PHYSICS 715
Problem Set 8 Due Friday, March 31, 2006
Reading: Landau and Lifshitz, Secs. 81–86
LD 26: The high temperature behavior of Zr ot: Euler-Maclaurin summation and
Mulholland’s formula.
The quantum corrections to the rotational partition function of a heteronuclear di-
atomic molecule with characteristic rotational temperature Θrcan be evaluated for
T > Θrby applying the Euler-Maclaurin summation formula to the series for Zrot.
(a) Derive the first three terms in the high-temperature expansion for Zrot (Mulhol-
land’s formula). [See, e.g., Abramowitz and Stegun, Handbook of Mathematical
Functions, Sec. 3.6.28, for the Euler-Maclaurin formula. Be sure to keep enough
of the derivative terms, and expand all results to a consistent order in Θrot/T .]
(b) Use your result to determine Srot and crot , the rotational specific heat per molecule,
to the same order.
LD 27: The rotational partition function and specific heat of H+
2: the remarkable
effect of statistics.
(a) The proton has spin s=1
2. The equilibrium separation of the protons in the H+
2
molecule is 1.06 ˚
A. Obtain an expression for the rotational partition function of H+
2
in terms of appropriate statistical weights and the rotational partition functions
Zeven and Zodd, defined as sums over even and odd values of ℓ. Explain your
reasoning carefully.
Calculate Zeven and Zodd numerically for 0 ≤T/Θr≤4 by using the series expan-
sions carried to a sufficient number of terms. Plot the results and compare with
that given by the appropriate multiple of Mulholland’s formula. Mark the boiling
points of H2(20.3 K) and N2(77.4 K) for reference. Plot in addition the ratio
ZS=1/ZS=0 of the equilibrium fractions of H+
2molecules with total nuclear spins
S= 0,1, and comment on the limiting behaviors for T→0,∞.
(b) Calculate the rotational specific heat per molecule crot numerically using the series
expansion for Zrot. Plot crot carefully for H+
2as a function of T/Θrfor 0 ≤T /Θr≤
4. Determine the limiting forms of crot (not just the limits) analytically for T≫Θr
and T≪Θr, and compare them with the numerical result. For what range of
temperatures (in K) is the system essentially classical? [Hint: the analytic result
for crot is simple when expressed in terms of Zrot and its derivatives with respect
to x= Θr/T . Write out the series defining the derivatives you need, and expand
as appropriate in getting the analytic result for T→0.]
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