Statistical Mechanics - Physics at Oregon State University

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248 APPENDIX A. SOLUTIONS TO SELECTED PROBLEMS. S(T, µ, V ) = NkB − fM (ɛo; T, µ) log(fM (ɛo; T, µ)) where the sum is over orbitals. There are several ways to do this. For example: S = − o < n >o= kBT ∂ log Zo ∂µ T,V < n >o= fM (ɛo; T, µ) = e 1 k B T (µ−ɛo) ∂ log Zo ∂µ T,V = 1 1 k e B T kBT (µ−ɛo) log Zo = e 1 k B T (µ−ɛo) = fM (ɛo; T, µ) Ω = −kBT log(Zo) = −kBT fM (ɛo; T, µ) o ∂Ω = kB fM (ɛo; T, µ) + kBT ∂T V,µ ∂ ∂T fM (ɛo; T, µ) o fM (ɛo; T, µ) = N o ∂ ∂T fM (ɛo; T, µ) = − 1 1 k (µ−ɛo)e B T kBT 2 (µ−ɛo) 1 = − T log(fM (ɛo; T, µ))fM (ɛo; T, µ) which gives the result we need Problem 5. Consider a system of independent particles. The number of orbitals with energy between E and E + dE is given by N(E)dE. The function N(E) is called the density of states. One measures the expectation value of a certain operator O. For a particle in an orbital o the value of the operator depends only on the energy of the orbital, or Oo = O(ɛo). Show that in the thermodynamic limit the ensemble average of the operator is given by ∞ < O >= O(E)N(E)f(E; T, µ)dE −∞ where f(E; T, µ) is the distribution function for these particles. o o
A.4. SOLUTIONS FOR CHAPTER 4. 249 < O > (T, µ, V ) = Oof(ɛo; T, µ) < O > (T, µ, V ) = O(ɛo)f(ɛo; T, µ) Define an energy mesh by Ei = idE where dE will be very small < O > (T, µ, V ) = O(ɛo)f(ɛo; T, µ) < O > (T, µ, V ) = which leads to the integral Problem 6. i i o o o,Ei
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Statistical Mechanics Henri J.F. Ja
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IV CONTENTS 4.3 Gas of poly-atomic
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VI CONTENTS
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VIII INTRODUCTION Introduction. The
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X INTRODUCTION In chapter nine we d
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2 CHAPTER 1. FOUNDATION OF STATISTI
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10 CHAPTER 1. FOUNDATION OF STATIST
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24 CHAPTER 2. THE CANONICAL ENSEMBL
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44 CHAPTER 3. VARIABLE NUMBER OF PA
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68 CHAPTER 4. STATISTICS OF INDEPEN
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90 CHAPTER 5. FERMIONS AND BOSONS
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120 CHAPTER 6. DENSITY MATRIX FORMA
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140 CHAPTER 7. CLASSICAL STATISTICA
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158 CHAPTER 8. MEAN FIELD THEORY: C
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192 CHAPTER 9. GENERAL METHODS: CRI
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Statistical Mechanics - Physics at Oregon State University

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