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Caltech | PH 127
58 sample documents related to PH 127
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Caltech PH 127
Physics 127c: Statistical Mechanics Fermi Liquid Theory: Thermodynamics Energy Expansion For a small number of excited quasiparticles the energy expanded about the ground state is E = E0 + p, p np, + O(n2 ) (1) where np, is plus one for every exci -
Caltech PH 127
Physics 127a: Class Notes Lecture 19: Molecular Gases Diatomic gases Consider a molecule made up of two atoms A and B, and at temperatures much lower than an electronic excitation energy (typically around 104 K). First let\'s suppose that the temperat -
Caltech PH 127
Physics 127a: Class Notes Lecture 2: A Simple Probability Example The \"equally likely\" of the fundamental postulate reminds us of a coin flip, and in fact a very simple probability problem actually gives us useful insights into statistical mechanics -
Caltech PH 127
Physics 127a: Statistical Mechanics Diamagnetism of the Electron Gas The Hamiltonian coupling the electron current to the magnetic eld B is H = i 1 e [pi + A(xi )]2 2m c (1) summing over the electrons i with position xi and momentum pi . (I will c -
Caltech PH 127
Physics 127b: Statistical Mechanics Linear Response Theory Useful references are Callen and Greene [1], and Chandler [2], chapter 16. Task To calculate the change in a measurement B (t) due to the application of a small \"field\" F (t) that gives a pe -
Caltech PH 127
Physics 127a: Class Notes Lecture 5: Energy, Heat and the Carnot Cycle Thermodynamic identity For the entropy of an isolated system S(E, N, V ) we can form the differential, evaluating the partials in terms of these expressions for T , , and P . This -
Caltech PH 127
Physics 127c: Statistical Mechanics Feynman Diagrams Using the language of second quantization it is now possible to develop the perturbation theory in the interaction. As in the classical case a diagrammatic formulation is found to be convenient. Th -
Caltech PH 127
Physics 127c: Statistical Mechanics Application of Path Integrals to Superuidity in He4 The path integral method, and its recent implementation using quantum Monte Carlo methods, provides both an intuitive understanding and a computational approach t -
Caltech PH 127
Physics 127b: Statistical Mechanics Fluctuations at a Second Order Transition We can use the Landau free energy to investigate fluctuations of the order parameter and so the validity of mean field theory, and the expansion itself. Remember that in th -
Caltech PH 127
Physics 127a: Class Notes Lecture 4: Entropy Second Law of Thermodynamics If we prepare an isolated system in a macroscopic conguration that is not the equilibrium one, the subsequent evolution to equilibrium will lead to an increase of the entropy S -
Caltech PH 127
Physics 127b: Statistical Mechanics Lecture:1 Classical Non-ideal Gas Partition Function We take the Hamiltonian to be the kinetic energy plus a potential energy U ({ri }) that is the sum of pairwise potentials pi2 1 H = u( ri - rj ). + (1) 2m 2 i,j -
Caltech PH 127
Physics 127c: Statistical Mechanics Quantum Monte Carlo Monte Carlo methods are good for evaluating probabilistic integrals. A key feature of quantum mechanics is that we must deal with complex amplitudes rather than real positive probabilities. Intr -
Caltech PH 127
Physics 127c: Statistical Mechanics Fermi Liquid Theory: Principles Landau developed the idea of quasiparticle excitations in the context of interacting Fermi systems. His theory is known as Fermi liquid theory. He introduced the idea phenomenologica -
Caltech PH 127
Physics 127a: Class Notes Lecture 14: Bose Condensation Ideal Bose Gas We consider an gas of ideal, spinless Bosons in three dimensions. The grand potential (T , , V ) is given by V 2 y 1/2 ln(1 - ze-y ) dy, (1) = 3 1/2 kT 0 with = h/ 2 mkT and -
Caltech PH 127
Physics 127b: Statistical Mechanics Fokker-Planck Equation The Langevin equation approach to the evolution of the velocity distribution for the Brownian particle might leave you uncomfortable. A more formal treatment of this type of problem is given -
Caltech PH 127
Physics 127b: Statistical Mechanics Scaling Hypothesis The scaling hypothesis allows us to relate all the power laws for the static, bulk thermodynamic quantities and the correlation function in terms of two basic exponents. The hypothesis was first -
Caltech PH 127
Physics 127b: Statistical Mechanics Lecture 3: First Order Phase Transitions The van der Waals equation for a gas is P+ a [V - b] = NkB T . V2 (1) (The variable a is proportional to N 2 and b to N, i.e. a = N 2 a and b = N b with a, b constants). -
Caltech PH 127
Physics 127a: Class Notes Lecture 7: Canonical Ensemble Simple Examples The canonical partition function provides the standard route to calculating the thermodynamic properties of macroscopic systemsone of the important tasks of statistical mechanic -
Caltech PH 127
Physics 127b: Statistical Mechanics Phase Transitions in Multicomponent Systems The Gibbs Phase Rule Consider a system with n components (different types of molecules) with r phases in equilibrium. The state of each phase is dened by P , T and then ( -
Caltech PH 127
Physics 127a: Class Notes Lecture 10: Other Ensembles/Thermodynamic Potentials Thermodynamic potentials We can define various other ensembles, by considering systems in equilibrium with reservoirs under various combinations of E, V , N transfer. This -
Caltech PH 127
Physics 127a: Class Notes Lecture 3: Motivation for Fundamental Postulate (Classical) Hamiltonian formulation of the dynamics For N particles there are 3N coordinates q1 , q2 . . . q3N and 3N conjugate momenta p1 , p2 . . . p3N . Usually these would -
Caltech PH 127
Physics 127c: Statistical Mechanics Statistical Mechanics of Superuidity Excitation Picture For the description in terms of a owing ground state plus excitations see Lecture 15 and problem 3 of Homework 7 for Ph127a. This approach has some disadvanta -
Caltech PH 127
Physics 127b: Statistical Mechanics Landau Theory of Second Order Phase Transitions Order Parameter Second order phase transitions occur when a new state of reduced symmetry develops continuously from the disordered (high temperature) phase. The orde -
Caltech PH 127
Physics 127a: Class Notes Lecture 16: Photons and Phonons The thermodynamics of electromagnetic radiation in a cavity or of the vibrational motion of a crystal can be treated in two equivalent ways: 1 1. an assembly of harmonic oscillators with quant -
Caltech PH 127
Physics 127a: Class Notes Lecture 12: Quantum Statistical Mechanics Basic Expressions The results we have found for classical statistical mechanics have corresponding ones when quantum mechanics is important. The average value of an observable is giv -
Caltech PH 127
Physics 127c: Statistical Mechanics Monte Carlo Methods Monte Carlo Integration Monte Carlo is most basically a way of doing integrals or sums. Consider first a one dimensional integral I= 0 1 f (x)dx = f (x) where we have added the suggestive mean -
Caltech PH 127
Physics 127c: Statistical Mechanics Superconductivity: Thermodynamics We could continue to evaluate the nite temperature thermodynamics by enumerating the states by hand. However it is more convenient to switch to the notation used in Homework 2. The -
Caltech PH 127
Physics 127c: Statistical Mechanics Vortex Lines Topological defects play a fundamental role in the properties of broken symmetry systems. In solids, for example, dislocations limit the maximum strain or stress that the solid can support, and initiat -
Caltech PH 127
Physics 127c: Statistical Mechanics Path Integral Methods The Trotter quantum Monte Carlo method leads easily into a discussion of path integral methods in statistical mechanics. Feynman introduced a \"sum over histories\" approach to quantum mechanics -
Caltech PH 127
Physics 127b: Statistical Mechanics Renormalization Group: Advanced Topics Revised February 9, 2004 First I discuss two extra details for the RNG procedure. The first is the notion of spin rescaling, an addition step in the RNG that was not needed i -
Caltech PH 127
Physics 127a: Class Notes Lecture 13: Ideal Quantum Gases Quantum States Consider an ideal gas in a box of sides L with periodic boundary conditions (x + L, y, z) = (x, y, z) etc. (1) For a single particle the wavefunction of the energy eigenstate s -
Caltech PH 127
Physics 127b: Statistical Mechanics Boltzmann Equation I: Scattering off xed impurities Scattering out v\' Scattering in v sc sc v v\' Figure 1: Scattering off a xed impurity: the scattering out and scattering in processes. The particle distrib -
Caltech PH 127
Physics 127b: Statistical Mechanics Langevin Equation To understand the Brownian motion more completely, we need to start from the basic physics, i.e. Newtons law of motion. The most direct way of implementing this is to recognize that there is a sto -
Caltech PH 127
Physics 127b: Statistical Mechanics Second Order Phase Transitions The Ising Ferromagnet Consider a simple d-dimensional lattice of N classical \"spins\" that can point up or down, si = 1. We suppose there is an interaction J between nearest neighbor s -
Caltech PH 127
Physics 127c: Statistical Mechanics Superconductivity: Microscopics For repulsive interactions the properties of an interacting Fermi system are not qualitatively different from the noninteracting system: the quantitative values of parameters are mod -
Caltech PH 127
Physics 127b: Statistical Mechanics Kinetic Theory Kinetic theory is the simplest approach that describes the dynamics of enormous numbers of particles. Often it is the only one where we can make progress in actually calculating numbers such as visco -
Caltech PH 127
Physics 127c: Statistical Mechanics Weakly Interacting Bose Gas Bogoliubov Theory The Hamiltonian is H = k + k b k b k + 1 2 + + u(q)bk+q bk -q bk bk . ~ kk q (1) We will look at the weakly interacting system at low temperatures. Already, without -
Caltech PH 127
Physics 127c: Statistical Mechanics Superconductivity: Ginzburg-Landau Theory Some of the key ideas for the Landau mean field description of phase transitions were developed in the context of superconductivity. It turns out that for conventional (low -
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Caltech PH 127
Physics 127c: Statistical Mechanics Kosterlitz-Thouless Transition The transition to superfluidity in thin films is an example of the Kosterlitz-Thouless transition, an exotic new type of phase transition driven by the unbinding of vortex pairs. Many -
Caltech PH 127
Physics 127a: Class Notes Lecture 8: Polymers The tools we have already allow us to study some interesting classical systems. One of these is the statistical mechanics of polymers-long molecules. Although this has been studied for decades, it has bec -
Caltech PH 127
Physics 127b: Statistical Mechanics Renormalization Group: General Case The steps in the renormalization group are 1. Eliminate degrees of freedom by a scale factor b so that N = N , bd (1) whilst preserving the free energy. This might be done by \"b -
Caltech PH 127
Physics 127b: Statistical Mechanics Lecture 2: Dense Gas and the Liquid State Mayer Cluster Expansion This is a method to calculate the higher order terms in the virial expansion. It introduces some general features of perturbation theory in many bod -
Caltech PH 127
Physics 127b: Statistical Mechanics Brownian Motion Brownian motion is the motion of a particle due to the buffeting by the molecules in a gas or liquid. The particle must be small enough that the effects of the discrete nature of matter are apparent -
Caltech PH 127
Physics 127c: Statistical Mechanics Bose Condensation in Trapped Alkali Gases Figure 1: Observation of Bose-Einstein condensation of Rubidium atoms by Anderson et al. Science 269, 198 (1995). The plots show the momentum distribution (measured by tur -
Caltech PH 127
Physics 127a: Class Notes Lecture 15: Statistical Mechanics of Superuidity Elementary excitations/quasiparticles In general, it is hard to list the energy eigenstates, needed to calculate the statistical mechanics of an interacting system. Indeed for -
Caltech PH 127
Physics 127b: Statistical Mechanics Renormalization Group: 1d Ising Model The ReNormalization Group (RNG) gives an understanding of scaling and universality, and provides various approximation schemes to calculate exponents etc. We will rst motivate -
Caltech PH 127
Physics 127c: Statistical Mechanics Weakly Interacting Fermi Gas Unlike the Boson case, there is usually no qualitative change in behavior going from the noninteracting to the weakly interacting Fermi gas for repulsive interactions. For example the e -
Caltech PH 127
Physics 127b: Statistical Mechanics Boltzmann Equation II: Binary Collisions Binary collisions in a classical gas Scattering out v\'1 Scattering in v\'2 v2 v1 R v2 v\'2 Center of Mass Frame V b R sc V\' v1 v\'1 sc V V\' Figure 1: Binary collisions i -
Caltech PH 127
Physics 127c: Statistical Mechanics Fermi Liquid Theory: Collective Modes Boltzmann Equation The quasiparticle energy including interactions p, = p + ~ 1 V f (p, p ; , )np , , p , (1) with p F + vF (p - pF ), acts as an effective Hamiltonian for t -
Caltech PH 127
Physics 127a: Class Notes Lecture 18: Gases with Internal Degrees of Freedom Partition Functions Most gases consist of atoms or molecules with internal degrees of freedom. For example atoms may have spin or orbital angular momenta, as well as electro -
Caltech PH 127
Physics 127a: Class Notes Lecture 9: Grand Canonical Ensemble This describes a system in contact with a reservoir with which it can exchange energy and particles. The equilibrium is characterized by the reservoir temperature T and chemical potential -
Caltech PH 127
Physics 127a: Class Notes Lecture 11: Entropy, Information and Maxwells Demon Gibbs Entropy For the canonical ensemble using Pn = Q1 eEn N A = kT ln QN = U T S it is easy to show S = k n (1a) (1b) Pn ln Pn (2) an expression directly relating the -
Caltech PH 127
Physics 127a: Class Notes Lecture 17: Ideal Fermi Gas General Properties For an ideal Fermi gas = kT ln Q = kT s ln 1 + zes (1) (2) N= s ns = s 1 z1 es + 1 where the sums run over single particle states s which are usually labelled by the wave -
Caltech PH 127
Ensemble Bath Fundamental Variables Thermodynamic Potentials Thermodynamic Relation Stationarity Principle Extensivity ( = E/N etc.) Euler Relation Applications Microcanonical Isolated E, N, V S (E, N, V ) , E (S, N, V ) Canonical Heat T, N, V Helm -
Caltech PH 127
Physics 127c: Statistical Mechanics Second Quantization Ladder Operators in the SHO It is useful to rst review the use of ladder operators in the simple harmonic oscillator. Here I present the bare bonesreview your favorite Quantum textbook for more -
Caltech PH 127
Physics 127a: Class Notes Lecture 1: The Fundamental Postulate After a general discussion of the scope of Statistical Mechanics I introduced the Fundamental Postulate of equal a priori probabilities: An isolated system in equilibrium is equally likel -
Caltech PH 127
Physics 127a: Class Notes Lecture 6: Canonical Ensemble Discussion and Derivation A system that can exchange energy via very weak contact with a temperature bath eventually comes to equilibrium. The canonical ensemble describes the statistical distri
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