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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