Thermodynamics, Statistical Physics, and Quantum Mechanics

196 SOLUTIONS as the mass of the neutrinodecreases. Equating the kineticenergyfrom (a) with(S.4.79.6), we see c) Substituting(S. ...

THERMODYNAMICS AND STATISTICAL PHYSICS 197 Substituting the radiation density into (S.4.80.4) gives or The ratios of coefficient ...

198 SOLUTIONS b) From (S.4.81.1) we may find the partition function Z: where we define The sum (S.4.81.2) may be easily calculat ...

THERMODYNAMICS AND STATISTICAL PHYSICS^199 For J = 1, c) For largeHthe magnetization saturates It is convenientto define theso-c ...

200 SOLUTIONS 4.82 Paramagnetism at High Temperature (Boston) a) The specific heat c of a system that has N energy states is giv ...

THERMODYNAMICS AND STATISTICAL PHYSICS 201 and we have where is the probability density. Therefore, For the quantum mechanical c ...

202 SOLUTIONS We wish to perform the sum from –J toJ, so and (S.4.82.5) gives c) For , and As in Problem 4.81 for where We then ...

THERMODYNAMICS AND STATISTICAL PHYSICS 203 4.83 One-Dimensional Ising Model (Tennessee) a) The partition function is defined as ...

204 SOLUTIONS 4.84 Three Ising Spins (Tennessee) a) Define and where The definition of the partition function is A direct calcul ...

THERMODYNAMICS AND STATISTICAL PHYSICS 205 4.86 N Independent Spins, Revisited (Tennessee) We use the expression where is the pr ...

206 SOLUTIONS ForBandMsmall we may rearrange (S.4.87.4),yielding where Thedivergence of at indicates theonset of ferromagnetism. ...

THERMODYNAMICS AND STATISTICAL PHYSICS 207 c) Given the form of the spin operator inpart (c),one immediatelyderives the equation ...

208 SOLUTIONS Since the moments are all independent, we may express the average mag- netization as On the other hand, For the en ...

THERMODYNAMICS AND STATISTICAL PHYSICS 209 4.90 Gas Fluctuations (Moscow Phys-Tech) a) We can disregard any particles from the h ...

210 SOLUTIONS 4.91 Quivering Mirror (MIT, Rutgers, Stony Brook) a) When themirror is in thermalequilibriumwith gas in the chambe ...

THERMODYNAMICS AND STATISTICAL PHYSICS 211 Since the chemical potential is expressed in P, and does not depend on N, we can writ ...

212 SOLUTIONS From (S.4.92.3) and SinceVis proportional to The relativefluctuation isgiven by 4.93 Energy Fluctuation in Canonic ...

THERMODYNAMICS AND STATISTICAL PHYSICS 213 Therefore, comparing (S.4.93.4) and (S.4.93.5), we deduce that, at constant volume, o ...

214 SOLUTIONS The converse is also true: If the variables and are statis- tically independent. Now for a closed system we can wr ...

THERMODYNAMICS AND STATISTICAL PHYSICS 215 where we used So, finally Using V and as independent variables we have Substituting ( ...