DOCSLIB.ORG

Mixtures of Ultracold Gases: Fermi Sea and Bose-Einstein Condensate of Lithium Isotopes Florian Schreck

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Mixtures of Ultracold Gases: Fermi Sea and Bose-Einstein Condensate of Lithium Isotopes Florian Schreck Mixtures of ultracold gases: Fermi sea and Bose-Einstein condensate of Lithium isotopes Florian Schreck To cite this version: Florian Schreck. Mixtures of ultracold gases: Fermi sea and Bose-Einstein condensate of Lithium isotopes. Atomic Physics [physics.atom-ph]. Université Pierre et Marie Curie - Paris VI, 2002. English. tel-00001340v2 HAL Id: tel-00001340 https://tel.archives-ouvertes.fr/tel-00001340v2 Submitted on 17 May 2002 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. PHYSICS DEPARTMENT OF THE ECOLE´ NORMALE SUPERIEURE´ LABORATOIRE KASTLER BROSSEL DOCTORAL THESIS OF THE UNIVERSITE´ PARIS VI speciality : quantum physics presented by Florian SCHRECK to obtain the title Docteur de l’Universit´e Paris VI Subject of the thesis : MIXTURES OF ULTRACOLD GASES: FERMI SEA AND BOSE-EINSTEIN CONDENSATE OF LITHIUM ISOTOPES - MELANGES DE GAZ ULTRAFROIDS: MER DE FERMI ET CONDENSAT DE BOSE-EINSTEIN DES ISOTOPES DU LITHIUM Defended the 21 january 2002 in front of the jury consisting of : M. A. ASPECT Rapporteur M. R. GRIMM Rapporteur M. C. COHEN-TANNOUDJI Pr´esident M. G. SHLYAPNIKOV Examinateur M. R. COMBESCOT Examinateur M. C. SALOMON Directeur de th`ese To my friends. Contents Introduction 1 1 Theory 9 1.1Theoryofquantumgases.......................... 10 1.1.1 Propertiesofaclassicalgas.................... 10 1.1.2 Propertiesofafermionicgas.................... 11 1.1.3 The effect of interactions on the degenerate Fermi gas . 18 1.1.4 Pauliblocking............................ 20 1.1.5 Detection of a degenerate Fermi gas . 21 1.1.6 ThedegenerateBosegas...................... 23 1.1.7 The effect of interactions and the Gross-Pitaevskii equation . 24 1.1.8 BEC with attractive interactions . 25 1.1.9 One-dimensionaldegenerategases................. 26 1.1.10The1Dcondensate......................... 27 1.1.11Thebrightsoliton.......................... 28 1.1.12 The effects of Bose statistics on the thermal cloud . 30 1.1.13Boson-fermionmixtures...................... 31 1.1.14TheBCStransition......................... 34 1.2Evaporativecooling............................. 39 1.2.1 Sympatheticcooling........................ 42 1.2.2 The Limits of sympathetic cooling . 44 1.2.3 TheBoltzmannequation...................... 47 1.2.4 Simulation of rethermalization including Pauli blocking . 49 1.2.5 Simulation of sympathetic cooling . 50 1.3 Collisions . 52 1.3.1 Energydependenceofthecrosssection.............. 54 1.3.2 Meanfieldpotential........................ 56 i ii CONTENTS 1.3.3 Resonanceenhancedscattering.................. 56 2 The experimental setup 59 2.1Overviewoftheexperiment........................ 60 2.2Propertiesoflithium............................ 60 2.2.1 Basicproperties........................... 60 2.2.2 Confinementinamagnetictrap.................. 61 2.2.3 Elasticscatteringcrosssections.................. 63 2.3 Other strategies to a degenerate Fermi gas . 65 2.4Thevacuumsystem............................. 67 2.4.1 Theatomicbeamsource...................... 67 2.4.2 Ovenbake-outprocedure...................... 68 2.4.3 Themainchamber......................... 70 2.5TheZeemanslower............................. 70 2.6Thelasersystem.............................. 73 2.7Themagnetictrap............................. 78 2.7.1 Theoryofmagnetictrapping.................... 78 2.7.2 Designparametersofthemagnetictrap.............. 80 2.7.3 Realizationofthemagnetictrap.................. 82 2.8Theopticaldipoletrap........................... 89 2.8.1 Principle of an optical dipole trap . 89 2.8.2 Trapping of an alkali atom in a dipole trap . 90 2.8.3 Setupoftheopticaltrap...................... 91 2.9Theradiofrequencysystem........................ 94 2.10Detectionoftheatoms........................... 95 2.10.1 Principle of absorption imaging . 96 2.10.2Imagingoptics........................... 97 2.10.3Probebeampreparation...................... 97 2.10.4Twoisotopeimaging........................ 100 2.10.5Detectionparameters........................ 101 2.10.6Usingabsorptionimages...................... 103 2.11 Experiment control and data acquisition . 104 3 Experimental results 107 3.1Ontheroadtoevaporativecooling.................... 108 3.1.1 ThetwoisotopeMOT....................... 108 CONTENTS iii 3.1.2 Opticalpumping.......................... 111 3.1.3 Transfer and capture in the Ioffe trap . 112 3.1.4 First trials of evaporative cooling . 115 3.1.5 DopplercoolingintheIoffetrap.................. 116 3.2Measurements................................ 121 3.2.1 Measurement of the trap oscillation frequencies . 121 3.2.2 Temperaturemeasurements.................... 122 3.2.3 Measurements of other parameters . 129 3.3ExperimentsinthehigherHFstates................... 130 3.3.1 Evaporative cooling of 7Li..................... 130 3.3.2 Sympathetic cooling of 6Li by 7Li................. 135 3.3.3 DetectionofFermidegeneracy................... 137 3.3.4 DetectionofFermipressure.................... 138 3.3.5 Thermalizationmeasurement................... 141 3.4ExperimentsinthelowerHFstates.................... 143 3.4.1 Statetransfer............................ 143 3.4.2 Evaporative cooling of 7Li..................... 146 3.4.3 Sympathetic cooling of 7Li by 6Li................. 147 3.4.4 Astablelithiumcondensate.................... 150 3.4.5 TheFermisea............................ 152 3.5Lossrates.................................. 154 3.6Experimentswiththeopticaltrap..................... 157 3.6.1 Adiabatictransfer.......................... 157 7 3.6.2 Detection of the Li |F =1,mF =1 Feshbach resonance . 158 7 3.6.3 Condensation in the Li |F =1,mF =1 state..........159 3.6.4 A condensate with tunable scattering length . 162 3.6.5 Thebrightsoliton.......................... 162 Conclusion and outlook 167 Thanks 171 A Article : Multiple-frequency laser source 177 B Article : Simultaneous magneto-optical trapping ... 183 C Article : Sympathetic cooling ... 189 iv CONTENTS D Article : Quasipure Bose-Einstein Condensate ... 195 Introduction All particles, elementary particles as well as composite particles such as atoms, belong to one of two possible classes: they are either fermions or bosons. Which class a particle belongs to is determined by its spin. If the spin is an odd multiple of /2, the particle is a fermion. For even multiples it is a boson. Examples of fermions are electrons or 6Li atoms. 7Li and photons are bosons. The quantum properties of a particle are influenced by its bosonic or fermionic nature. For a system of identical particles the many particle wavefunction must be symmetric under the exchange of two particles for bosons and anti-symmetric under the exchange of two particles for fermions. A direct consequence of this (anti-)symmetrization postulate is that it is impossible for fermions to occupy the same quantum state. This is called the Pauli exclusion principle. No process can add a fermion to an already occupied state. The process is inhibited by Pauli blocking. For bosons it is favorable to occupy the same state and the more bosons that are already in this state, the higher the probability that another boson is transferred to it. This property is called Bose enhancement. 1 2 INTRODUCTION Bose enhancement leads to a phase transition at high phase-space densities, corre- sponding to low temperatures and/or high densities. When the phase-space density is increased past a certain critical value, the occupation of the ground-state rapidly becomes macroscopic. This effect, called Bose-Einstein condensation was predicted in 1924 by S. Bose and A. Einstein [1, 2, 3]. In cold atom experiments, the temperature of a harmonically trapped, weakly interacting gas is lowered until the phase transition occurs at a critical temperature TC 1 3 ω¯ N / TC = , (1) kB 1.202 whereω ¯ is the geometrical mean of the trap oscillation frequencies, kB the Boltzmann constant and N the number of trapped atoms. Below this temperature a macroscopic fraction of the atoms occupy the ground state of the trap [4]. The behavior of each of these atoms can be described by the same wavefunction. Thus a gas under these conditions is called a degenerate gas. The atoms in the ground state are called the Bose-Einstein condensate (in the following called condensate or BEC). Atoms in higher energy levels belong to what is termed the thermal cloud. For temperatures below the temperature corresponding to the harmonic oscillator energy level splitting Tω¯ = ω/k¯ B, even a classical gas would occupy the ground state macroscopically. But the critical temperature can be much higher than this temperature (Tc >Tω¯ ). Therefore the phenomenon is not a classical one. The first experiments involving Bose-Einstein condensates were performed with liq- uid 4He [5]. Below 2 K, the liquid becomes superfluid. Low-temperature supercon- ductivity can be explained similarly by the condensation of paired electrons [6]. In these liquid and solid systems the bosons interact strongly with their neighbors, mak- ing a theoretical description much more complicated than Einstein’s first work. In 1995 the group of C. Wieman and E. Cornell
Load more

Recommended publications
  • Competition Between Paramagnetism and Diamagnetism in Charged
    Competition between paramagnetism and diamagnetism in charged Fermi gases Xiaoling Jian, Jihong Qin, and Qiang Gu∗ Department of Physics, University of Science and Technology Beijing, Beijing 100083, China (Dated: November 10, 2018) The charged Fermi gas with a small Lande-factor g is expected to be diamagnetic, while that with a larger g could be paramagnetic. We calculate the critical value of the g-factor which separates the dia- and para-magnetic regions. In the weak-field limit, gc has the same value both at high and low temperatures, gc = 1/√12. Nevertheless, gc increases with the temperature reducing in finite magnetic fields. We also compare the gc value of Fermi gases with those of Boltzmann and Bose gases, supposing the particle has three Zeeman levels σ = 1, 0, and find that gc of Bose and Fermi gases is larger and smaller than that of Boltzmann gases,± respectively. PACS numbers: 05.30.Fk, 51.60.+a, 75.10.Lp, 75.20.-g I. INTRODUCTION mK and 10 µK, respectively. The ions can be regarded as charged Bose or Fermi gases. Once the quantum gas Magnetism of electron gases has been considerably has both the spin and charge degrees of freedom, there studied in condensed matter physics. In magnetic field, arises the competition between paramagnetism and dia- the magnetization of a free electron gas consists of two magnetism, as in electrons. Different from electrons, the independent parts. The spin magnetic moment of elec- g-factor for different magnetic ions is diverse, ranging trons results in the paramagnetic part (the Pauli para- from 0 to 2.
    [Show full text]
  • Solid State Physics 2 Lecture 5: Electron Liquid
    Physics 7450: Solid State Physics 2 Lecture 5: Electron liquid Leo Radzihovsky (Dated: 10 March, 2015) Abstract In these lectures, we will study itinerate electron liquid, namely metals. We will begin by re- viewing properties of noninteracting electron gas, developing its Greens functions, analyzing its thermodynamics, Pauli paramagnetism and Landau diamagnetism. We will recall how its thermo- dynamics is qualitatively distinct from that of a Boltzmann and Bose gases. As emphasized by Sommerfeld (1928), these qualitative di↵erence are due to the Pauli principle of electons’ fermionic statistics. We will then include e↵ects of Coulomb interaction, treating it in Hartree and Hartree- Fock approximation, computing the ground state energy and screening. We will then study itinerate Stoner ferromagnetism as well as various response functions, such as compressibility and conduc- tivity, and screening (Thomas-Fermi, Debye). We will then discuss Landau Fermi-liquid theory, which will allow us understand why despite strong electron-electron interactions, nevertheless much of the phenomenology of a Fermi gas extends to a Fermi liquid. We will conclude with discussion of electrons on the lattice, treated within the Hubbard and t-J models and will study transition to a Mott insulator and magnetism 1 I. INTRODUCTION A. Outline electron gas ground state and excitations • thermodynamics • Pauli paramagnetism • Landau diamagnetism • Hartree-Fock theory of interactions: ground state energy • Stoner ferromagnetic instability • response functions • Landau Fermi-liquid theory • electrons on the lattice: Hubbard and t-J models • Mott insulators and magnetism • B. Background In these lectures, we will study itinerate electron liquid, namely metals. In principle a fully quantum mechanical, strongly Coulomb-interacting description is required.
    [Show full text]
  • Lecture 24. Degenerate Fermi Gas (Ch
    Lecture 24. Degenerate Fermi Gas (Ch. 7) We will consider the gas of fermions in the degenerate regime, where the density n exceeds by far the quantum density nQ, or, in terms of energies, where the Fermi energy exceeds by far the temperature. We have seen that for such a gas μ is positive, and we’ll confine our attention to the limit in which μ is close to its T=0 value, the Fermi energy EF. ~ kBT μ/EF 1 1 kBT/EF occupancy T=0 (with respect to E ) F The most important degenerate Fermi gas is 1 the electron gas in metals and in white dwarf nε()(),, T= f ε T = stars. Another case is the neutron star, whose ε⎛ − μ⎞ exp⎜ ⎟ +1 density is so high that the neutron gas is ⎝kB T⎠ degenerate. Degenerate Fermi Gas in Metals empty states ε We consider the mobile electrons in the conduction EF conduction band which can participate in the charge transport. The band energy is measured from the bottom of the conduction 0 band. When the metal atoms are brought together, valence their outer electrons break away and can move freely band through the solid. In good metals with the concentration ~ 1 electron/ion, the density of electrons in the electron states electron states conduction band n ~ 1 electron per (0.2 nm)3 ~ 1029 in an isolated in metal electrons/m3 . atom The electrons are prevented from escaping from the metal by the net Coulomb attraction to the positive ions; the energy required for an electron to escape (the work function) is typically a few eV.
    [Show full text]
  • Chapter 13 Ideal Fermi
    Chapter 13 Ideal Fermi gas The properties of an ideal Fermi gas are strongly determined by the Pauli principle. We shall consider the limit: k T µ,βµ 1, B � � which defines the degenerate Fermi gas. In this limit, the quantum mechanical nature of the system becomes especially important, and the system has little to do with the classical ideal gas. Since this chapter is devoted to fermions, we shall omit in the following the subscript ( ) that we used for the fermionic statistical quantities in the previous chapter. − 13.1 Equation of state Consider a gas ofN non-interacting fermions, e.g., electrons, whose one-particle wave- functionsϕ r(�r) are plane-waves. In this case, a complete set of quantum numbersr is given, for instance, by the three cartesian components of the wave vector �k and thez spin projectionm s of an electron: r (k , k , k , m ). ≡ x y z s Spin-independent Hamiltonians. We will consider only spin independent Hamiltonian operator of the type ˆ 3 H= �k ck† ck + d r V(r)c r†cr , �k � where thefirst and the second terms are respectively the kinetic and th potential energy. The summation over the statesr (whenever it has to be performed) can then be reduced to the summation over states with different wavevectork(p=¯hk): ... (2s + 1) ..., ⇒ r � �k where the summation over the spin quantum numberm s = s, s+1, . , s has been taken into account by the prefactor (2s + 1). − − 159 160 CHAPTER 13. IDEAL FERMI GAS Wavefunctions in a box. We as- sume that the electrons are in a vol- ume defined by a cube with sidesL x, Ly,L z and volumeV=L xLyLz.
    [Show full text]
  • Electron-Electron Interactions(Pdf)
    Contents 2 Electron-electron interactions 1 2.1 Mean field theory (Hartree-Fock) ................ 3 2.1.1 Validity of Hartree-Fock theory .................. 6 2.1.2 Problem with Hartree-Fock theory ................ 9 2.2 Screening ..................................... 10 2.2.1 Elementary treatment ......................... 10 2.2.2 Kubo formula ............................... 15 2.2.3 Correlation functions .......................... 18 2.2.4 Dielectric constant ............................ 19 2.2.5 Lindhard function ............................ 21 2.2.6 Thomas-Fermi theory ......................... 24 2.2.7 Friedel oscillations ............................ 25 2.2.8 Plasmons ................................... 27 2.3 Fermi liquid theory ............................ 30 2.3.1 Particles and holes ............................ 31 2.3.2 Energy of quasiparticles. ....................... 36 2.3.3 Residual quasiparticle interactions ................ 38 2.3.4 Local energy of a quasiparticle ................... 42 2.3.5 Thermodynamic properties ..................... 44 2.3.6 Quasiparticle relaxation time and transport properties. 46 2.3.7 Effective mass m∗ of quasiparticles ................ 50 0 Reading: 1. Ch. 17, Ashcroft & Mermin 2. Chs. 5& 6, Kittel 3. For a more detailed discussion of Fermi liquid theory, see G. Baym and C. Pethick, Landau Fermi-Liquid Theory : Concepts and Ap- plications, Wiley 1991 2 Electron-electron interactions The electronic structure theory of metals, developed in the 1930’s by Bloch, Bethe, Wilson and others, assumes that electron-electron interac- tions can be neglected, and that solid-state physics consists of computing and filling the electronic bands based on knowldege of crystal symmetry and atomic valence. To a remarkably large extent, this works. In simple compounds, whether a system is an insulator or a metal can be deter- mined reliably by determining the band filling in a noninteracting cal- culation.
    [Show full text]
  • Schedule for Rest of Quarter • Review: 3D Free Electron Gas • Heat Capacity
    Lecture 16 Housekeeping: schedule for rest of quarter Review: 3D free electron gas Heat capacity of free electron gas Semi classical treatment of electric conductivity Review: 3D fee electron gas This model ignores the lattice and only treats the electrons in a metal. It assumes that electrons do not interact with each other except for pauli exclusion. 풌∙풓 Wavelike-electrons defined by their momentum eigenstate k: 휓풌(풓) = 푒 Treat electrons like quantum particle in a box: confined to a 3D box of dimensions L in every direction Periodic boundary conditions on wavefunction: 휓(푥 + 퐿, 푦, 푧) = 휓(푥, 푦, 푧) 휓(푥, 푦 + 퐿, 푧) = 휓(푥, 푦, 푧) 휓(푥, 푦, 푧 + 퐿) = 휓(푥, 푦, 푧) Permissible values of k of the form: 2휋 4휋 푘 = 0, ± , ± , … 푥 퐿 퐿 Free electron schrodinger’s eqn in 3D + boundary conditions gives energy eigenvalues: ℏ2 휕2 휕2 휕2 − ( + + ) 휓 (푟) = 휖 휓 (푟) 2푚 휕푥2 휕푦2 휕푧2 푘 푘 푘 ℏ2 ℏ2푘2 (푘2 + 푘2 + 푘2) = 휖 = 2푚 푥 푦 푧 푘 2푚 There are N electrons and we put them into available eigenstates following these rules: o Every state is defined by a unique quantized value of (푘푥, 푘푦, 푘푧) o Every state can hold one spin up and one spin down electron o Assume that single-electron solution above applies to collection of electrons (i.e. electrons do not interact and change the eigenstates) o Fill low energy states first. In 3D, this corresponds to filling up a sphere in k space, one 2 2 2 2 ‘shell’ at a time.
    [Show full text]
  • Chapter 6 Free Electron Fermi
    Chapter 6 Free Electron Fermi Gas Free electron model: • The valence electrons of the constituent atoms become conduction electrons and move about freely through the volume of the metal. • The simplest metals are the alkali metals– lithium, sodium, potassium, Na, cesium, and rubidium. • The classical theory had several conspicuous successes, notably the derivation of the form of Ohm’s law and the relation between the electrical and thermal conductivity. • The classical theory fails to explain the heat capacity and the magnetic susceptibility of the conduction electrons. M = B • Why the electrons in a metal can move so freely without much deflections? (a) A conduction electron is not deflected by ion cores arranged on a periodic lattice, because matter waves propagate freely in a periodic structure. (b) A conduction electron is scattered only infrequently by other conduction electrons. Pauli exclusion principle. Free Electron Fermi Gas: a gas of free electrons subject to the Pauli Principle ELECTRON GAS MODEL IN METALS Valence electrons form the electron gas eZa -e(Za-Z) -eZ Figure 1.1 (a) Schematic picture of an isolated atom (not to scale). (b) In a metal the nucleus and ion core retain their configuration in the free atom, but the valence electrons leave the atom to form the electron gas. 3.66A 0.98A Na : simple metal In a sea of conduction of electrons Core ~ occupy about 15% in total volume of crystal Classical Theory (Drude Model) Drude Model, 1900AD, after Thompson’s discovery of electrons in 1897 Based on the concept of kinetic theory of neutral dilute ideal gas Apply to the dense electrons in metals by the free electron gas picture Classical Statistical Mechanics: Boltzmann Maxwell Distribution The number of electrons per unit volume with velocity in the range du about u 3/2 2 fB(u) = n (m/ 2pkBT) exp (-mu /2kBT) Success: Failure: (1) The Ohm’s Law , (1) Heat capacity Cv~ 3/2 NKB the electrical conductivity The observed heat capacity is only 0.01, J = E , = n e2 / m, too small.
    [Show full text]
  • Fermi and Bose Gases
    Fermi and Bose gases Anders Malthe-Sørenssen 25. november 2013 1 1 Thermodynamic identities and Legendre transforms 2 Fermi-Dirac and Bose-Einstein distribution When we discussed the ideal gas we assumed that quantum effects were not important. This was implied when we introduced the term 1=N! for the partition function for the ideal gas, because this term only was valid in the limit when the number of states are many compared to the number of particles. Now, we will address the general case. We start from an individual particle. For a single particle, we address its orbitals: Its possible quantum states, each of which corresponds to a solution of the Schrodinger equation for a single particle. For many particles we will address how the “fill up" the various orbitals { where the underlying assump- tion is that we can treat a many-particle system as a system with the same energy levels as for a single particular system, that is as a system with no interactions between the states of the various particles. However, the number of particles that can occupy the same state depends on the type of particle: For Fermions (1/2 spin) Only 0 or 1 fermion can be in a particular state. For Bosons (integer spin) Any number of bosons can be in a particular state. We will now apply a Gibbs ensemble approach to study this system. This means that we will consider a system where µ, V; T is constant. We can use the Gibbs sum formalism to address the behavior of this system.
    [Show full text]
  • One-Dimensional Multicomponent Fermions with Δ-Function Interaction in Strong- and Weak-Coupling Limits: Κ-Component Fermi Gas
    PHYSICAL REVIEW A 85, 033633 (2012) One-dimensional multicomponent fermions with δ-function interaction in strong- and weak-coupling limits: κ-component Fermi gas Xi-Wen Guan,1 Zhong-Qi Ma,2 and Brendan Wilson1 1Department of Theoretical Physics, Research School of Physics and Engineering, Australian National University, Canberra ACT 0200, Australia 2Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China (Received 23 January 2012; published 26 March 2012) We derive the first few terms of the asymptotic expansion of the Fredholm equations for one-dimensional κ- component fermions with repulsive and attractive δ-function interaction in strong- and weak-coupling regimes. We thus obtain a highly accurate result for the ground-state energy of a multicomponent Fermi gas with polarization for these regimes. This result provides a unified description of the ground-state properties of the Fermi gas with higher spin symmetries. However, in contrast to the two-component Fermi gas, there does not exist a mapping that can unify the two sets of Fredholm equations as the interacting strength vanishes. Moreover, we find that the local pair-correlation functions shed light on the quantum statistic effects of the κ-component interacting fermions. For the balanced spin case with repulsive interaction, the ground-state energy obtained confirms Yang and You’s result [Chin. Phys. Lett. 28, 020503 (2011)] that the energy per particle as κ →∞is the same as for spinless Bosons. DOI: 10.1103/PhysRevA.85.033633 PACS number(s): 03.75.Ss, 03.75.Hh, 02.30.Ik, 34.10.+x I. INTRODUCTION Although the model was solved long ago by Sutherland [4], solutions of the Fredholm equations for the model are far from The recently established experimental control over the being thoroughly investigated except for the two-component effective spin-spin interaction between atoms is opening up Fermi gas [16].
    [Show full text]
  • Ideal Quantum Gases from Statistical Physics Using Mathematica © James J
    Ideal Quantum Gases from Statistical Physics using Mathematica © James J. Kelly, 1996-2002 The indistinguishability of identical particles has profound effects at low temperatures and/or high density where quantum mechanical wave packets overlap appreciably. The occupation representation is used to study the statistical mechanics and thermodynamics of ideal quantum gases satisfying Fermi- Dirac or Bose-Einstein statistics. This notebook concentrates on formal and conceptual developments, liberally quoting more technical results obtained in the auxiliary notebooks occupy.nb, which presents numerical methods for relating the chemical potential to density, and fermi.nb and bose.nb, which study the mathematical properties of special functions defined for Ferm-Dirac and Bose-Einstein systems. The auxiliary notebooks also contain a large number of exercises. Indistinguishability In classical mechanics identical particles remain distinguishable because it is possible, at least in principle, to label them according to their trajectories. Once the initial position and momentum is determined for each particle with the infinite precision available to classical mechanics, the swarm of classical phase points moves along trajectories which also can in principle be determined with absolute certainty. Hence, the identity of the particle at any classical phase point is connected by deterministic relationships to any initial condition and need not ever be confused with the label attached to another trajectory. Therefore, in classical mechanics even identical particles are distinguishable, in principle, even though we must admit that it is virtually impossible in practice to integrate the equations of motion for a many-body system with sufficient accuracy. Conversely, in quantum mechanics identical particles are absolutely indistinguishable from one another.
    [Show full text]
  • 8.044 Lecture Notes Chapter 9: Quantum Ideal Gases
    8.044 Lecture Notes Chapter 9: Quantum Ideal Gases Lecturer: McGreevy 9.1 Range of validity of classical ideal gas . 9-2 9.2 Quantum systems with many indistinguishable particles . 9-4 9.3 Statistical mechanics of N non-interacting (indistinguishable) bosons or fermions 9-9 9.4 Classical ideal gas limit of quantum ideal gas . 9-13 2 9.5 Ultra-high temperature (kBT mc and kBT µ) limit . 9-15 9.6 Fermions at low temperatures . 9-18 9.7 Bosons at low temperatures . 9-33 Reading: Baierlein, Chapters 8 and 9. 9-1 9.1 Range of validity of classical ideal gas For a classical ideal gas, we derived the partition function N 3=2 Z1 V 2πmkBT Z = ;Z1 = 3 = V 2 ; N! λth h h where the length scale λth ≡ is determined by the particle mass and the temperature. 2πmkB T When does this break down? 1. If `idealness' fails, i.e. if interactions become important. This is interesting and impor- tant but out of bounds for 8.044. We'll still assume non-interacting particles. 2. If `classicalness' fails. Even with no interactions, at low enough temperatures, or high enough densities, quantum effects modify Z. An argument that classicalness must fail comes by thinking harder about λth, the \thermal de Broglie wavelength". Why is it called that? Recall from 8.04 that the de Broglie wavelength h for a particle with momentum p is λdB = p . For a classical ideal gas, we know the RMS momentum from equipartition ~p2 3 p h i = k T =) p ≡ ph~p2i = 3mk T: 2m 2 B RMS B We could have defined ~ h h λth ≡ = p ; pRMS 3mkBT there's no significance to the numerical prefactor of p1 instead of p1 .
    [Show full text]
  • Density of States
    Density of States Franz Utermohlen February 13, 2018 Contents 1 Preface 1 2 Summation definition 2 2.1 Examples of Eqs. 2 and 3 for a fermionic system . 2 2.2 Example: Ideal Fermi gas . 3 3 Derivative definition 4 3.1 Example: Ideal Fermi gas . 4 4 Useful expressions 6 1 Preface In these notes, the expressions Vd denotes the volume of a d-dimensional sphere of radius R and Ωd−1 denotes the solid angle a d-dimensional sphere (or equivalently, the surface area of a d-dimensional sphere of unit radius).1 These expressions are given by d=2 d=2 π d 2π Vd = d R ; Ωd−1 = d : (1) Γ(1 + 2 ) Γ( 2 ) For example, for d = 1; 2; and 3, these are d = 1 : V1 = 2R; Ω0 = 2 ; 2 d = 2 : V2 = πR ; Ω1 = 2π ; 4 d = 3 : V = πR3 ; Ω = 4π : 3 3 2 1The d − 1 subscript refers to the fact that the fact that the surface area of a d-dimensional sphere is (d − 1)-dimensional. It's also useful to use d − 1 as the subscript, since the surface area of a d-dimensional d−1 sphere of radius R is given by Sd−1 = Ωd−1R . 1 2 Summation definition Intensive quantities A can generally be expressed in the form 1 X A = a( ) ; (2) V i d i where Vd is the d-dimensional volume of the system, the sum is over all possible single- particle states i, and i is the energy of the single-particle state i.
    [Show full text]