布朗大学统计力学2014年期末考试题

PHYS2140:Statistical Mechanics

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Final Exam,Thursday,May15,2014,BH141

This is a closed-book,closed-note,two-hour exam.There are formula sheets on the last two pages.Please show all work in your yellow exam book,and turn in a neat, concise exam.

1.(10points)Classical statistical mechanics:equipartition.A long,thin

(needle-shaped)dust grainfloats in a boxfilled with gas at a constant tem-perature T.On average,is the angular momentum vector nearly parallel or perpendicular to the long axis of the grain?

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2.(15points)Photon gas.Consider a photon gas enclosed in a volume V and

in equilibrium at temperature T.The photon is a massless particle,soε=pc.

(a)(5points)What is the chemical potential of the gas?Explain.

(b)(5points)Determine how the number of photons in the volume depends

upon the temperature.Your answer can be in the form N∝T n(ie what is n?).

(c)(5points)Determine how the energy depends on temperature.Again,I

just want the exponent m in U∝T m.

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3.(25points)Quantum rotators.The quantum energy levels of a rigid rotator

are

εj=j(j+1)h2/(8π2ma2),(1) where j=0,1,2,...;m and a are positive constants.The degeneracy of each level is g j=2j+1.

(a)(7points)Find the general expression for the partition function Q.Do not

attempt to evaluate the sum.

(b)(6points)Approximate the partition function as an integral at high tem-

perature,and evaluate the integral.Hint:a convenient change of variables in the integral is x=j(j+1).

(c)(6points)Using your answer in(b),evaluate the internal energy and heat

capacity at high temperature.

(d)(6points)Find the internal energy and heat capacity at low temperature.

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4.(25points)Virial coefficient.Consider a gas of particles in3-dimensional

space interacting through a pairwise central potential V(r),where

V(r)=

∞for0<r<a

− for a<r<b

0for b<r<∞

(2)

(a)(10points)Calculate the second virial coefficient B2(T).

(b)(5points)Calculate the pressure and the density to second order in z=

exp(βµ).Leave your answer in terms of B2(i.e.not in terms of a,b,and ).

(c)(10points)Calculate the Helmholtz free energy A in terms of N,V,T,

and the virial coefficient B2.

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5.(25points)Quantum gas.Consider a gas of non-interacting non-relativistic

particles of spin1/2in two dimensions,confined to an area A=L2.The single particle energy isε=¯h2k2/(2m).

(a)(4points)Find the density of states g(ε).Hint:recall that

states (...)=

∞

dεg(ε)(...).(3)

(b)(4points)Find the Fermi energyεF.

(c)(4points)Find the average energy per particle at T=0.

(d)(5points)For T=0,find the relation between the chemical potentialµ

and the density n=N/A.The formula sheet at the end of the exam may help you with the integral you will encounter.

(e)(4points)What is the limit of the chemical potential at high temperature?

(f)(4points)What is the limit of the chemical potential at low temperature?

Compare with your answer in(b).

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