Physics 6011, Spring 2015, Homework 11, Version 1

Due: Thursday, March 26, 10pm.Important note: solutions for some or all of the problems in this assignment can befound in various sources, including the class textbooks. It is of course reasonable andencouraged to read other portions of the class textbooks (or other books if you sodesire), to discuss with classmates, or to ask questions to the instructor. However,please resist the temptation to read (or copy) the solutions to any of the problems,and try your best to solve the problems by your own effort. In some cases, however,you may be asked to compare your results with the results shown in the textbook;in that case, make sure you only look at the textbook results after you have writtenyour own solution.

0. Time spent (5 pts.)

At the top of the first page of your solution, please leave 5 lines of space for the grades (andany general comments) to be written.

Just after that, please write a table containing the following columns. Each column shouldreport only one time. In all cases, the times reported should refer to work done after thedeadline for the previous homework. Columns: i) One column for the total time spent onreading assignments ii) One column for the total time spent reading/studying in preparationto work on the homework iii) One column per problem containing the time spent workingon that problem (please answer individually for each problem), iv) One column for the totaltime spent on the course (in hours), including everything you did for the course except the3 hours of lectures and the discussion hour.

1. Oscillation modes and normal mode density

In this problem, we use the density of normal modes g(ω) discussed in lectures. As areminder, g(ω) dω is the number of modes i with frequencies ωi satisfying ω ≤ ωi < ω+ dω.

(a) This question is answered in full in Sec. 7.3 of the textbook. Don’t look at the answeruntil you have written your own solution in full.

In the last two parts of the previous homework assignment, you determined the energy〈E〉(T, V,N) and the heat capacity C(T, V,N) for the case when g(ω) = V Cωr−1, andyou determined g(ω) for electromagnetic waves in a rectangular 3D cavity of volume V .Use those results, together with the Planck distribution, to obtain the following proper-ties of the electromagnetic field in a cavity of volume V , at temperature T (“blackbodyradiation”): i) the energy density per unit volume u(ω)dω associated with a frequencyinterval [ω, ω+ dω), ii) the total energy density E/V for the cavity, iii) the heat capac-ity, and iv) the pressure of the radiation (Hint: compute the Helmholtz free energy anduse that µ = 0 for photons.)

After writing your solution, compare your results with those in the textbook.

(b) Consider a crystalline solid in d dimensions containing N atoms. Assume that all

vibrational normal modes are plane waves that have frequencies given by ωr(~k) =

cs|~k|, r = 1, · · · , d (acoustic modes), where cs > 0 is the speed of sound, and that thereis a total of Nd normal modes. Show that these assumptions lead to the Debye modelof the next item.

(c) Debye model

Assume that the low frequency modes of a crystalline lattice are simple plane waves sothat a good approximation for the density of vibrational modes is g(ω) ≈ (Nd2/ωd

D)ωd−1

for 0 ≤ ω ≤ ωD and g(ω) = 0 outside that interval. The frequency ωD is called theDebye frequency. Determine the thermodynamic behavior of this model, i.e. find thenumber 〈N〉, the energy 〈E〉, and the heat capacity C(T,N). At some point you willfind an integral that is not easy to solve explicitly. To go beyond this, focus on theheat capacity C(T,N). Define a characteristic temperature θD, obtain C(T,N) in thelow temperature (T � θD) and high temperature (T � θD) limits, and sketch it for alltemperatures.

(d) Do the results in Section 7.3 for the blackbody radiation correspond to the high temper-ature limit of the Debye model? or to the low temperature limit of the Debye model?or neither? Explain why this happens (Hint: start by comparing the density of modesg(ω) for blackbody radiation in a cavity and for the Debye model.)

2. Noninteracting Fermions

Consider a system of noninteracting fermions such that there are V D(ε)dε = g(ε)dε singleparticle states with energies εi in the interval ε ≤ εi < ε + dε. For this problem, the onlyresults about ideal Fermi systems that you should use, are Eqs. (1) and (2) from Sec. 8.1 inthe textbook, and the definition of Fermi energy εF given in class.

(a) By using the above definition of D(ε), convert the rhs of Eqs. (1) and (2) in Sec. 8.1 ofthe textbook into integrals over the single particle energy ε.

(b) Starting from the formulas obtained before, write a formula for the pressure of a systemof noninteracting fermions, in term of an integral containing the density of states D(ε)and the Fermi function f(ε) = 1/(exp(β(ε− µ) + 1).

(c) By using the expressions obtained before, show that for a system of free fermions ofspin 1/2 in 3 dimensions at any temperature, 〈E〉 = 3

2pV .

(d) By using the expressions obtained before, write the pressure at T = 0 for a system offree fermions of mass m in 3 dimensions, in terms of m, h̄ and the Fermi energy εF .

3. Bose-Einstein condensation, part I (based on Prob. 7.14 in the textbook)

Consider an n-dimensional gas of ideal bosons of mass m with spin S = 0, whose single-particle energy spectrum is given by ε ∝ ps, where s is some positive number. Assume thatthe system is in a hypercubic box of side L, define V = Ln as the generalization of thevolume to dimension n, and use periodic boundary conditions.

(a) Compute the (intensive) density of states D(ε) ≡ g(ε)/V .

(b) Find generalizations to the case of general n and s of Eqs. (5), (6), (7), and (8) inSec. 7.1 of the textbook. In the integrals that you write, make sure that the powerof the variable of integration (which is 1/2 in the case of the original equations in thetextbook) is written explicitly in your expression.

(c) Save a copy of your work for next week. In the next assignment, you will be asked tocontinue working on this problem (see below).

4. Preview of a problem that will be in homework assignment 12: Bose-Einstein condensation,part II

You are encouraged to start working on this problem, but don’t turn it in yet.

Consider an n-dimensional gas of ideal bosons of mass m with spin S = 0, whose single-particle energy spectrum is given by ε ∝ ps, where s is some positive number. Assume thatthe system is in a hypercubic box of side L, define V = Ln as the generalization of thevolume to dimension n, and use periodic boundary conditions.

Discuss the onset of Bose-Einstein condensation in this system, especially its dependence onthe numbers n and s. Find generalizations to the case of general n and s of Eqs. (5), (6),(7) (8), (11), (12), (16), (23), (24) in Sec. 7.1 of the textbook. (In some cases, the equationwill be trivial, for example you may find Tc = 0). Study the thermodynamic behavior of thissystem and show that,

P =s

n

U

V, CV (T →∞) =

n

sNk, CP (T →∞) = (

n

s+ 1)Nk.

Consider the cases when there is a Bose-Einstein condensation with Tc > 0 and write theequation for the particle density,

〈N〉V

=1

V

z

1− z+

1

λnTg̃(z),

where g̃(z) is a function of the fugacity that you need to determine. Multiply both sides ofthis equation by λnT , and show how to solve the resulting equation graphically, by plottingthe lhs, the second term in the rhs, and the sum of the two terms in the rhs as functionsof βµ at fixed β. The point where the horizontal line representing the lhs crosses the curverepresenting the sum of the two terms in the rhs gives the solution for the equation. Bydrawing three different horizontal lines at appropriate places in the same plot, show how tographically solve the equation in the following three cases: i) T > Tc, ii) T = Tc, and iii)T < Tc. Sketch the corresponding diagram for a generic case where there is no Bose-Einsteincondensation, and explain how one can see in the diagram that the transition is absent.