課程名稱︰統計物理導論 課程性質︰物理系必修 課程教師︰陳義裕老師 開課學院：理學院 開課系所︰物理系 考試日期（年月日）︰June 6, 2012 考試時限（分鐘）：180 min 是否需發放獎勵金： 是 （如未明確表示，則不予發放） 試題 : 1.(20pts) (a)(10pts) Please prove the equipartition theorem, which states that, in the formalism of classical statictical physics, the thermally averaged energy of a system whose Hamiltonian is given by H = C1*q1^2 + C2*q2^2 + ... + Cl*ql^2 must be equal to l*(kB*T/2) which is independent of the constants C1,C2,C3,...,Cl. In the above, each of the qj is a "canonical coordinate" discussed in class. (b)(10pts) A simple harmonic oscillator has a quantized energy given by εj = jhν, j=0,1,2,... Please show that the thermally averaged energy of this oscillator is <ε> = hν/(exp(hν/(kB*T))-1) 2.(10pts) A puzzle troubling 19-th century pyhsicists concerning the specific heat of diatomic gas molecules was that, for some unknown reason, only the trnaslational motion of the molecules is "excited" when the temperature T is low, and the rotational motion of the molecules is next "excited" when T is increased by quite a bit, whereas the vibrational motion is "excited" only when T is extremely high, which in practice is rarely met in experiments. Please describe in detail how quantum theory manages to explain it all. 3.(25pts) Borrowing Debye's idea for a 3-D crystal, that is, introducing a Debye temperature θD and considering only the dispersion relation ω^2 = c^2 * k^2 for elastic waves, please compute the spdecific heat (per mole) for a 2-D crystal in the following limits: (a)(10pts) T/θD >> 1 (b)(10pts) T/θD << 1 (c)(5pts) A major weakness to Einstein's model of specific heat compared with Debye's model is that, when the temperature is low , Einstein predicted a specific heat that is way too small compared with that of Debye's. Please explain in words why this is expected. 4.(20pts) Dr. Planckenstein once proposed that energy of a mode of electromagnetic radiation in a cavity should be quantized according to εj = j^2 * hν j = 0,1,2,... Then, borrowing the idea of Rayleigh-Jeans, Planckenstein derived the spectral radiant energy density of the blackbody radiation in a cavity to be given by u = (8πν^2/c^3) < ε > where < ε > is the thermally averaged energy of εj. (a)(10pts) In the limit hν/(kB*T) >> 1, does Planckenstein's formula reduce to the Wien form for some constants a and b ? You must give a valid proof to your claim to receive the credits. u(Wien) = aν^3*exp(-bν/T) (b)(10pts) In the limit h/(kB*T) << 1, does Planckenstein's formula reduce to the standard Rayleigh-Jeans form? You must give a valid proof to your claim to receive the credits. u(Rayleigh-Jeans) = (8πν^2/c^3)*kB*T 5.(15pjs) For a huge number N of non-interactin bosons confined in some given small volume V satisfying the Bose-Einstein distribution nj = 1/(exp(β(εj-μ))-1) please show that the majority of the particles will be in the ground state if the temperature T is well below a certain temperature TB that is determined by N/V, the mass m of the particles, and the two universal constants h and kB. (Please derive the expression for TB.) This phenomenon is called Bose-Einstein condensation. 6.(10pts) Let pj be a probabilistic distribution to be determined by the maximum entropy formalism. Suppose it is know that Σpj = 1 Σpj*Ej = U = given number for given Ej and U. Please prove that the maximization of S = -Σpj*ln(pj) subject to the above two constraints necessarily yields exp(-βEj) pj = ------------------------------ Σexp(-βEk) k for some number β. --

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