 1 Thermodynamics 3 1.1 State Variables.......................... 3 1.2 Inexact Differentials....................... 5 1.3 Work and Heat..........................
More informationProblem: Calculate the entropy change that results from mixing 54.0 g of water at 280 K with 27.0 g of water at 360 K in a vessel whose walls are
Problem: Calculate the entropy change that results from mixing 54.0 g of water at 280 K with 27.0 g of water at 360 K in a vessel whose walls are perfectly insulated from the surroundings. Is this a spontaneous
More informationPHYSICS DEPARTMENT, PRINCETON UNIVERSITY PHYSICS 301 FINAL EXAMINATION. January 13, 2005, 7:30 10:30pm, Jadwin A10 SOLUTIONS
PHYSICS DEPARTMENT, PRINCETON UNIVERSITY PHYSICS 301 FINAL EXAMINATION January 13, 2005, 7:30 10:30pm, Jadwin A10 SOLUTIONS This exam contains five problems. Work any three of the five problems. All problems
More information2. Thermodynamics. Introduction. Understanding Molecular Simulation
2. Thermodynamics Introduction Molecular Simulations Molecular dynamics: solve equations of motion r 1 r 2 r n Monte Carlo: importance sampling r 1 r 2 r n How do we know our simulation is correct? Molecular
More informationEntropy in Macroscopic Systems
Lecture 15 Heat Engines Review & Examples p p b b Hot reservoir at T h p a a c adiabats Heat leak Heat pump Q h Q c W d V 1 V 2 V Cold reservoir at T c Lecture 15, p 1 Review Entropy in Macroscopic Systems
More informationBoltzmann Distribution Law (adapted from Nash)
Introduction Statistical mechanics provides a bridge between the macroscopic realm of classical thermodynamics and the microscopic realm of atoms and molecules. We are able to use computational methods
More informationStatistical thermodynamics L1L3. Lectures 11, 12, 13 of CY101
Statistical thermodynamics L1L3 Lectures 11, 12, 13 of CY101 Need for statistical thermodynamics Microscopic and macroscopic world Distribution of energy  population Principle of equal a priori probabilities
More informationBasic Concepts and Tools in Statistical Physics
Chapter 1 Basic Concepts and Tools in Statistical Physics 1.1 Introduction Statistical mechanics provides general methods to study properties of systems composed of a large number of particles. It establishes
More informationStatistical thermodynamics Lectures 7, 8
Statistical thermodynamics Lectures 7, 8 Quantum Classical Energy levels Bulk properties Various forms of energies. Everything turns out to be controlled by temperature CY1001 T. Pradeep Ref. Atkins 9
More informationLecture 25 Goals: Chapter 18 Understand the molecular basis for pressure and the idealgas
Lecture 5 Goals: Chapter 18 Understand the molecular basis for pressure and the idealgas law. redict the molar specific heats of gases and solids. Understand how heat is transferred via molecular collisions
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department Statistical Physics I Spring Term 2013 Notes on the Microcanonical Ensemble
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department 8.044 Statistical Physics I Spring Term 2013 Notes on the Microcanonical Ensemble The object of this endeavor is to impose a simple probability
More informationLecture 20: Spinodals and Binodals; Continuous Phase Transitions; Introduction to Statistical Mechanics
Lecture 20: 11.28.05 Spinodals and Binodals; Continuous Phase Transitions; Introduction to Statistical Mechanics Today: LAST TIME: DEFINING METASTABLE AND UNSTABLE REGIONS ON PHASE DIAGRAMS...2 Conditions
More informationInternal Degrees of Freedom
Physics 301 16Oct2002 151 Internal Degrees of Freedom There are several corrections we might make to our treatment of the ideal gas If we go to high occupancies our treatment using the MaxwellBoltzmann
More informationChapter 3  First Law of Thermodynamics
Chapter 3  dynamics The ideal gas law is a combination of three intuitive relationships between pressure, volume, temp and moles. David J. Starling Penn State Hazleton Fall 2013 When a gas expands, it
More informationAtoms, Molecules and Solids. From Last Time Superposition of quantum states Philosophy of quantum mechanics Interpretation of the wave function:
Essay outline and Ref to main article due next Wed. HW 9: M Chap 5: Exercise 4 M Chap 7: Question A M Chap 8: Question A From Last Time Superposition of quantum states Philosophy of quantum mechanics Interpretation
More informationCONTENTS 1. In this course we will cover more foundational topics such as: These topics may be taught as an independent study sometime next year.
CONTENTS 1 0.1 Introduction 0.1.1 Prerequisites Knowledge of di erential equations is required. Some knowledge of probabilities, linear algebra, classical and quantum mechanics is a plus. 0.1.2 Units We
More informationSpontaneity: Second law of thermodynamics CH102 General Chemistry, Spring 2012, Boston University
Spontaneity: Second law of thermodynamics CH102 General Chemistry, Spring 2012, Boston University three or four forces and, as capstone, a minimalist cosmic constitution to legislate their use: Article
More informationIntroduction to Statistical Thermodynamics
Cryocourse 2011 Chichilianne Introduction to Statistical Thermodynamics Henri GODFRIN CNRS Institut Néel Grenoble http://neel.cnrs.fr/ Josiah Willard Gibbs worked on statistical mechanics, laying a foundation
More informationChapter 20 The Second Law of Thermodynamics
Chapter 20 The Second Law of Thermodynamics When we previously studied the first law of thermodynamics, we observed how conservation of energy provided us with a relationship between U, Q, and W, namely
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department Statistical Physics I Spring Term 2013 Exam #1
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department 8.044 Statistical Physics I Spring Term 2013 Exam #1 Problem 1 (30 points) Quantum Dots A complicated process creates quantum dots (also called
More informationPhysics 342 Lecture 27. Spin. Lecture 27. Physics 342 Quantum Mechanics I
Physics 342 Lecture 27 Spin Lecture 27 Physics 342 Quantum Mechanics I Monday, April 5th, 2010 There is an intrinsic characteristic of point particles that has an analogue in but no direct derivation from
More informationLecture 5. HartreeFock Theory. WS2010/11: Introduction to Nuclear and Particle Physics
Lecture 5 HartreeFock Theory WS2010/11: Introduction to Nuclear and Particle Physics Particlenumber representation: General formalism The simplest starting point for a manybody state is a system of
More informationQuantum control of dissipative systems. 1 Density operators and mixed quantum states
Quantum control of dissipative systems S. G. Schirmer and A. I. Solomon Quantum Processes Group, The Open University Milton Keynes, MK7 6AA, United Kingdom S.G.Schirmer@open.ac.uk, A.I.Solomon@open.ac.uk
More informationi=1 n i, the canonical probabilities of the microstates [ βǫ i=1 e βǫn 1 n 1 =0 +Nk B T Nǫ 1 + e ǫ/(k BT), (IV.75) E = F + TS =
IV.G Examples The two examples of sections (IV.C and (IV.D are now reexamined in the canonical ensemble. 1. Two level systems: The impurities are described by a macrostate M (T,. Subject to the Hamiltonian
More information18.13 Review & Summary
5/2/10 10:04 PM Print this page 18.13 Review & Summary Temperature; Thermometers Temperature is an SI base quantity related to our sense of hot and cold. It is measured with a thermometer, which contains
More informationComparing and Improving Quark Models for the Triply Bottom Baryon Spectrum
Comparing and Improving Quark Models for the Triply Bottom Baryon Spectrum A thesis submitted in partial fulfillment of the requirements for the degree of Bachelor of Science degree in Physics from the
More information140a Final Exam, Fall 2006., κ T 1 V P. (? = P or V ), γ C P C V H = U + PV, F = U TS G = U + PV TS. T v. v 2 v 1. exp( 2πkT.
40a Final Exam, Fall 2006 Data: P 0 0 5 Pa, R = 8.34 0 3 J/kmol K = N A k, N A = 6.02 0 26 particles/kilomole, T C = T K 273.5. du = TdS PdV + i µ i dn i, U = TS PV + i µ i N i Defs: 2 β ( ) V V T ( )
More informationPhysics 132 Fundamentals of Physics for Biologists II. Statistical Physics and Thermodynamics
Physics 132 Fundamentals of Physics for Biologists II Statistical Physics and Thermodynamics QUIZ 2 25 Quiz 2 20 Number of Students 15 10 5 AVG: STDEV: 5.15 2.17 0 0 2 4 6 8 10 Score 1. (4 pts) A 200
More informationarxiv: v2 [hepth] 7 Apr 2015
Statistical Mechanics Derived From Quantum Mechanics arxiv:1501.05402v2 [hepth] 7 Apr 2015 YuLei Feng 1 and YiXin Chen 1 1 Zhejiang Institute of Modern Physics, Zhejiang University, Hangzhou 310027,
More informationMidTerm. Phys224 Spring 2008 Dr. P. Hanlet
MidTerm Name: Show your work!!! If I can read it, I will give you partial credit!!! Correct answers without work will NOT get full credit. Concept 5 points) 1. In terms of the First Law of Thermodynamics
More informationPhonon II Thermal Properties
Phonon II Thermal Properties Physics, UCF OUTLINES Phonon heat capacity Planck distribution Normal mode enumeration Density of states in one dimension Density of states in three dimension Debye Model for
More informationPhysics 3700 Introduction to Quantum Statistical Thermodynamics Relevant sections in text: Quantum Statistics: Bosons and Fermions
Physics 3700 Introduction to Quantum Statistical Thermodynamics Relevant sections in text: 7.1 7.4 Quantum Statistics: Bosons and Fermions We now consider the important physical situation in which a physical
More informationdv = adx, where a is the active area of the piston. In equilibrium, the external force F is related to pressure P as
Chapter 3 Work, heat and the first law of thermodynamics 3.1 Mechanical work Mechanical work is defined as an energy transfer to the system through the change of an external parameter. Work is the only
More informationHarmonic Oscillator I
Physics 34 Lecture 7 Harmonic Oscillator I Lecture 7 Physics 34 Quantum Mechanics I Monday, February th, 008 We can manipulate operators, to a certain extent, as we would algebraic expressions. By considering
More informationSommerfeldDrude model. Ground state of ideal electron gas
SommerfeldDrude model Recap of Drude model: 1. Treated electrons as free particles moving in a constant potential background. 2. Treated electrons as identical and distinguishable. 3. Applied classical
More informationStatistical Physics. The Second Law. Most macroscopic processes are irreversible in everyday life.
Statistical Physics he Second Law ime s Arrow Most macroscopic processes are irreversible in everyday life. Glass breaks but does not reform. Coffee cools to room temperature but does not spontaneously
More informationPart II: Statistical Physics
Chapter 6: Boltzmann Statistics SDSMT, Physics Fall Semester: Oct.  Dec., 2014 1 Introduction: Very brief 2 Boltzmann Factor Isolated System and System of Interest Boltzmann Factor The Partition Function
More informationThermodynamics of nuclei in thermal contact
Thermodynamics of nuclei in thermal contact KarlHeinz Schmidt, Beatriz Jurado CENBG, CNRS/IN2P3, Chemin du Solarium B.P. 120, 33175 Gradignan, France Abstract: The behaviour of a dinuclear system in
More informationIf electrons moved in simple orbits, p and x could be determined, but this violates the Heisenberg Uncertainty Principle.
CHEM 2060 Lecture 18: Particle in a Box L181 Atomic Orbitals If electrons moved in simple orbits, p and x could be determined, but this violates the Heisenberg Uncertainty Principle. We can only talk
More informationQuantum Theory of Angular Momentum and Atomic Structure
Quantum Theory of Angular Momentum and Atomic Structure VBS/MRC Angular Momentum 0 Motivation...the questions Whence the periodic table? Concepts in Materials Science I VBS/MRC Angular Momentum 1 Motivation...the
More informationStatistical Mechanics and Information Theory
1 MultiUser Information Theory 2 Oct 31, 2013 Statistical Mechanics and Information Theory Lecturer: Dror Vinkler Scribe: Dror Vinkler I. INTRODUCTION TO STATISTICAL MECHANICS In order to see the need
More informationThe 3 dimensional Schrödinger Equation
Chapter 6 The 3 dimensional Schrödinger Equation 6.1 Angular Momentum To study how angular momentum is represented in quantum mechanics we start by reviewing the classical vector of orbital angular momentum
More informationHW posted on web page HW10: Chap 14 Concept 8,20,24,26 Prob. 4,8. From Last Time
HW posted on web page HW10: Chap 14 Concept 8,20,24,26 Prob. 4,8 From Last Time Philosophical effects in quantum mechanics Interpretation of the wave function: Calculation using the basic premises of quantum
More informationThe Northern California Physics GRE Bootcamp
The Northern California Physics GRE Bootcamp Held at UC Davis, Sep 89, 2012 Damien Martin Big tips and tricks * Multiple passes through the exam * Dimensional analysis (which answers make sense?) Other
More informationWhat is thermal equilibrium and how do we get there?
arxiv:1507.06479 and more What is thermal equilibrium and how do we get there? Hal Tasaki QMath 13, Oct. 9, 2016, Atlanta 40 C 20 C 30 C 30 C about the talk Foundation of equilibrium statistical mechanics
More informationA Brief Introduction to Statistical Mechanics
A Brief Introduction to Statistical Mechanics E. J. Maginn, J. K. Shah Department of Chemical and Biomolecular Engineering University of Notre Dame Notre Dame, IN 46556 USA Monte Carlo Workshop Universidade
More informationDefinite Integral and the Gibbs Paradox
Acta Polytechnica Hungarica ol. 8, No. 4, 0 Definite Integral and the Gibbs Paradox TianZhi Shi College of Physics, Electronics and Electrical Engineering, HuaiYin Normal University, HuaiAn, JiangSu, China,
More informationG : Statistical Mechanics
G25.2651: Statistical Mechanics Notes for Lecture 1 Defining statistical mechanics: Statistical Mechanics provies the connection between microscopic motion of individual atoms of matter and macroscopically
More informationQuantumMechanical Carnot Engine
QuantumMechanical Carnot Engine Carl M. Bender 1, Dorje C. Brody, and Bernhard K. Meister 3 1 Department of Physics, Washington University, St. Louis MO 63130, USA Blackett Laboratory, Imperial College,
More informationThe finegrained Gibbs entropy
Chapter 12 The finegrained Gibbs entropy 12.1 Introduction and definition The standard counterpart of thermodynamic entropy within Gibbsian SM is the socalled finegrained entropy, or Gibbs entropy. It
More informationThe Particle in a Box
Page 324 Lecture 17: Relation of Particle in a Box Eigenstates to Position and Momentum Eigenstates General Considerations on Bound States and Quantization Continuity Equation for Probability Date Given:
More informationPhys Midterm. March 17
Phys 7230 Midterm March 17 Consider a spin 1/2 particle fixed in space in the presence of magnetic field H he energy E of such a system can take one of the two values given by E s = µhs, where µ is the
More informationPhysics 221B Spring 2018 Notes 30 The ThomasFermi Model
Copyright c 217 by Robert G. Littlejohn Physics 221B Spring 218 Notes 3 The ThomasFermi Model 1. Introduction The ThomasFermi model is a relatively crude model of multielectron atoms that is useful
More informationHarmonic Oscillator with raising and lowering operators. We write the Schrödinger equation for the harmonic oscillator in one dimension as follows:
We write the Schrödinger equation for the harmonic oscillator in one dimension as follows: H ˆ! = "!2 d 2! + 1 2µ dx 2 2 kx 2! = E! T ˆ = "! 2 2µ d 2 dx 2 V ˆ = 1 2 kx 2 H ˆ = ˆ T + ˆ V (1) where µ is
More information2. The Schrödinger equation for oneparticle problems. 5. Atoms and the periodic table of chemical elements
1 Historical introduction The Schrödinger equation for oneparticle problems 3 Mathematical tools for quantum chemistry 4 The postulates of quantum mechanics 5 Atoms and the periodic table of chemical
More informationLecture 12: Phonon heat capacity
Lecture 12: Phonon heat capacity Review o Phonon dispersion relations o Quantum nature of waves in solids Phonon heat capacity o Normal mode enumeration o Density of states o Debye model Review By considering
More informationProblem Set 10: Solutions
University of Alabama Department of Physics and Astronomy PH 253 / LeClair Fall 21 Problem Set 1: Solutions 1. For a onedimensional infinite square well of length l the allowed energies for noninteracting
More informationAn Inverse Mass Expansion for Entanglement Entropy. Free Massive Scalar Field Theory
in Free Massive Scalar Field Theory NCSR Demokritos National Technical University of Athens based on arxiv:1711.02618 [hepth] in collaboration with Dimitris Katsinis March 28 2018 Entanglement and Entanglement
More informationPhase Transitions. µ a (P c (T ), T ) µ b (P c (T ), T ), (3) µ a (P, T c (P )) µ b (P, T c (P )). (4)
Phase Transitions A homogeneous equilibrium state of matter is the most natural one, given the fact that the interparticle interactions are translationally invariant. Nevertheless there is no contradiction
More informationIrreversible Processes
Irreversible Processes Examples: Block sliding on table comes to rest due to friction: KE converted to heat. Heat flows from hot object to cold object. Air flows into an evacuated chamber. Reverse process
More informationChem 467 Supplement to Lecture 19 Hydrogen Atom, Atomic Orbitals
Chem 467 Supplement to Lecture 19 Hydrogen Atom, Atomic Orbitals PreQuantum Atomic Structure The existence of atoms and molecules had long been theorized, but never rigorously proven until the late 19
More informationC. Show your answer in part B agrees with your answer in part A in the limit that the constant c 0.
Problem #1 A. A projectile of mass m is shot vertically in the gravitational field. Its initial velocity is v o. Assuming there is no air resistance, how high does m go? B. Now assume the projectile is
More informationLecture Notes 2014March 13 on Thermodynamics A. First Law: based upon conservation of energy
Dr. W. Pezzaglia Physics 8C, Spring 2014 Page 1 Lecture Notes 2014March 13 on Thermodynamics A. First Law: based upon conservation of energy 1. Work 1 Dr. W. Pezzaglia Physics 8C, Spring 2014 Page 2 (c)
More informationalthough Boltzmann used W instead of Ω for the number of available states.
Lecture #13 1 Lecture 13 Obectives: 1. Ensembles: Be able to list the characteristics of the following: (a) icrocanonical (b) Canonical (c) Grand Canonical 2. Be able to use Lagrange s method of undetermined
More informationChapter 14. Ideal Bose gas Equation of state
Chapter 14 Ideal Bose gas In this chapter, we shall study the thermodynamic properties of a gas of noninteracting bosons. We will show that the symmetrization of the wavefunction due to the indistinguishability
More informationTwo and ThreeDimensional Systems
0 Two and ThreeDimensional Systems Separation of variables; degeneracy theorem; group of invariance of the twodimensional isotropic oscillator. 0. Consider the Hamiltonian of a twodimensional anisotropic
More informationQuantum Field Theory and Condensed Matter Physics: making the vacuum concrete. Fabian Essler (Oxford)
Quantum Field Theory and Condensed Matter Physics: making the vacuum concrete Fabian Essler (Oxford) Oxford, June 2013 Lev Landau This work contains many things which are new and interesting. Unfortunately,
More informationPhysics 576 Stellar Astrophysics Prof. James Buckley. Lecture 2 Radiation
Physics 576 Stellar Astrophysics Prof. James Buckley Lecture 2 Radiation Reading/Homework Assignment Read chapter 1, sections 1.1, 1.2, 1.5 Homework will be assigned on Thursday. Radiation Radiation A
More informationarxiv: v3 [condmat.statmech] 7 Jan 2015
Entanglement Entropies of NonEquilibrium FiniteSpin Systems Koichi Nakagawa Hoshi University, Tokyo 148501, Japan arxiv:1410.6988v3 [condmat.statmech] 7 Jan 015 Abstract For the purpose of clarifying
More informationCoupling of Angular Momenta Isospin NucleonNucleon Interaction
Lecture 5 Coupling of Angular Momenta Isospin NucleonNucleon Interaction WS0/3: Introduction to Nuclear and Particle Physics,, Part I I. Angular Momentum Operator Rotation R(θ): in polar coordinates the
More informationGrandcanonical ensembles
Grandcanonical ensembles As we know, we are at the point where we can deal with almost any classical problem (see below), but for quantum systems we still cannot deal with problems where the translational
More informationLectures 21 and 22: Hydrogen Atom. 1 The Hydrogen Atom 1. 2 Hydrogen atom spectrum 4
Lectures and : Hydrogen Atom B. Zwiebach May 4, 06 Contents The Hydrogen Atom Hydrogen atom spectrum 4 The Hydrogen Atom Our goal here is to show that the twobody quantum mechanical problem of the hydrogen
More informationFebruary 18, In the parallel RLC circuit shown, R = Ω, L = mh and C = µf. The source has V 0. = 20.0 V and f = Hz.
Physics Qualifying Examination Part I 7 Minute Questions February 18, 2012 1. In the parallel RLC circuit shown, R = 800.0 Ω, L = 160.0 mh and C = 0.0600 µf. The source has V 0 = 20.0 V and f = 2400.0
More informationPHY413 Quantum Mechanics B Duration: 2 hours 30 minutes
BSc/MSci Examination by Course Unit Thursday nd May 4 :  :3 PHY43 Quantum Mechanics B Duration: hours 3 minutes YOU ARE NOT PERMITTED TO READ THE CONTENTS OF THIS QUESTION PAPER UNTIL INSTRUCTED TO DO
More informationPhysics 228 Today: Ch 41: 13: 3D quantum mechanics, hydrogen atom
Physics 228 Today: Ch 41: 13: 3D quantum mechanics, hydrogen atom Website: Sakai 01:750:228 or www.physics.rutgers.edu/ugrad/228 Happy April Fools Day Example / Worked Problems What is the ratio of the
More information1. For the case of the harmonic oscillator, the potential energy is quadratic and hence the total Hamiltonian looks like: d 2 H = h2
15 Harmonic Oscillator 1. For the case of the harmonic oscillator, the potential energy is quadratic and hence the total Hamiltonian looks like: d 2 H = h2 2mdx + 1 2 2 kx2 (15.1) where k is the force
More informationIn the case of a nonrotating, uncharged black hole, the event horizon is a sphere; its radius R is related to its mass M according to
Black hole General relativity predicts that when a massive body is compressed to sufficiently high density, it becomes a black hole, an object whose gravitational pull is so powerful that nothing can escape
More informationBrief Review of Statistical Mechanics
Brief Review of Statistical Mechanics Introduction Statistical mechanics: a branch of physics which studies macroscopic systems from a microscopic or molecular point of view (McQuarrie,1976) Also see (Hill,1986;
More informationOptical Lattices. Chapter Polarization
Chapter Optical Lattices Abstract In this chapter we give details of the atomic physics that underlies the Bose Hubbard model used to describe ultracold atoms in optical lattices. We show how the ACStark
More informationThe Wave Function. Chapter The Harmonic Wave Function
Chapter 3 The Wave Function On the basis of the assumption that the de Broglie relations give the frequency and wavelength of some kind of wave to be associated with a particle, plus the assumption that
More informationLine spectrum (contd.) Bohr s Planetary Atom
Line spectrum (contd.) Hydrogen shows lines in the visible region of the spectrum (red, bluegreen, blue and violet). The wavelengths of these lines can be calculated by an equation proposed by J. J. Balmer:
More information