Statistical Mechanics Notes. Ryan D. Reece

Transcription

1 Statistical Mechanics Notes Ryan D. Reece August 11, 2006

2 Contents 1 Thermodynamics State Variables Inexact Differentials Work and Heat Entropy Thermodynamic Potentials Maxwell Relations Heat Capacities Ideas of Statistical Mechanics The Goal Boltzmann s Hypothesis Entropy and Probability The Procedure Canonical Ensemble Definition The Bridge to Thermodynamics Example: A Particle in a Box Maxwell-Boltzmann Statistics Density of States The Maxwell-Boltzmann Distribution Example: Particles in a Box Black Body Radiation Molecular Beams Cavity Radiation

3 5.3 The Planck Distribution Thermodynamics of Multiple Particles Definition Chemical Potential The Gibbs-Duhem Relation Identical Particles Exchange Symmetry of Quantum States Spin Over Counting Grand Canonical Ensemble Definition The Bridge to Thermodynamics Fermions Bosons Non-relativistic Cold Electron Gas Example: White Dwarf Stars Relativistic Cold Electron Gas Example: Collapse to a Neutron Star Questions 45 2

4 Chapter 1 Thermodynamics 1.1 State Variables Given a function f(x, y), its differential is df = f f dx + dy (1.1) x y The function is called analytic if its mixed partials are equal. 2 f x y = 2 f y x (1.2) In this case, the differential of f, equation 1.1, is called an exact differential. Because the mixed partials are equal, Green s Theorem says that any closed line integral of f in x-y-space is zero. ( 2 f 0 ) ( ) x y 2 f f f dx dy = dx + y x x y dy (1.3) 0 = df (1.4) Suppose we break some closed path into two paths α and β which begin at point Λ and end at point Γ (see Figure 1.1). If f is analytic, the path integrals of f along the two paths α and β must be equal because the integral along the closed path, equal to the difference of the the two path integrals, is zero. df = df df = 0 (1.5) α 3 β

5 Figure 1.1: Path Independence of State Variables This means the value of f, is a function of the endpoints alone and independent of the path. f = df = f(λ, Γ) f( x, y) (1.6) Concerning thermodynamics, if x and y are macroscopic variables that we can measure (temperature, pressure, volume, etc.), and we can find some variable f with an exact differential in terms of x and y, then f is called a state variable. State variables have defined values when a system has come to equilibrium, regardless of what processes (paths) brought the system there. The macroscopic variables x and y, are themselves state variables. An equation of state can be derived relating state variables independent of path, for example the equation of state for an ideal gas 1 is p V = N k T (1.7) where pressure (p), volume (V ), number of particles (N), and temperature (T ) are all state variables and k is Boltzmann s constant. 1 We prove that this is the equation of state for an ideal gas, using statistical mechanics in Section 3.3 4

6 1.2 Inexact Differentials Suppose we had derived the following differential equation for f and f is not analytic. δf = A(x, y)dx + B(x, y)dy (1.8) A y B x (1.9) Then δf is an inexact differential, denoted by δ. It may be possible to multiply δf by some expression called an integrating factor to make an exact differential. Consider the following example. We see that δf is indeed inexact. δf = (2x + yx 2 )dx + x 3 dy (1.10) 2 f y x = x2, 2 f x y = 3x2 (1.11) Choosing an integrating factor of e xy gives ds is exact. ds e xy δf = e xy (2x + yx 2 )dx + e xy x 3 dy (1.12) 2 S x y = 2 S y x = (x3 y + 3x 2 )e xy (1.13) We will use this method when we define entropy in Section Work and Heat Naïve application conservation of energy requires that the change in the energy of a system, U, is the work done on it, W. But this ignores thermal processes as a means to take and give energy. Introducing a quantity for the heat added to a system, Q, accounts for these thermal processes. This is the First Law of Thermodynamics. U = W + Q (1.14) 5

7 Considering the traditional example of a cylinder of gas with a piston such that we may vary the volume, V, and pressure, p, of the gas, we note that a differential change in volume, dv, does the following differential work on the system δw = p dv (1.15) For some less ideal system, this can be generalized. If some system has X j variables that when varied do work on th system, then there are p j generalized pressures that satisfy δw = j p j dx j (1.16) But, equation 1.15 is an inexact differential meaning that the work is dependent on the path through P -V -space and not just on the endpoints. That is, it is also dependent on how much heat comes in or out of the system. This implies that the differential heat is also an inexact differential. 1.4 Entropy du = δw + δq (1.17) Our goal is to make equation 1.17 an exact differential in terms of state variables. We have seen that in our traditional system of a cylinder of gas, we can write δw in terms of state variables p and V. Now we need to find a way of writing δq in terms of state variables. δq = du + p dv (1.18) From the equation of state for an ideal gas (equation 1.7) we have P = NkT V (1.19) P dv = NkT dv (1.20) V Then knowing that the energy of an ideal gas 2 is U = 3 NkT (1.21) 2 2 We prove the equation of state and equation 1.21 for an ideal gas in Section 3.3 6

8 Plugging eqautions 1.20 and 1.22 into 1.18 gives Indeed, δq is inexact. du = 3 Nk dt (1.22) 2 δq = 3 NkT Nk dt + dv (1.23) 2 V 2 Q V T = V 2 Q T V = T ( ) 3 2 Nk = 0 (1.24) ( ) NkT = Nk (1.25) V V If equation 1.25 had not had that factor of T before taking the derivative, then both mixed partials would equal zero, and the differential would be exact. This motivates us to pick an integrating factor of T 1. We define the entropy, S, to have the differential It is an exact differential ds δq (1.26) T = 3 Nk Nk dt + dv (1.27) 2 T V 2 S V T = 2 S T V = 0 (1.28) meaning S is a state variable. With entropy defined, we can now write du in terms of state variables. du = T ds p dv (1.29) 1.5 Thermodynamic Potentials Sometimes it is convenient to create other state variables from our existing ones, depending on which variables are held constant in a process. The thermodynamic potentials, other than the total energy (U), are enthalpy (H), Helmholtz Free Energy (F ), and Gibbs Free Energy (G). They are created by adding pv and/or subtracting T S from U (see Figure 1.2). 7

9 Figure 1.2: Mnemonic for the thermodynamic potentials We can derive the exact differentials of these potentials, simply by differentiating their definitions and plugging in our exact differential for du. dh = T ds p dv + p dv + V dp = T ds + V dp (1.30) df dg = T ds p dv T ds S dt = S dt p dv (1.31) = T ds p dv + p dv + V dp T ds S dt (1.32) = V dp S dt (1.33) 8

10 1.6 Maxwell Relations By recognizing that each of the thermodynamic potentials has an exact differential, we can derive the Maxwell Relations by equating the mixed partials. ( ) ( ) 2 U V S = 2 U T p = (1.34) S V V S S T ( ) ( ) 2 H S p = 2 H V T p S = (1.35) S p p S ( ) ( ) 2 F T V = 2 F p S V T = (1.36) T V V T ( ) ( ) 2 G p T = 2 G S V T p = (1.37) p T 1.7 Heat Capacities Heat capacity is defined as the amount of differential heat that must be supplied to a system to change the system s temperature, per the differential change in temperature. C δq (1.38) dt Because gases expand significantly upon heating, their heat capacity depends on whether pressure or volume is held fixed. For most solids and liquids the differences between C V and C p is less pronounced. From we see that when V is held fixed ( ) U C V = T T δq = du + p dv (1.39) V = T ( ) S T V To express C p, it is convenient to note the differential enthalpy. p (1.40) dh = δq + V dp (1.41) Therefore, when pressure is held fixed, dh = δq, and we have that ( ) ( ) H S C p = = T T T p p (1.42) 9

11 Chapter 2 Ideas of Statistical Mechanics 2.1 The Goal The goal of statistical mechanics is to form a theoretical foundation for the empirical laws of thermodynamics based on more fundamental physics (i.e. quantum mechanics). 2.2 Boltzmann s Hypothesis Boltzmann s insightful hypothesis to define entropy in terms of the number of microstates. S = k B ln Ω (2.1) One way to see why this is a convenient definition is to recognize that this makes entropy extensive, that is, it is proportional to the number of particles or subsystems in our system, N. Other extensive state variables are U, V, and trivially N. If subsystem 1 has Ω 1 microstates and entropy S 1, and subsystem 2 has Ω 2 microstates and entropy S 2, then the combined system has Ω T = Ω 1 Ω 2 different microstates. The entropy of the combined system is S = k B ln Ω T (2.2) = k B ln(ω 1 Ω 2 ) (2.3) = k B ln Ω 1 + k B ln Ω 2 (2.4) = S 1 + S 2 (2.5) 10

12 All extensive quantities are proportional to each other, with the constant of proportionality equal to the corresponding partial derivative. Therefore, instead of writing U in differential form as in equation 1.29, we may write ( ) ( ) U U U = V + S (2.6) V S S V = p V + T S (2.7) 2.3 Entropy and Probability Suppose a huge number of systems, N, are in thermal contact. N 1 of which are in mircrostate 1, N 2 in microstate 2, etc. The number of ways to make this arrangement is N! Ω = (2.8) N 1! N 2! N 3!... The entropy of the entire ensemble of systems is S N = k ln Ω = k ln N! k i ln N i! (2.9) Using Stirling s approximation that ln N N ln N N for large N gives S N = k(n ln N N i N i ln N i + i N i ) (2.10) Noting that N = i N i simplifies this to ( S N = k N i ln N ) N i ln N i (2.11) i i = k N i (ln N ln N i ) (2.12) i = k i N i ln N i N (2.13) = k N i N i N ln N i N (2.14) The probability of finding any given system in the i-th microstate is P i = N i /N. Therefore, S N = k N P i ln P i (2.15) i 11

13 The entropy per system, S = S N /N, is S = k i P i ln P i (2.16) 2.4 The Procedure Determine the spectrum of energy levels and their degeneracies using quantum mechanics or some other means. Calculate the partition function. Use the bridge to thermodynamics (F = k B T ln Z in Canonical Ensemble or Φ = k B T ln Ξ in the Grand Canonical Ensemble) to calculate any desired thermodynamic variable. 12

14 Chapter 3 Canonical Ensemble 3.1 Definition Suppose system A is in contact with a reservoir (heat bath) R and has come to equilibrium with a temperature T. The total number of microstates in the combined system is Ω = Ω A (U A ) Ω R (U R ) (3.1) where U A and U R denote the energy in system A and the reservoir respectively. We choose to write quantities in terms of the energy in system A. E i is the energy of the i-th energy level of system A. If the energy levels of system A have degeneracy g i, the total number of microstates with U A = E i is Ω(E i ) = g i Ω R (U T E i ) (3.2) where U T = U A + U R is the total energy of the combined system. From equation 1.29, we have that du = T ds pdv (3.3) ( ) S = 1 (3.4) U T Combining this with the statistical definition of entropy, S = k ln Ω, gives ( ) 1 ln kt = ΩR (3.5) U R 13 V V

15 When energy is added to the reservoir, the temperature does not change. This is the property that defines a heat bath. Therefore, we can integrate equation 3.5 to get C U R kt = ln Ω R (3.6) where C is some integration constant. The total number of states is Ω R = e C e U R/kT Ω(E i ) = g i e C e (U T E i )/kt (3.7) (3.8) Ω total = j Ω(E j ) (3.9) = e C e U T /kt j g j e E j/kt (3.10) where j indexes the energy levels of A. The probability that A has an energy E i is P (E i ) = Ω(E i) Ω total (3.11) = = g i e C e ( U T E i )/kt e C e U T /kt j g j e E j/kt (3.12) g i e E i/kt j g j e E j/kt (3.13) = g i e E i/kt Z where we defined the partition function, Z, as (3.14) Z = j g j e E j/kt (3.15) We could have chosen to index the microstates instead of the energy levels, in which case the probability of the system being in the i-th microstate would simply be P i = ee i /kt Z (3.16) 14

16 Figure 3.1: Boltzmann Distribution for nondegenerate energy levels and the partition function would be Z = j e E j/kt (3.17) This probability distribution is called the Boltzmann Distribution. The partition function serves as a normalization factor for the distribution. For nondegenerate states and finite T, lower energy levels are always more likely. As T approaches infinity, all states are equally likely. A large number of distinguishable systems in thermal contact can be treated as a heat bath for any one of the individual systems in the group. This is called the Canonical Ensemble, in which case the the distribution of energies should fit the Boltzmann Distribution. 15

17 3.2 The Bridge to Thermodynamics Plugging in the Boltzmann Distribution to equation 2.16 gives S = k i = k i = k i = 1 T = 1 T i i P i ln P i (3.18) ( e E i/kt e E i ) /kt ln Z Z ( ) Ei Z kt + ln Z e E i/kt e E i/kt Z e E i/kt Z e 1 E i/kt E i + k ln Z Z i (3.19) (3.20) (3.21) E i + k ln Z (3.22) Recognizing that the first term in equation 3.22 contains an express for the average energy of any one of the systems U = E = i equation 3.22 can be rewritten as P i E i = i e E i/kt Z E i (3.23) S = U T + k ln Z (3.24) We can now calculate the Helmholtz Free Energy, F, from equation 3.24 U T S = k T ln Z (3.25) F = k T ln Z (3.26) It is now apparent that the partition function is more than just a normalization factor for the probability distribution of energies. If one can calculate the partition function of some system, then F can be calculated using equation From F one can determine relations between the thermodynamic state variables. 16

18 Another way to see how Z is directly related to state variables, specifically U, is the following. For convenience, define β 1/kT. Noting that U = E (3.27) = 1 E i P i (3.28) Z i = 1 E i e βe i (3.29) Z i Z = i e βe i (3.30) Z β = i E i e βe i (3.31) means that equation 3.29 can be written as U = 1 Z Z β = ln Z β (3.32) (3.33) 3.3 Example: A Particle in a Box Imagine our system is a single particle confined to a one-dimensional box. This system is replicated a huge number of times and the systems are all in thermal contact making a Canonical Ensemble. The time independent Schödinger wave equation in one dimension is 2 h2 ψ(x) + V (x)ψ(x) = Eψ(x) (3.34) 2m 2 x For a particle in a square potential of width L and with infinite barriers, the energy eigenvalues are E n = h2 π 2 2mL 2 n2, n 1, 2, 3, 4... (3.35) 17

19 For convenience, we define the constant ε to be ε h2 π 2 2mL, E 2 n = ε n 2 (3.36) Now that we know the spectrum of energies of the system, we can calculate 1 the partition funciton. Z = e En/kT (3.37) = = = n=1 n=1 e εn2 /kt If we define the de Broglie wavelength 2 as 2π h 2 λ D mkt then we can write the partition function as (3.38) e εn2 /kt dn (3.39) 0 ε π (3.40) kt 2 L2 mkt 2π h 2 (3.41) (3.42) Z = L λ D (3.43) We can now calculate the Helmholtz Free Energy, and other macroscopic thermodynamic quantities. ( ) L2 mkt F = kt ln(z) = kt ln 2π h 2 (3.44) = k ( ) L 2 2 T ln mkt 2π h 2 (3.45) 1 In equation 3.40, we use the integral e x2 = π According to de Broglie, a particle in an eigenstate of momentum with momentum p has a corresponding wavelength λ = h/p. A thermal distribution of particles therefore has a distribution of wavelengths. λ D represents the average wavelength. 18

20 The entropy is ( ) ( ) F F S = = T V T L = k ( ) L 2 2 ln mkt 2π h 2 + k ( 2π h 2 ) ( ) L 2 2 T mk L 2 mkt 2π h 2 = k [ ( ) ] L 2 mkt ln 2 2π h Using equation 1.40, the heat capacity is ( ) ( ) S S C V = T = T T V T L = T k ( 2π h 2 )( ) L 2 mk 2 L 2 mkt 2π h 2 = k 2 (3.46) (3.47) (3.48) (3.49) (3.50) (3.51) What if the particle is in a three-dimensional box? Solving this basic quantum mechanical problem gives the spectrum of energies. E n1,n 2,n 3 = ε (n n n 2 3) (3.52) where ε is the same as we defined in equation The partition function is Z = e ε(n2 1 +n2 2 +n2 3 )/kt (3.53) = n 1 =1 n 2 =1 n 3 =1 ( ) ( ) ( ) e εn2 1 /kt e εn2 2 /kt e εn2 3 /kt n 1 =1 n 2 =1 n 3 =1 (3.54) ( ) L 2 3/2 mkt = 2π h 2, V = L 3 (3.55) ( ) 3/2 mkt = V 2π h 2 = V (3.56) The Helmholtz Free Energy is λ 3 D F = kt ln Z (3.57) = kt ln V + 3 ( ) mkt 2 ln 2π h 2 (3.58) 19

21 The pressure is ( ) F p = = kt V T V We have derived an equation of state for this system! (3.59) p V = k T (3.60) Imagine our box contains N particles instead of one. If the particles do not interact, this defines an ideal gas, then the pressure of the system is just N times the pressure due to a single particle, giving equation 3.59 an extra factor of N. This gives the equation of state for an ideal gas. p V = N k T (3.61) Back to the case of only one particle, the entropy of the system is ( ) [ F S = = k ln V + 3 ( ) mkt T 2 ln 2π h ] (3.62) 2 The heat capacity is ( ) S C V = T T V = T 3k ( 2π h 2 )( L 2 mk ) 2 L 2 mkt 2π h 2 V (3.63) (3.64) = 3k (3.65) 2 We see that the heat capacity of a particle in a three-dimensional box is three times that for a a particle in a one-dimensional box. This makes sense because each dimension of translational motion independently adds to the capacity of the system to store energy in thermal motion. Each translation degree of freedom gives a k term to the heat capacity for a system with 2 constant volume. Because the heat capacity is not dependent on temperature, we can easily integrate it from zero to the some temperature to find out how much thermal energy is stored in the system. T ( ) U T dt = C T V dt (3.66) 0 V 0 U = C V T (3.67) U = 3 2 kt (3.68) 20

22 Chapter 4 Maxwell-Boltzmann Statistics 4.1 Density of States We know that a free particle in a three-dimensional box has energy eigenstates: ( ) ( ) ( ) n1 πx n2 πy n3 πz ψ n1,n 2,n 3 (x, y, z) = C sin sin sin (4.1) with energies L x = C sin (k x x) sin (k y y) sin (k z z) (4.2) ( E n1,n 2,n 3 = h2 π 2 n 2 1 2m L 2 x + n2 2 L 2 y L y = h2 ( k 2 2m x + ky 2 + kz) 2 ) + n2 3 L 2 z L z (4.3) (4.4) These states correspond to lattice points in a space, k-space, whose axises are k x, k y, and k z. A vector in this space is called a wave vector with a magnitude, or wavenumber 1 k = kx ˆx + k y ŷ + k z ẑ (4.5) k = k 2 x + k 2 y + k 2 z (4.6) 1 We will now use k B to denote Boltzmann s constant in order not to confuse it with the magnitude of a wave vector k. 21

23 Figure 4.1: k-space In polar coordinates, θ = arccos ( ) kz, φ = arctan k ( ky k x ) (4.7) volume element in k-space = k 2 sin θ dk dθ dφ (4.8) density of points in k-space = L xl y L z π 3 = V π 3 (4.9) Therefore, the number of states in a differential volume is dn = V π 3 k2 sin θ dk dθ dφ (4.10) Integrating dθ and dφ gives the number of states in a differential shell of thickness dk. dn = V π π 2 2 π 3 k2 dk dφ sin θ dθ 0 0 (4.11) = V π 3 k2 dk π 2 1 = V 2π 2 k2 dk (4.12) 22

24 Figure 4.2: Density of points in k-space We define that density of states, D(k), such that dn = V 2π 2 k2 dk = D(k) dk (4.13) In this discussion, we have ignored the possibility of there being multiple spin states at a single point in k-space. For future reference, we define D 0 (k) to be the density of states ignoring any spin degeneracy. D 0 (k) V 2π 2 k 2 (4.14) In general, if there are g s spin states with the same energy at the same point in k-space, the density of states is D(k) = g s D 0 (k) (4.15) We can now write down the partition function, using k to integrate over all states. Z = 0 D(k) e E k/k B T dk (4.16) 4.2 The Maxwell-Boltzmann Distribution If we allow multiple particles to occupy the same state, the probability the i-th state is occupied if we toss N particles among the states at random 23

25 is 2 P ocp = 1 P (N i = 0) (4.17) = 1 (1 P i ) N (4.18) N(N 1) = NP i Pi (4.19) 2 where N i denotes the number of particles occupying the i-th state and P i is the probability that a particle gets tossed into the i-th states. If the density of states is sufficiently large, and the temperature high, then P i 1, and we can ignore higher terms in P i. In this regime, it does not even matter if we allow multiple particles to occupy the same state, be them fermions or bosons, because the probability of multiple occupancies is so low. P ocp N P i (4.20) Using this approximation, the number of states occupied in a differential k-shell is dn ocp = P ocp dn (4.21) = N e E k/k B T D(k) dk (4.22) Z = f MB (k) dk (4.23) where we have defined the Maxwell-Boltzmann Distribution, f MB (k), to be f MB (k) = N D(k) e E k /k B T Z (4.24) 4.3 Example: Particles in a Box Back to our example of particles in a box, we can plug in the partition function from equation 3.56 and the density of states from equation 4.13 into equation f MB (k) dk = N e E k/k B T ( ) ( λ 3 D V V 2π 2 k2 ) dk (4.25) = Nλ3 D 2π 2 k 2 e E k/k B T dk (4.26) 2 We have used that the probability of r successful tosses out of N tosses, with a probability p for the success of each toss, is given by the binomial distribution. P (x = r) = ( N r ) p r (1 p) (N r) 24

26 Let us say we want to calculate the average velocity squared of the particles. We first have to calculate k 2 k 2 f 0 MB (k) dk = (4.27) f 0 MB (k) dk k 4 e E k/k B T dk 0 = (4.28) e 0 E k/k BT dk k 4 e h2 k 2 /2mk B T dk 0 = (4.29) e 0 h2 k 2 /2mk BT dk Substitute 3 x 2 h2 k 2 2mk B T In the non-relativistic limit, k 2 = 2mk BT h 2 = 2mk B T h 2 x 4 e x2 dx 0 e 0 x2 dx ( ) ( ) 3 π 4 π 8 (4.30) (4.31) = 3mk BT h 2 (4.32) v 2 = p2 m 2 = 3k BT m For each particle, the average energy is = h2 m 2 k 2 (4.33) (4.34) U = 1 2 m v 2 = 3 2 k BT (4.35) which is the same result we got for the ideal gas using a different method in equation We used the integrals 0 x 4 e x2 dx = 3 π 8 and 0 e x2 dx = π 4. 25

27 Chapter 5 Black Body Radiation 5.1 Molecular Beams 5.2 Cavity Radiation 5.3 The Planck Distribution 26

28 Chapter 6 Thermodynamics of Multiple Particles 6.1 Definition Now we will consider systems consisting of distinguishable types of particles, where all the particles in a given type are identical. In addition to the exchange of work and heat between subsystems, now the number of particles of one type can be transfered to another type. One example of this is a cylinder of gas with a semipermeable membrane dividing the cylinder. There are two types of gas molecules that are distinguishable by which side of the membrane they are on. Another example is a single container of gas holding A type molecules, B type molecules, and a third AB type that is the product of some kind of combination of A and B type molecules. 6.2 Chemical Potential Introducing the ability for particles to change type within a system gives a new way to exchange energy. We define the chemical potential, µ, to be the internal energy gained by a system when the average number of particles, N, of some type increases by one. This means we must add a term to equation 2.7. U = p V + T S + µ N (6.1) 27

29 Therefore, µ = ( ) U N V,S and we can add this term to equation 1.29, giving (6.2) du = T ds p dv + µ dn (6.3) This also leads to a noteworthy equation for the Gibbs Free Energy: G U T S + p V = µ N (6.4) ( ) G µ = = G (6.5) N N Flipping around equation 6.3 gives a relation for µ that is often convenient. T,P ds = 1 T du + p T dv µ T ( ) S µ = T N U,V dn (6.6) (6.7) 6.3 The Gibbs-Duhem Relation Differentiating equation 6.1 directly gives du = T ds + S dt p dv V dp + µ dn + N dµ (6.8) but then using equation 6.3, we cancel some terms, leaving 0 = S dt V dp + N dµ (6.9) This is called the Gibbs-Duhem Relation. We can generalize this for a system with various work variables X i and particle types indexed by j. 0 = S dt i X i dp i + j N j dµ j (6.10) 28

30 Chapter 7 Identical Particles 7.1 Exchange Symmetry of Quantum States Suppose we have a system of two identical particles. Because they are identical, the square of the wavefunction that describes the state should be unchanged if two particles are exchanged. X denotes the exchange operator. ψ(x 1, x 2 ) 2 = X ψ(x 1, x 2 ) 2 (7.1) = ψ(x 2, x 1 ) 2 (7.2) This implies that the exchanged wavefunctions differ only by a complex factor whose modulus squared is one. ψ(x 1, x 2 ) = e iθ ψ(x 2, x 1 ) (7.3) If we exchange the particles twice, we have done nothing. wavefunctions should be identical. Therefore the ψ(x 1, x 2 ) = X 2 ψ(x 1, x 2 ) (7.4) e i2θ = 1 (7.5) which only has solutions θ =... 2π, π, 0, π, 2π... e iθ = ±1. Particles with an exchange factor of +1, i.e. symmetric under exchange, are called bosons. Those with an exchange factor of 1, i.e. antisymmetric under exchange, are called fermions. 29

31 Larger numbers of identical bosons or fermions must still obey the appropriate exchange symmetry. The ways to construct a wavefunction for several identical particles with the correct symmetry are the following: ψ Bose (x 1, x 2, x 3...) = ψ i (x 1 ) ψ j (x 2 ) ψ k (x 3 )... + permutations of i, j, k... (7.6) ψ i (x 1 ) ψ i (x 2 ) ψ i (x 3 ) ψ j (x 1 ) ψ j (x 2 ) ψ j (x 3 )... ψ Fermi (x 1, x 2, x 3...) = ψ k (x 1 ) ψ k (x 2 ) ψ k (x 3 ) (7.7).... The determinate in equation 7.7 is called a Slater Determinate and gives the correct signs for the total wavefunction to be asymmetric under exchange. Consider a system of two fermions, for example two electrons bound to an atom. The Slater Determinate gives ψ Fermi (x 1, x 2 ) = ψ i (x 1 ) ψ j (x 2 ) ψ i (x 2 ) ψ j (x 1 ) (7.8) Immediately one can see that if the two particles are in the same state (i = j), ψ Fermi 0. This explains the Pauli Exclusion Principle, that no two fermions can exist in the same quantum state. 7.2 Spin There is a very important result from relativistic quantum mechanics called the Spin-Statistics Theorem, which is beyond the scope of this work. It holds that fermions have half-integer spin and bosons have integer spin. fermions: s = 1/2, 3/2, 5/2... (7.9) bosons: s = 0, 1, 2, 3... (7.10) The total wavefunction must obey the exchange symmetry. ψ total = φ(space) χ(spin) (7.11) This allows fermions to exist in a symmetric spacial state, as long as their combined spin state is antisymmetric. For two spin 1/2 fermions, the combined spin states with spin, s, and z-projection of the spin, m, are χ = s m. 30

32 s = 1 symmetric triplet: s = 0 antisymmetric singlet: 1 1 = (7.12) 1 0 = + (7.13) 1 1 = (7.14) 0 0 = (7.15) The spin singlet allows two electrons (which are fermions) to exist in the same spacial state (orbital) without violating Pauli s Exclusion Principle. ψ ee (x 1, x 2 ) = φ i (x 1 ) φ i (x 2 ) 0 0 (7.16) 7.3 Over Counting 31

33 Chapter 8 Grand Canonical Ensemble 8.1 Definition Suppose system A is in contact with a with a huge reservoir with which the system can exchange both heat and particles. We use Ω R to denote the number of microstates in the some macrostate specified by a number of particles and a temperature. Plugging S = k B ln Ω R into equation 6.7 gives ( ) µ ln k B T = ΩR (8.1) N R U R,V R Similar to the Canonical Ensemble in equation 3.5, we also have that ( ) 1 ln k B T = ΩR (8.2) U R V R,N R The reservoir is so large that its temperature and chemical potential do not significantly change when heat or particles are added to it. Therefore, we may trivially integrate both equations. They have the simultaneous solution (where C is some combined constant of integration): ln Ω R = C + U R µn R k B T Ω R = e C e (U R µn R )/k B T The total number of microstates of the combined system is (8.3) (8.4) Ω = Ω A (U A, N A ) Ω R (U R, N R ) (8.5) 32

34 The number of microstates with system A in its i-th microstate is Ω i = 1 Ω R (U T U A, N T N A ) = Ω R (U T E i, N T N i ) (8.6) where U T = U A + U R and N T = N A + N R. Ω i = e C e (U T E i µn T +µn i )/k B T (8.7) Since U T and N T are conserved, they can be absorbed into the leading constant. Ω i = C e (E i µn i )/k B T (8.8) The total number of states is Ω total = j Ω j (8.9) The probability that A is in the i-th state is P i = = Ω i Ω total (8.10) C e (E i µn i )/k B T C j e (E j µn j )/k B T (8.11) = e (E i µn i )/k B T Ξ where we defined the Grand Partition Function as (8.12) Ξ = j e (E j µn j )/k B T (8.13) 8.2 The Bridge to Thermodynamics Plugging in the distribution function in equation 8.12 into equation 2.16 gives S = k B P i ln P i (8.14) = k B = k B i i i ( e (E i µn i )/k B ) T P i ln Ξ P i [ (Ei µn i ) k B T 33 ] ln Ξ (8.15) (8.16)

35 Recognizing that U = i P i E i and N = i P i N i (8.17) we have that S = U T µn T + k B ln Ξ (8.18) Rearranging this, we define the Grand Potential to be Φ U T S µ N (8.19) Φ = k B T ln Ξ (8.20) Differentiating Φ and plugging in the differential of U from equation 6.3 gives dφ = du T ds S dt µ dn N dµ (8.21) = T ds p dv + µ dn T ds S dt µ dn N dµ (8.22) = p dv S dt N dµ (8.23) By taking various partial derivatives, we can calculate p, S, and N from Φ. We can then calculate any thermodynamic state variable we want using the various Maxwell Relations. Most notably, we can calculate the average number of particles in the system. ( ) Φ N = (8.24) µ T,V 8.3 Fermions For our system we choose the i-th single quantum state of a fermion, a state with a specific wave vector and spin state. Pauli s exclusion principle forbids more than one particle to exist in this state. Therefore, the spectrum of energies for this system is only {0, E i }, 0 for the state being unoccupied, and E i, being the energy of that state, for when it is occupied. Then the grand partition function for this system is Ξ = 1 + e (E i µ)/k B T (8.25) 34

36 The grand potential is Φ = k B T ln ( 1 + e (E i µ)/k B T ) (8.26) The average number of particles in this state is N = ( ) Φ µ T,V = 1 e (E i µ)/k BT + 1 (8.27) (8.28) Now considering all of the quantum states, if we use the magnitude of the wave vector, k, to index the energy levels, and let the density of states, D(k), handle the spin and other degeneracies, we can write the distribution of occupied states for fermions, called the Fermi-Dirac Distribution. f FD (k) = D(k) e (E k µ)/k B T +1 (8.29) 8.4 Bosons We choose our system to be the i-th single quantum state of a boson, a state with a specific wave vector and spin state. Any nonnegative integer number of bosons can occupy a single state. Therefore, the spectrum of energies for our system is {0, E i, 2E i, 3E i...}. The grand partition function is 1 Ξ = 1 + e (E i µ)/k B T + e 2(E i µ)/k B T + e 3(E i µ)/k B T +... (8.30) = ( ) e (E i µ)/k B T j (8.31) = j=0 1 1 e (E i µ)/k B T (8.32) where (E i µ) must be positive for the series to converge. The grand potential is Φ = k B T ln ( 1 e (E i µ)/k B T ) (8.33) 1 Recognize that the sum in equation 8.31 is a geometric series. 35

37 The average number of particles in this state is N = ( ) Φ µ T,V = 1 e (E i µ)/k BT 1 (8.34) (8.35) Just like we did for fermions, if we use k to index the energy levels, and let D(k) handle the spin and other degeneracies, we can write the distribution of occupied states for bosons, called the Bose-Einstein Distribution. f BE (k) = D(k) e (E k µ)/k B T 1 (8.36) 8.5 Non-relativistic Cold Electron Gas Electrons have a spin degeneracy of 2. Therefore, the density of states of an electron gas is D(k) = 2 D 0 (k) = V π 2 k2 (8.37) As temperature approaches zero, a the particles in a fermion gas stack in the allowed states, starting at the ground states. This means that the particles pile up in the states in the corner of k-space closest to the origin, making an eighth of a sphere. The surface of that sphere is called the Fermi surface. The states on that surface all have the same radius or wavenumber, k F, called the Fermi wavenumber. In order to calculate the Fermi wavenumber, we equate the number of electrons,n e, with number of states, stacking them in the corner. N e = kf 0 k F = D(k) dk = V π 2 kf 0 k 2 dk = V k F 3π 2 (8.38) ( 3π 2 N ) 1/3 e = (3π 2 n) 1/3 (8.39) V where n is the number density of electrons. In the non-relativistic limit, the energy of a state with a wavenumber k is E k = p2 2m = h2 k 2 2m e (8.40) 36

38 Therefore, the energy of the states on the Fermi surface, called the Fermi energy, is E F = h2 2m e (3π 2 n) 2/3 (8.41) The total internal energy of the electron gas at T = 0 is U = = From equations 8.39 and 8.41 kf 0 kf 0 E k D(k) dk (8.42) ( h 2 ) ( ) k 2 V k 2 dk (8.43) 2m e kf π 2 = h2 V k 4 dk (8.44) 2π 2 m e 0 h 2 V = k 10π 2 F 5 (8.45) m e k 3 F = 3π 2 n = 3π 2 N e V (8.46) Therefore, U = h 2 V 10 π 2 m e k 2 F = 2m ee F h 2 (8.47) ( 3 π 2 N ) ( ) e 2 m e E F V h 2 (8.48) = 3 5 E F N e (8.49) Even at T=0, an electron gas has positive internal energy and therefore a positive pressure. To calculate this degeneracy pressure, we write the explicit dependence of U on V. Plugging equation 8.39 into 8.45 gives U = = h 2 ( V 3π 2 N ) 5/3 e (8.50) 10π 2 m e V h 2 ( ) 3π 2 5/3 N 10π 2 e V 2/3 (8.51) m e 37

39 ( ) U p = (8.52) V T,N e = 2 h 2 ( 3π 2 N ) 5/3 e (8.53) 3 10π 2 m e V = 2 h 2 k 2 ( ) F 3 π 2 n (8.54) 3 10 π 2 m e = 2 5 E F n (8.55) Notice that the degeneracy pressure increases with the number density, and that the Fermi energy also increases with the number density. 8.6 Example: White Dwarf Stars When a star can no longer perform fusion of its matter, creating heat and pressure to suppress its collapse, the star collapses until the electron degeneracy pressure increases enough to halt the collapse. Such a star is called a white dwarf. In order to calculate the stable radius, R, of a white dwarf, we write U as a function of R. h 2 ( ) U = 3π 2 5/3 N 10π 2 e V 2/3, V = 4 m e 3 πr3 (8.56) h 2 ( ) ( ) 2/3 = 3π 2 5/3 4 N 10π 2 e m e 3 πr3 (8.57) = 9 ( ) 1/3 3 π 2/3 h 2 (8.58) 20 2 m e R 2 Now we calculate the gravitational potential energy for the star. du G = Gm r dm (8.59) m = 4 3 πr3 ρ, dm = 4πr 2 ρ dr (8.60) du G = G r ( ) 4 (4πr 3 πr3 ρ 2 ρ ) dr (8.61) = 16π2 Gρ 2 r 4 dr (8.62) 3 38

40 R U G = 16π2 Gρ 2 r 4 dr (8.63) 3 0 = 16π2 15 Gρ 2 R 5 (8.64) The density in terms of the number of nucleons, N N, and the mass of a nucleon, m N, is Therefore, ρ = N N m N 4πR 3 /3 (8.65) U G = 16 π 2 15 ( ) 2 NN m N G R 5 (8.66) 4 πr 3 /3 The total energy, U T, is = 3GN 2 N m2 N 5R (8.67) For convenience, we define the constants A 9 20 U T = U + U G (8.68) ( ) 1/3 3 2/3 h2 π (8.69) 2 m e B 3 5 GN 2 Nm 2 N (8.70) U T = A R 2 B R U T = A R 2 B R (8.71) (8.72) The stable minimum in energy is found by finding the critical point. du T dr = 2A R 3 + B R 2 = 0 (8.73) R = 2A B = ( 9π 4 ) 2/3 h 2 N 5/3 e GN 2 N m2 N m e (8.74) 39

41 Figure 8.1: Potential well for white dwarf with non-relativistic electrons Most elements have approximately as many neutrons as protons, and will have the same number of electrons as protons. Therefore, N N e 1 2 N N (8.75) The mass of the star, M, is almost entirely due to nucleons M N N m n, m m e m N (8.76) N = M 2m N = M 2(1800 m) = M 3600 m (8.77) R = = = ( ) 2/3 9π h 2 N 5/3 (8.78) 4 GM 2 m ( ) 2/3 9π h ( ) 2 M 5/ m (8.79) 4 GM 2 m ( ) 2/3 9π (3600) 5/3 h 2 (8.80) 4 G M 1/3 m 8/3 For a star with the mass of the sun, M = m kg, and R m. 40

42 The Fermi energy of such a white dwarf can be calculated by combining the following equations E F = h2 ( 3π 2 n ) 2/3 2m n = N 4πR 3 /3 N = M 2(1800 m) (8.81) (8.82) (8.83) which gives that E F 0.18 MeV. Note that the rest mass energy of an electron is 0.51 MeV, indicating that the most energetic electrons in our example white dwarf star (those on the Fermi surface) are getting near relativistic. 8.7 Relativistic Cold Electron Gas As we confine an electron gas into a smaller and smaller volume, the allowed quantum states get higher and higher in energy. In a small enough volume, even the lowest energy electrons will be relativistic. In the relativistic limit, the energy of an electron with a wavenumber k is So the Fermi energy is E k = pc = hkc (8.84) E F = hk F c = hc ( 3π 2 n ) 1/3 (8.85) 41

43 The total internal energy of the electron gas at T = 0 is U = = kf 0 kf 0 = hcv π 2 E k D(k) dk (8.86) ( ) V k 2 ( hkc) dk (8.87) kf 0 π 2 k 3 dk (8.88) = hcv 4π 2 k4 F (8.89) = E F V k 3 4π 2 F (8.90) = E F V ( ) 3 π 2 n (8.91) 4 π 2 = 3 4 E F N e (8.92) 8.8 Example: Collapse to a Neutron Star Following the same procedure we used in Section 8.6, we write the the internal energy as a function of radius of the star. U = hcv 4π 2 k4 F (8.93) = hcv ( 3π 2 N ) 4/3 e (8.94) 4π 2 V = hc 4π 2 ( 3π 2 N e ) 4/3 V 1/3 = hc ( ) 3π 2 4/3 N 4π 2 e = 3 ( For convenience, we define the constant C 3 4 (8.95) ( ) 1/3 4 3 πr3 (8.96) ) 2/3 π 1/3 h c N 4/3 e R 1 (8.97) ( ) 2/3 3 π 1/3 h c Ne 4/3 (8.98) 2 42

44 The gravitational potential energy is the same as we calculated in Section 8.6. The total energy is U T = U + U G (8.99) = C R B R = C B R (8.100) (8.101) There is no stable radius for a white dwarf star with relativistic electrons. The critical point is when C = B. Using the same assumptions from equations 8.75, 8.76, and 8.77, we have that ( ) 2/3 3 π 1/3 h c Ne 4/3 = GN Nm 2 2 N (8.102) ( 3 2 M 2/3 = 5 4 ) 2/3 ) ( ) 4/3 ( hc M π 1/3 = M 2 (8.103) G 2 m N ( ) 2/3 ( ) 4/3 ) 3 1 ( hc π 1/3 m 4/3 N (8.104) 2 ( 5 M = 3 16 The Planck Mass is defined as 2 ) 3/2 ( hc π G G ) 3/2 m 2 N (8.105) m P = hc G (8.106) We call the critical mass, M, the Chandrasekhar limit, M Ch. M Ch = ( ) 3/2 5 3 π m 3 P m 2 N 16 (8.107) m 3 P m 2 N (8.108) kg (8.109) 1.5 m (8.110) If mass of the star is less than M Ch, then C B is positive and the star will expand, lowering its energy levels until it is in the non-relativistic regime. If 43

45 the mass of the star is greater than M Ch, then C B is negative and the star will continue to collapse. The star is forced to perform an enormous number of inverse beta-decays, converting its electrons into electron neutrinos that radiate away and converting its protons into neutrons. Then the star can reach a new equilibrium supported by neutron degeneracy pressure. Such a star is called a neutron star. It is virtually a huge single nucleus consisting of only neutrons. If the neutron degeneracy pressure is not enough to stop the collapse, the star becomes a black hole. 44

46 Chapter 9 Questions How are the thermodynamic and statistical mechanical definitions of entropy equivalent? U = i P i E i du = i E i dp i + i P i de i = δq + δw? When do we use Z N = Z N 1 /N!. Why don t we in f MB? 45