DOCSLIB.ORG

The Spin-Statistics Theorem and Identical Particle Distribution Functions

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Spin-Statistics Theorem and Identical Particle Distribution Functions Elegant Connections in Physics The Spin-Statistics Theorem and Identical Particle Distribution Functions by Dwight E. Neuenschwander In the restaurants of some countries, the maître d’ will seat your party at a table already occupied by strangers if the table has sufficient empty chairs. In other cultures, new diners expect to be seated at empty tables even when occupied tables are surrounded by sur- plus seats. Elementary particles, and composites made of them, fall analogously into two categories: bosons and fermions. Bosons readily share the same state, analogous to the first group of diners. Fermions prefer solitude, like the second group of diners. This article offers a simple argument for their respective distribution functions that describe how a system of identical particles populates the states available to them. Since the most stable configuration of the sequence of closed-shell electron num- of gases, it may be sufficient to model a a system typically finds it in the state of bers 2, 8, 18,…is equivalent to the rule that hydrogen atom as a solid point mass mov- lowest energy, why don’t all the electrons no state can hold more than one electron. ing with momentum p and located at posi- in an atom reside in the 1s orbital? If they To implement this rule he had to invent an tion r. Here the atom’s microstate consists did, there would be no chemistry, and thus extra quantum number. It was soon iden- of six numbers, the momentum and posi- no biology and no life—nor would metals tified with electron spin, when Samuel tion coordinates in phase space. conduct electricity, and white dwarf stars Goudsmit and George Uhlenbeck pro- Between macrostates and single-par- would collapse into black holes. The uni- posed it in the autumn of 1925.[3] Pauli’s ticle microstates are states of a system of verse would be very different indeed, and spin-statistics theorem of 1940 general- N particles. For N = 2, with one particle we would not be around to appreciate it! ized the argument about spin and statistics. in the single-particle state ψa and the other The Pauli exclusion principle formally ar- [4] Its flip side, that bosons can coalesce, in microstate ψb, how simple it would be ticulates the hypothesis that electrons, as finds dramatic illustration in superfluidity mathematically if the two-particle state fermions, do not all collapse into an atom’s and superconductivity, variations of Bose- was merely the product of the two one- 1s orbital; only two of them can live there, Einstein condensation. particle wave functions, ψ = ψaψb. This and then only with opposite spins, to not The word “state” carries three contexts works fine for systems of distinguishable be in the same space-spin state. in this discussion: macroscopic thermo- particles, such as the deuteron—a proton The exclusion principle grew out of the dynamic states, and microscopic states in and a neutron bound together.[5] But the realization that the discrete quantum states two varieties—single-particle and mul- plot thickens when the particles are indis- for electrons could not, by themselves, tiparticle microstates. The macroscopic tinguishable. If ψ = ψaψb was the whole explain the periodic table of the elements. state of a bottle of hydrogen gas in thermal story for identical particles, the universe The crucial clue towards a resolution came equilibrium is characterized by observ- would be rather sterile. Simplicity is beau- in 1924, when Edmund Stoner published ables such as pressure, temperature, and tiful, but complexity can be essential. a paper noting a correlation between the volume. Microscopically, each atom has a Statistical mechanics aims to under- number of electron states of alkali metals set of available internal states, such as the stand macroscopic states in terms of mi- placed in a magnetic field and the number atomic orbitals with spin, ψ that de- crostates. The next section reviews how nlml ms of closed-shell electrons in noble gases. scribe the electron relative to the nucleus. this game is played. [1,2] From this observation Pauli realized For other purposes, as in the kinetic theory Fall 2013 Radiations 27 Elegant Connections in Physics StatisticalStatistical Mechanics Mechanics Review Review Indistinguishable Particles Particles in QuantumQuantum Mechanics Mechanics Consider a system of N molecules, each in one of the many possible microstates available to it (“molecule” here is a generic Now let’s enlarge our perspective and speak of the states of term which could also denote a nucleus or a neutrino, depending an N-particle system. As before, each individual particle will be on the problem). Label the various single-particle microstates as in a single-particle microstate of energy Ei. A possible N-particle state a, state b, and so on. Let ni denote the number of state is specified by a list molecules in state i. If I were to reach into a box of these molecules and pick one at random, the probability Pi of selecting = (n1, n2, n3, …), (7) one in state i is ni /N. Since every molecule must be in one microstate or another, !where . Each labels one possible state of the N- particle system. The energy of state is the sum ! ! (1) ! = ! ! ! !! . ! ! (8) ! ! ! ! If E! denote= s a= molecule’s1. energy when in state i, and U the total i !! = ! !! !! energy carried by all the molecules (the internal energy of In this context the partition function of Eq. (4) becomes the sum thermodynamics), then over the allowed : ! ; (2) = . (9) !!!! !! ! !!!! ! ! ! ! ! ! !! = ! ! ! ! ! = denotes ! ! the= average! ! !molecular= ! ! energy. For example, suppose N = 2 and each particle can be in one of A primary responsibility of statistical mechanics expresses Pi three one-particle microstates. The possible states of the two- in! terms of Ei and the ambient thermal energy kT in which the particle system are: molecules find themselves, when in thermal equilibrium with their surroundings at temperature T. Boltzmann’s constant k = (1,1,0) (1, 0, 1) = (0, 1, 1) 1.381×10─23 J/K forms a conversion factor between temperature 1 = 2 = 3 and energy. A derivation of Pi as a function of T can be = (2, 0, 0) = (0, 2, 0) = (0, 0, 2) approached at least two ways: through the “method of most probable distribution,”[6] or with phase space arguments based with4 respective5 energies 6 on Liouville’s theorem.[7] One finds = E1 + E2 = E1 + E3 = E2 + E3 (3) ! ! ! ! ! ! = 2E =! 2E = 2E! !! /!" 1 2 3. !! = ! ! , where the normalization factor Z, determined by Eq. (1), is the ! ! From! Eq. (!9!) the partition! function reads “partition function,” (4) !!!! !!!! !!!! !!!/!" ! ! ! = ! + ! + ⋯ + ! In ! terms= !of Z, the. computation of U can be expressed succinctly. = ! ! ! ! ! From Eqs. (2) and (3), !!! !!! !!! !!! !!!! ! ! + ! ! + ⋯ + ! = , (10) ! ! ! , (5) ! ! ! ! ! ! ! ! ! where! ! + ! ! + ! ! + ! + ! + ! ! ! !! ! ! = ! ! !! ! where β = 1/kT. Thanks to a nice property of the exponential, . (11) !!!! ! = Eq. (5) becomes !With= N ! held fixed some do not occur, as in our illustration ! !" !" where N = 2 excludes (0,0,0) and (1,1,1). To work with fixed N ! ! !" ! , means the set of all allowed! must be known before ZN can be (6) ! evaluated. That is feasible when N = 2, but statistical mechanics ! 23 ! = −! !" ln !. deals with systems that contain on the order of 10 particles. The partition function offers a window from the macroscopic However, if N were not fixed, then each ni in every could world of thermometers and pressure gauges into the microscopic range over 0, 1, 2,…, Nmax. To determine Nmax for identical world of atomic structures and interactions. For example, particles, the spin-statistics theorem steps in. Such generality! consider a system of N identical switches, each one either off makes Z summable. (with energy E1 = 0) or on (with energy E2 = ε), so that Z = 1 + “Identical” here means that no method exists in principle to −εβ e . To see if a real macroscopic system can be well modeled distinguish one particle from another. This raises concern about by such a set of switches, we could compare the measured heat possibly double-counting microscopic states. If particles 1 and 2 capacity C = dU/dT to the prediction of it calculated from Z. in a two-particle system exchange places in their one-particle microstates, should that swapped configuration be counted as the same, or as distinct, from the original? According to the 28 Radiations Fall 2013 Elegant Connections in Physics rules of quantum mechanics,[8] one sums over both possibilities, values s, s − 1,…, −s. The quantum number s can be one of the 2 which interfere with one another through the cross terms in |ψ| possible values Particles with integer s are = |ψ + ψ |2. The N = 2 wave function therefore has original exchanged bosons, and particles! that! carry! half-odd-integer s are fermions. two options for including one particle in state a and an identical ! ! ! Pauli’s 19400 proof, , 1, of, 2the, ,s3pin, …-statistics theorem[4] makes one in state b: consistent with relativistic quantum field theory the assertion that identical fermions use the minus sign in Eq. (20), and ψ (1,2) = c ψ (1)ψ (2) + c ψ (2)ψ (1). (12) ab orig a b exch a b identical bosons use the plus sign. The Pauli exclusion principle emerges as a consequence: If two fermions are identical, they If these exhaust the possibilities for arranging the two particles, cannot be in the same state because, by Eq. (20), ψ (1,2) = 0. 2 2 aa then |corig| + |cexch| = 1. But |corig| = |cexch| because the particles But for identical bosons, the total wave function is enhanced: iδ 2 2 2 are identical. Thus cexch = corig e , where i = −1 with δ a real |ψaa(1,2)| = 2|ψa| .
Load more

Recommended publications
  • Identical Particles
    8.06 Spring 2016 Lecture Notes 4. Identical particles Aram Harrow Last updated: May 19, 2016 Contents 1 Fermions and Bosons 1 1.1 Introduction and two-particle systems .......................... 1 1.2 N particles ......................................... 3 1.3 Non-interacting particles .................................. 5 1.4 Non-zero temperature ................................... 7 1.5 Composite particles .................................... 7 1.6 Emergence of distinguishability .............................. 9 2 Degenerate Fermi gas 10 2.1 Electrons in a box ..................................... 10 2.2 White dwarves ....................................... 12 2.3 Electrons in a periodic potential ............................. 16 3 Charged particles in a magnetic field 21 3.1 The Pauli Hamiltonian ................................... 21 3.2 Landau levels ........................................ 23 3.3 The de Haas-van Alphen effect .............................. 24 3.4 Integer Quantum Hall Effect ............................... 27 3.5 Aharonov-Bohm Effect ................................... 33 1 Fermions and Bosons 1.1 Introduction and two-particle systems Previously we have discussed multiple-particle systems using the tensor-product formalism (cf. Section 1.2 of Chapter 3 of these notes). But this applies only to distinguishable particles. In reality, all known particles are indistinguishable. In the coming lectures, we will explore the mathematical and physical consequences of this. First, consider classical many-particle systems. If a single particle has state described by position and momentum (~r; p~), then the state of N distinguishable particles can be written as (~r1; p~1; ~r2; p~2;:::; ~rN ; p~N ). The notation (·; ·;:::; ·) denotes an ordered list, in which different posi­ tions have different meanings; e.g. in general (~r1; p~1; ~r2; p~2)6 = (~r2; p~2; ~r1; p~1). 1 To describe indistinguishable particles, we can use set notation.
    [Show full text]
  • Magnetic Materials I
    5 Magnetic materials I Magnetic materials I ● Diamagnetism ● Paramagnetism Diamagnetism - susceptibility All materials can be classified in terms of their magnetic behavior falling into one of several categories depending on their bulk magnetic susceptibility χ. without spin M⃗ 1 χ= in general the susceptibility is a position dependent tensor M⃗ (⃗r )= ⃗r ×J⃗ (⃗r ) H⃗ 2 ] In some materials the magnetization is m / not a linear function of field strength. In A [ such cases the differential susceptibility M is introduced: d M⃗ χ = d d H⃗ We usually talk about isothermal χ susceptibility: ∂ M⃗ χ = T ( ∂ ⃗ ) H T Theoreticians define magnetization as: ∂ F⃗ H[A/m] M=− F=U−TS - Helmholtz free energy [4] ( ∂ H⃗ ) T N dU =T dS− p dV +∑ μi dni i=1 N N dF =T dS − p dV +∑ μi dni−T dS−S dT =−S dT − p dV +∑ μi dni i=1 i=1 Diamagnetism - susceptibility It is customary to define susceptibility in relation to volume, mass or mole: M⃗ (M⃗ /ρ) m3 (M⃗ /mol) m3 χ= [dimensionless] , χρ= , χ = H⃗ H⃗ [ kg ] mol H⃗ [ mol ] 1emu=1×10−3 A⋅m2 The general classification of materials according to their magnetic properties μ<1 <0 diamagnetic* χ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ B=μ H =μr μ0 H B=μo (H +M ) μ>1 >0 paramagnetic** ⃗ ⃗ ⃗ χ μr μ0 H =μo( H + M ) → μ r=1+χ μ≫1 χ≫1 ferromagnetic*** *dia /daɪə mæ ɡˈ n ɛ t ɪ k/ -Greek: “from, through, across” - repelled by magnets. We have from L2: 1 2 the force is directed antiparallel to the gradient of B2 F = V ∇ B i.e.
    [Show full text]
  • 5.1 Two-Particle Systems
    5.1 Two-Particle Systems We encountered a two-particle system in dealing with the addition of angular momentum. Let's treat such systems in a more formal way. The w.f. for a two-particle system must depend on the spatial coordinates of both particles as @Ψ well as t: Ψ(r1; r2; t), satisfying i~ @t = HΨ, ~2 2 ~2 2 where H = + V (r1; r2; t), −2m1r1 − 2m2r2 and d3r d3r Ψ(r ; r ; t) 2 = 1. 1 2 j 1 2 j R Iff V is independent of time, then we can separate the time and spatial variables, obtaining Ψ(r1; r2; t) = (r1; r2) exp( iEt=~), − where E is the total energy of the system. Let us now make a very fundamental assumption: that each particle occupies a one-particle e.s. [Note that this is often a poor approximation for the true many-body w.f.] The joint e.f. can then be written as the product of two one-particle e.f.'s: (r1; r2) = a(r1) b(r2). Suppose furthermore that the two particles are indistinguishable. Then, the above w.f. is not really adequate since you can't actually tell whether it's particle 1 in state a or particle 2. This indeterminacy is correctly reflected if we replace the above w.f. by (r ; r ) = a(r ) (r ) (r ) a(r ). 1 2 1 b 2 b 1 2 The `plus-or-minus' sign reflects that there are two distinct ways to accomplish this. Thus we are naturally led to consider two kinds of identical particles, which we have come to call `bosons' (+) and `fermions' ( ).
    [Show full text]
  • Identical Particles in Classical Physics One Can Distinguish Between
    Identical particles In classical physics one can distinguish between identical particles in such a way as to leave the dynamics unaltered. Therefore, the exchange of particles of identical particles leads to di®erent con¯gurations. In quantum mechanics, i.e., in nature at the microscopic levels identical particles are indistinguishable. A proton that arrives from a supernova explosion is the same as the proton in your glass of water.1 Messiah and Greenberg2 state the principle of indistinguishability as \states that di®er only by a permutation of identical particles cannot be distinguished by any obser- vation whatsoever." If we denote the operator which interchanges particles i and j by the permutation operator Pij we have ^ y ^ hªjOjªi = hªjPij OPijjÃi ^ which implies that Pij commutes with an arbitrary observable O. In particular, it commutes with the Hamiltonian. All operators are permutation invariant. Law of nature: The wave function of a collection of identical half-odd-integer spin particles is completely antisymmetric while that of integer spin particles or bosons is completely symmetric3. De¯ning the permutation operator4 by Pij ª(1; 2; ¢ ¢ ¢ ; i; ¢ ¢ ¢ ; j; ¢ ¢ ¢ N) = ª(1; 2; ¢ ¢ ¢ ; j; ¢ ¢ ¢ ; i; ¢ ¢ ¢ N) we have Pij ª(1; 2; ¢ ¢ ¢ ; i; ¢ ¢ ¢ ; j; ¢ ¢ ¢ N) = § ª(1; 2; ¢ ¢ ¢ ; i; ¢ ¢ ¢ ; j; ¢ ¢ ¢ N) where the upper sign refers to bosons and the lower to fermions. Emphasize that the symmetry requirements only apply to identical particles. For the hydrogen molecule if I and II represent the coordinates and spins of the two protons and 1 and 2 of the two electrons we have ª(I;II; 1; 2) = ¡ ª(II;I; 1; 2) = ¡ ª(I;II; 2; 1) = ª(II;I; 2; 1) 1On the other hand, the elements on earth came from some star/supernova in the distant past.
    [Show full text]
  • The Conventionality of Parastatistics
    The Conventionality of Parastatistics David John Baker Hans Halvorson Noel Swanson∗ March 6, 2014 Abstract Nature seems to be such that we can describe it accurately with quantum theories of bosons and fermions alone, without resort to parastatistics. This has been seen as a deep mystery: paraparticles make perfect physical sense, so why don't we see them in nature? We consider one potential answer: every paraparticle theory is physically equivalent to some theory of bosons or fermions, making the absence of paraparticles in our theories a matter of convention rather than a mysterious empirical discovery. We argue that this equivalence thesis holds in all physically admissible quantum field theories falling under the domain of the rigorous Doplicher-Haag-Roberts approach to superselection rules. Inadmissible parastatistical theories are ruled out by a locality- inspired principle we call Charge Recombination. Contents 1 Introduction 2 2 Paraparticles in Quantum Theory 6 ∗This work is fully collaborative. Authors are listed in alphabetical order. 1 3 Theoretical Equivalence 11 3.1 Field systems in AQFT . 13 3.2 Equivalence of field systems . 17 4 A Brief History of the Equivalence Thesis 20 4.1 The Green Decomposition . 20 4.2 Klein Transformations . 21 4.3 The Argument of Dr¨uhl,Haag, and Roberts . 24 4.4 The Doplicher-Roberts Reconstruction Theorem . 26 5 Sharpening the Thesis 29 6 Discussion 36 6.1 Interpretations of QM . 44 6.2 Structuralism and Haecceities . 46 6.3 Paraquark Theories . 48 1 Introduction Our most fundamental theories of matter provide a highly accurate description of subatomic particles and their behavior.
    [Show full text]
  • Talk M.Alouani
    From atoms to solids to nanostructures Mebarek Alouani IPCMS, 23 rue du Loess, UMR 7504, Strasbourg, France European Summer Campus 2012: Physics at the nanoscale, Strasbourg, France, July 01–07, 2012 1/107 Course content I. Magnetic moment and magnetic field II. No magnetism in classical mechanics III. Where does magnetism come from? IV. Crystal field, superexchange, double exchange V. Free electron model: Spontaneous magnetization VI. The local spin density approximation of the DFT VI. Beyond the DFT: LDA+U VII. Spin-orbit effects: Magnetic anisotropy, XMCD VIII. Bibliography European Summer Campus 2012: Physics at the nanoscale, Strasbourg, France, July 01–07, 2012 2/107 Course content I. Magnetic moment and magnetic field II. No magnetism in classical mechanics III. Where does magnetism come from? IV. Crystal field, superexchange, double exchange V. Free electron model: Spontaneous magnetization VI. The local spin density approximation of the DFT VI. Beyond the DFT: LDA+U VII. Spin-orbit effects: Magnetic anisotropy, XMCD VIII. Bibliography European Summer Campus 2012: Physics at the nanoscale, Strasbourg, France, July 01–07, 2012 3/107 Magnetic moments In classical electrodynamics, the vector potential, in the magnetic dipole approximation, is given by: μ μ × rˆ Ar()= 0 4π r2 where the magnetic moment μ is defined as: 1 μ =×Id('rl ) 2 ∫ It is shown that the magnetic moment is proportional to the angular momentum μ = γ L (γ the gyromagnetic factor) The projection along the quantification axis z gives (/2Bohrmaμ = eh m gneton) μzB= −gmμ Be European Summer Campus 2012: Physics at the nanoscale, Strasbourg, France, July 01–07, 2012 4/107 Magnetization and Field The magnetization M is the magnetic moment per unit volume In free space B = μ0 H because there is no net magnetization.
    [Show full text]
  • Spontaneous Symmetry Breaking and Mass Generation As Built-In Phenomena in Logarithmic Nonlinear Quantum Theory
    Vol. 42 (2011) ACTA PHYSICA POLONICA B No 2 SPONTANEOUS SYMMETRY BREAKING AND MASS GENERATION AS BUILT-IN PHENOMENA IN LOGARITHMIC NONLINEAR QUANTUM THEORY Konstantin G. Zloshchastiev Department of Physics and Center for Theoretical Physics University of the Witwatersrand Johannesburg, 2050, South Africa (Received September 29, 2010; revised version received November 3, 2010; final version received December 7, 2010) Our primary task is to demonstrate that the logarithmic nonlinearity in the quantum wave equation can cause the spontaneous symmetry break- ing and mass generation phenomena on its own, at least in principle. To achieve this goal, we view the physical vacuum as a kind of the funda- mental Bose–Einstein condensate embedded into the fictitious Euclidean space. The relation of such description to that of the physical (relativis- tic) observer is established via the fluid/gravity correspondence map, the related issues, such as the induced gravity and scalar field, relativistic pos- tulates, Mach’s principle and cosmology, are discussed. For estimate the values of the generated masses of the otherwise massless particles such as the photon, we propose few simple models which take into account small vacuum fluctuations. It turns out that the photon’s mass can be naturally expressed in terms of the elementary electrical charge and the extensive length parameter of the nonlinearity. Finally, we outline the topological properties of the logarithmic theory and corresponding solitonic solutions. DOI:10.5506/APhysPolB.42.261 PACS numbers: 11.15.Ex, 11.30.Qc, 04.60.Bc, 03.65.Pm 1. Introduction Current observational data in astrophysics are probing a regime of de- partures from classical relativity with sensitivities that are relevant for the study of the quantum-gravity problem [1,2].
    [Show full text]
  • International Centre for Theoretical Physics
    INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS OUTLINE OF A NONLINEAR, RELATIVISTIC QUANTUM MECHANICS OF EXTENDED PARTICLES Eckehard W. Mielke INTERNATIONAL ATOMIC ENERGY AGENCY UNITED NATIONS EDUCATIONAL, SCIENTIFIC AND CULTURAL ORGANIZATION 1981 MIRAMARE-TRIESTE IC/81/9 International Atomic Energy Agency and United Nations Educational Scientific and Cultural Organization 3TERKATIOHAL CENTRE FOR THEORETICAL PHTSICS OUTLINE OF A NONLINEAR, KELATIVXSTTC QUANTUM HECHAHICS OF EXTENDED PARTICLES* Eckehard W. Mielke International Centre for Theoretical Physics, Trieste, Italy. MIRAHABE - TRIESTE January 198l • Submitted for publication. ABSTRACT I. INTRODUCTION AMD MOTIVATION A quantum theory of intrinsically extended particles similar to de Broglie's In recent years the conviction has apparently increased among most theory of the Double Solution is proposed, A rational notion of the particle's physicists that all fundamental interactions between particles - including extension is enthroned by realizing its internal structure via soliton-type gravity - can be comprehended by appropriate extensions of local gauge theories. solutions of nonlinear, relativistic wave equations. These droplet-type waves The principle of gauge invarisince was first formulated in have a quasi-objective character except for certain boundary conditions vhich 1923 by Hermann Weyl[l57] with the aim to give a proper connection between may be subject to stochastic fluctuations. More precisely, this assumption Dirac's relativistic quantum theory [37] of the electron and the Maxwell- amounts to a probabilistic description of the center of a soliton such that it Lorentz's theory of electromagnetism. Later Yang and Mills [167] as well as would follow the conventional quantum-mechanical formalism in the limit of zero Sharp were able to extend the latter theory to a nonabelian gauge particle radius.
    [Show full text]
  • Quantum Statistics (Quantenstatistik)
    Chapter 7 Quantum Statistics (Quantenstatistik) 7.1 Density of states of massive particles Our goal: discover what quantum effects bring into the statistical description of systems. Formulate conditions under which QM-specific effects are not essential and classical descriptions can be used. Consider a single particle in a box with infinitely high walls U ε3 ε2 Figure 7.1: Levels of a single particle in a 3D box. At the ε1 box walls the potential is infinite and the wave-function is L zero. A one-particle Hamiltonian has only the momentum p^2 ~2 H^ = = − r2 (7.1) 2m 2m The infinite potential at the box boundaries imposes the zero-wave-function boundary condition. Hence the eigenfunctions of the Hamiltonian are k = sin(kxx) sin(kyy) sin(kzz) (7.2) ~ 2π + k = ~n; ~n = (n1; n2; n3); ni 2 Z (7.3) L ( ) ~ ~ 2π 2 ~p = ~~k; ϵ = k2 = ~n2;V = L3 (7.4) 2m 2m L 4π2~2 ) ϵ = ~n2 (7.5) 2mV 2=3 Let us introduce spin s (it takes into account possible degeneracy of the energy levels) and change the summation over ~n to an integration over ϵ X X X X X X Z X Z 1 ··· = ··· = · · · ∼ d3~n ··· = dn4πn2 ::: (7.6) 0 ν s ~k s ~n s s 45 46 CHAPTER 7. QUANTUM STATISTICS (QUANTENSTATISTIK) (2mϵ)1=2 V 1=3 1 1 V m3=2 n = V 1=3 ; dn = 2mdϵ, 4πn2dn = p ϵ1=2dϵ (7.7) 2π~ 2π~ 2 (2mϵ)1=2 2π2~3 X = gs = 2s + 1 spin-degeneracy (7.8) s Z X g V m3=2 1 ··· = ps dϵϵ1=2 ::: (7.9) 2~3 ν 2π 0 Number of states in a given energy range is higher at high energies than at low energies.
    [Show full text]
  • Lecture 7: Ensembles
    Matthew Schwartz Statistical Mechanics, Spring 2019 Lecture 7: Ensembles 1 Introduction In statistical mechanics, we study the possible microstates of a system. We never know exactly which microstate the system is in. Nor do we care. We are interested only in the behavior of a system based on the possible microstates it could be, that share some macroscopic proporty (like volume V ; energy E, or number of particles N). The possible microstates a system could be in are known as the ensemble of states for a system. There are dierent kinds of ensembles. So far, we have been counting microstates with a xed number of particles N and a xed total energy E. We dened as the total number microstates for a system. That is (E; V ; N) = 1 (1) microstatesk withsaXmeN ;V ;E Then S = kBln is the entropy, and all other thermodynamic quantities follow from S. For an isolated system with N xed and E xed the ensemble is known as the microcanonical 1 @S ensemble. In the microcanonical ensemble, the temperature is a derived quantity, with T = @E . So far, we have only been using the microcanonical ensemble. 1 3 N For example, a gas of identical monatomic particles has (E; V ; N) V NE 2 . From this N! we computed the entropy S = kBln which at large N reduces to the Sackur-Tetrode formula. 1 @S 3 NkB 3 The temperature is T = @E = 2 E so that E = 2 NkBT . Also in the microcanonical ensemble we observed that the number of states for which the energy of one degree of freedom is xed to "i is (E "i) "i/k T (E " ).
    [Show full text]
  • INDISTINGUISHABLE PARTICLES: Where Does the N! Come From? Is It
    INDISTINGUISHABLE PARTICLES: Where does the N! come from? Is it correct? How to deal with non-ideal gases? What does this have to do with Keq? The origin of the N! For distinguishable non-interacting particles, it is easy to see that the total partition function Q is the product of the one-particle partition functions q. However, to understand entropy of mixing (and why opening a valve between two containers of nitrogen at the same P,T does not increase the entropy) Gibbs found that for indistinguishable non-interacting particles (like nitrogen molecules) one wants instead A very rough argument about why you might expect an N! from combinatorics is given in Atkins and many other undergraduate P. Chem. textbooks. However, this argument is not rigorous, in fact it is easy to see problems with it. Here we outline a more detailed derivation (following Kittel & Kroemer's Thermal Physics ). Fundamentally, the N! comes from the Pauli condition on the wavefunction (that the wavefunction must be either symmetric or antisymmetric for permutation of identical particles) but this is pretty complicated to explain; for an attempt to derive the partition function directly from Slater determinants see Pathria's Statistical Mechanics . Note that the Gibbs N! factor which fixes the entropy also shifts the Helmholtz free energy F = U - TS: F = -kT ln Q = -NkT ln q + kT ln(N!) ~ -NkT ln q + NkT ln N - NkT (using Stirling's approximation for the last part - only valid for N>>1). It does not affect the energy U. Recall that the probability that a system which can exchange particles and heat with a large reservoir will be found with N particles in the quantum state s(N) is where Z is the grand canonical partition function the sum is over all possible quantum states of the system, with all possible numbers of particles.
    [Show full text]
  • Ei Springer Contents
    Wolfgang Nolting • Anupuru Ramakanth Quantum Theory of Magnetism ei Springer Contents 1 Basic Facts 1 1.1 Macroscopic Maxwell Equations 1 1.2 Magnetic Moment and Magnetization 7 1.3 Susceptibility 13 1.4 Classification of Magnetic Materials 15 1.4.1 Diamagnetism 15 1.4.2 Paramagnetism 15 1.4.3 Collective Magnetism 17 1.5 Elements of Thertnodynamics 19 1.6 Problems 22 2 Atomic Magnetism 25 2.1 Hund's Rules 25 2.1.1 Russell—Saunders (LS-) Coupling 25 2.1.2 Hund's Rules for LS Coupling 27 2.2 Dirac Equation 28 2.3 Electron Spin 34 2.4 Spin—Orhit Coupling 40 2.5 Wigner—Eckart Theorem 45 2.5.1 Rotation 45 2.5.2 Rotation Operator 47 2.5.3 Angular Momentum 48 2.5.4 Rotation Matrices 50 2.5.5 Tensor Operators 52 2.5.6 Wigner—Eckart Theorem 53 2.5.7 Examples of Application 55 2.6 Electron in an External Magnetic Field 56 2.7 Nuclear Quadrupole Field 62 2.8 Hyperfine Field 67 2.9 Magnetic Ham ilton i an of the Atomic Electron 72 2.10 Many-Electron Systems 74 2.10.1 Coulomh Interaction 74 ix x Contents 2.10.2 Spin—Orbit Coupling 75 2.10.3 Further Couplings 76 2.11 Problems 81 3 Diamagnetism 85 3.1 Bohr—van Leeuwen Theorem 85 3.2 Larmor Diamagnetism (Insulators) 87 3.3 The Sommerfeld Model of a Metal 90 3.3.1 Properties of the Model 91 3.3.2 Sommerfeld Expansion 99 3.4 Landau Diamagnetism (Metals) 104 3.4.1 Free Electrons in Magnetic Field (Landau Levels) 104 3.4.2 Grand Canonical Potential of the Conduction Electrons 109 3.4.3 Susceptibility of the Conduction Electrons 117 3.5 The de Haas—Van Alphen Effect 121 3.5.1 Oscillations in the Magnetic
    [Show full text]