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Returning to the Bose system, we see that Bose-Einstein condensation corresponds to a spontaneous breaking of the global gauge invariance. In analogy with ferromagnetism, we imagine subjecting the system to an external field coupled to If;(x), calculate the ensemble average of If;(x) in the thermodynamic
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*J. Goldstone, op. cit. A brief discussion of this phenomenon will be given in Section 16.6.
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Note that in calculating the spontaneous magnetization in the model in Section 11.6, we in effect used the correct average (12.98) instead of (12.99), because we ignored the - M solution (by common sense).
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The only essential difference with the ferromagnetic case is that, unlike the magnetic field, the external field 1J(x) here is a mathematical device that cannOI be realized experimentally. *
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12.1 (a) Show that the entropy per photon in blackbody radiation is independent of the temperature, and in d spatial dimensions is given by
n- d n- d
(d + 1)_n---;~00::;cl_
(b) Show that the answer would have been d
+ 1 if the photons obeyed Boltzman
statistics.
12.2 Some experimental valuest for the specific heat of liquid He 4 are given in the
accompanying table. The values are obtained along the vapor pressure curve of liquid He 4 , but we may assume that they are not very different from the values of C v at the same temperatures.
Temperature (K)
0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00
Specific Heat (joulejg-deg)
0.0051 0.0068 0.0098 0.0146 0.0222 0.0343 0.0510 0.0743 0.1042
*The Bose-Einstein condensation of the ideal gas has been reanalyzed in terms of the superfluid order parameter by J. D. Gunton and M. J. Buckingham, Phys. Rev. 166, 152 (1968). tTaken from H. C. Kramers, "Some Properties of Liquid Helium below 1 0 K," Dissertation. Leiden (1955).
BOSE SYSTEMS
(a) Show that the behavior of the specific heat at very low temperatures is characteristic
of that of a gas of phonons.
(b) Find the velocity of sound in liquid He 4 at low temperature.
12.3 Equation (12.64) states that C = 0 for v < vc. Using the formula S = - (aC/aT)p, we would obtain S = 0 for v < vc' in contradiction to (12.65). What is wrong with the previous statement 12.4 In the neighborhood of z = 1 the following expansion may be obtained (F. London, loc. cit.):
gS/2(Z) =
2.363,,3/2 + 1.342 - 2.612" - 0.730,,2 + ...
where" == -log z. From this the corresponding expansions for g3/2' gl/2, and g-l/2 maybe obtained by the recursion formula gn-l = -agn/a". Using this expansion show that for the ideal Bose gas the discontinuity of aCv/aT at T = ~. is given by
a c v) ( a c v) ( aT Nk T~T/ aT Nk T~Tc-
12.5 Show that the equation of state of the ideal Bose gas in the gas phase can be written in the form of a virial expansion, Le.,
412 ---;;
1 ("}'.3) + (1 - 913 )("},.3)2 2 ---;; "8
lim -V log 2( z, V, T)
V---+oo
"~.6 (a) Calculate the grand partition function 2(z, V, T) for a two-dimensional ideal Bose gas and obtain the limit
where V = L 2 is the area available to the system. (b) Find the average number of particles per unit area as a function of z and T. (c) Show that there is no Bose-Einstein condensation for a two-dimensional ideal Bose gas.
12.7 Consider two free bosons contained in a box of volume V with periodic boundary conditions. Let the momenta of the two particles be p and q.
(a) Write down the normalized wave function l/;pq(r1 ,r2 ) for both p (b) Show that for p
*" q
*" q and p =
(c) Explain the meaning of the statement" spatial repulsion leads to momentum space
attraction."
12.8 For the imperfect Bose gas discussed in Section 12.4, show that in the gas phase
:; =
1 + (-
+ 2 A
a) ~ + ( ~ - 3~ )( ~
Thus we can conclude that the third and higher virial coefficients, if they depend on a, must involve orders of a 2 or higher.
STATISTICAL MECHANICS
12.9 Consider an ideal Bose gas. Let l/;(x) be the boson field operator.
(a) Show
iftt(x)ift(y)
where
+ f(lx - yl)
mkT e- r/
f(r)
with ro =
(b) Let T as t --> O.
d k f-- e ikor (n (2'1T)3
= -2- -
iii y'2mkTllog zl
T,. from the high-temperature side. Find ro as a function of t
(T - T,.)/T,.
( c) The density-density correlation function is defined as
f(x) == (p(x)p(O) - (NIV)2
where p(x)
I/;t(x)l/;(x) is the density operator. Show
f(x)
L ei(k-q)ox(nq(nk+1)_ -Leikox(nk) 1
-> 00
Work out f (x) more explicitly, using the results of (a) and (b).