Statistical Mechanics - Physics at Oregon State University

More documents

Recommendations

Info

240 APPENDIX A. SOLUTIONS TO SELECTED PROBLEMS. and the average dipole moment is which simplifies to or = s1,···,sN = N P (sj)P rob(s1, · · · , sN) j 1 eaE ( 2ea)e+ eaE 2ea)e− 2kT eaE eaE e + 2kT + e− 2kT 2kT + (− 1 = 1 Nae tanh(eaE 2 2kT ) which reduces for small fields, or eaE ≪ 2kT to Problem 7. ≈ NE a2 e 2 The probability of finding a system in a state s is Ps. In this case the entropy of the system is S = −kB Ps log(Ps). Assume that the system consists of two s independent subsystems and that the state s is the combination of subsystem 1 being in state s1 and system 2 being in state s2. Use the formula for the entropy given above to show that S = S1 + S2. The state of the system is given by the states of the subsystems, s = (s1, s2). Because the systems are independent we have Ps = Ps1Ps2 and hence: 4kT S = −kB Ps log(Ps) = −kB Ps1Ps2 log(Ps1Ps2) With Psi = 1 this gives Problem 8. s s1,s2 S = −kB Ps1Ps2 (log(Ps1) + log(Ps1)) s1 s2 S = −kB Ps1 log(Ps1) + s1 s2 Ps2 log(Ps1) = S1 + S2 The energy eigenvalues for a single particle in a given potential are ɛn. These energy values are independent of how many other particles are present. Show that the partition function Z(T, N) obeys the relation Z(T, N) = (Z(T, 1)) N .
A.2. SOLUTIONS FOR CHAPTER 2. 241 For one particle we have: Z(T, 1) = e −βɛn1 When we have N particles they are in states n1, n2, · · · , nN and we have Z(T, N) = · · · e −β(ɛn1 +ɛn2 +···+ɛnN ) n1 n2 Which is equal to : Z(T, N) = e −βɛn 1 e −βɛn2 · · · e −βɛnN = (Z(T, 1)) N Problem 9. n1 n2 nN The energy eigenvalues of a system are given by 0 and ɛ + n∆ for n = 0, 1, 2, · · ·. We have both ɛ > 0 and ∆ > 0. Calculate the partition function for this system. Calculate the internal energy and the heat capacity. Plot the heat capacity as a function of temperature for 0 < kBT < ɛ for (a) ∆ ≫ ɛ, (b) ∆ = ɛ , and (c) ∆ ≪ ɛ. Z = e β0 + n1 nN ∞ e −β(ɛ+n∆) −βɛ = 1 + e e −nβ∆ n=0 The sum can be done using n rn = 1 1−r U = ɛ We have three cases: and gives: Z = 1 + e−βɛ 1 − e−β∆ = 1 + e−βɛ − e−β∆ 1 − e−β∆ 2 ∂ ∂ U = kBT log(Z) = − ∂T ∂β log(Z) U = ɛe−βɛ − ∆e−β∆ 1 + e−βɛ ∆e−β∆ + − e−β∆ 1 − e−β∆ e−βɛ −β∆ e − 1 − e−β∆ 1 + e−βɛ + ∆ − e−β∆ e ∆ ≫ ɛ ⇒ U ≈ ɛ −βɛ 1 = ɛ 1 + e−βɛ eβɛ + 1 e ∆ = ɛ ⇒ U = ɛ −βɛ 1 = ɛ 1 − e−βɛ eβɛ − 1 ∆ ≪ ɛ ⇒ U ≈ ɛ + kBT (for kBT ≫ ∆) n e −β∆ 1 + e −βɛ − e −β∆
Page 1:
Statistical Mechanics Henri J.F. Ja
Page 4 and 5:
IV CONTENTS 4.3 Gas of poly-atomic
Page 6 and 7:
VI CONTENTS
Page 8 and 9:
VIII INTRODUCTION Introduction. The
Page 10 and 11:
X INTRODUCTION In chapter nine we d
Page 12 and 13:
2 CHAPTER 1. FOUNDATION OF STATISTI
Page 14 and 15:
Page 16 and 17:
Page 18 and 19:
Page 20 and 21:
10 CHAPTER 1. FOUNDATION OF STATIST
Page 22 and 23:
Page 24 and 25:
Page 26 and 27:
Page 28 and 29:
Page 30 and 31:
Page 32 and 33:
Page 34 and 35:
24 CHAPTER 2. THE CANONICAL ENSEMBL
Page 36 and 37:
Page 38 and 39:
Page 40 and 41:
Page 42 and 43:
Page 44 and 45:
Page 46 and 47:
Page 48 and 49:
Page 50 and 51:
Page 52 and 53:
Page 54 and 55:
44 CHAPTER 3. VARIABLE NUMBER OF PA
Page 56 and 57:
Page 58 and 59:
Page 60 and 61:
Page 62 and 63:
Page 64 and 65:
Page 66 and 67:
Page 68 and 69:
Page 70 and 71:
Page 72 and 73:
Page 74 and 75:
Page 76 and 77:
Page 78 and 79:
68 CHAPTER 4. STATISTICS OF INDEPEN
Page 80 and 81:
Page 82 and 83:
Page 84 and 85:
Page 86 and 87:
Page 88 and 89:
Page 90 and 91:
Page 92 and 93:
Page 94 and 95:
Page 96 and 97:
Page 98 and 99:
Page 100 and 101:
90 CHAPTER 5. FERMIONS AND BOSONS
Page 102 and 103:
92 CHAPTER 5. FERMIONS AND BOSONS t
Page 104 and 105:
94 CHAPTER 5. FERMIONS AND BOSONS w
Page 106 and 107:
96 CHAPTER 5. FERMIONS AND BOSONS T
Page 108 and 109:
98 CHAPTER 5. FERMIONS AND BOSONS S
Page 110 and 111:
Page 112 and 113:
Page 114 and 115:
Page 116 and 117:
Page 118 and 119:
Page 120 and 121:
Page 122 and 123:
Page 124 and 125:
Page 126 and 127:
Page 128 and 129:
Page 130 and 131:
120 CHAPTER 6. DENSITY MATRIX FORMA
Page 132 and 133:
Page 134 and 135:
Page 136 and 137:
Page 138 and 139:
Page 140 and 141:
Page 142 and 143:
Page 144 and 145:
Page 146 and 147:
Page 148 and 149:
Page 150 and 151:
140 CHAPTER 7. CLASSICAL STATISTICA
Page 152 and 153:
Page 154 and 155:
Page 156 and 157:
Page 158 and 159:
Page 160 and 161:
Page 162 and 163:
Page 164 and 165:
Page 166 and 167:
Page 168 and 169:
158 CHAPTER 8. MEAN FIELD THEORY: C
Page 170 and 171:
Page 172 and 173:
Page 174 and 175:
Page 176 and 177:
Page 178 and 179:
Page 180 and 181:
Page 182 and 183:
Page 184 and 185:
Page 186 and 187:
Page 188 and 189:
Page 190 and 191:
Page 192 and 193:
Page 194 and 195:
Page 196 and 197:
Page 198 and 199:
Page 200 and 201: 190 CHAPTER 8. MEAN FIELD THEORY: C
Page 202 and 203: 192 CHAPTER 9. GENERAL METHODS: CRI
Page 240 and 241: 230 APPENDIX A. SOLUTIONS TO SELECT
Page 270: 260 APPENDIX A. SOLUTIONS TO SELECT
show all

Statistical Mechanics - Physics at Oregon State University

Create successful ePaper yourself

Delete template?

Save as template?