Transcript Philips 170 p

Chapter 7 Fermi and Bose Gas
(費米和玻色氣體)
1. Fermi Gas (費米氣體)
For a system of N noninteracting free fermions (費米氣體) in volume V with
concentration n, quantum concentration nQ  (Mt/2pħ2)3/2 and Fermi-Dirac
distribution function
f(e,t,m) = 1/{exp[(e – m)/t] + 1} = 1/(x-1 + 1)
Fermi gas is in classical regime (classical gas 古典氣體) if
n << nQ, f(e,t,m) ~ exp[(m - e)/t] = x << 1
Fermi gas is in quantum regime (quantum gas 量子氣體) if
n ~ nQ
For fixed concentration n, the temperature T is an important variable system is in
quantum regime
T < T0  (2pħ2/kM)n2/3 ~ TF  (ħ2/2kM)(3p2n)2/3 (TF: Fermi temperature for s = 1/2)
where T0 is defined by n = nQ
A Fermi gas in the quantum regime with T << T0 is a degenerate Fermi gas
(簡併費米氣體)
(7-1)(12/16)
(Ground state of Fermi gas in three dimension)(三維費米氣體之基態)
For a system of N free non-interacting electrons (free electron gas 自由電子氣體)
(mass m, spin s = ½) in a cube of V = L3 at ground state T = 0 K
for one free electron (N = 1), energy levels with quantum number (nx,ny,nz,ms)
en = (ħ2/2m)(p/L)2(nx2 + ny2 + nz2) (nx, ny, nz: positive integers) (see Chap 1)
= (ħ2/2m)(p/L)2n2
single electron quantum states (orbitals) are specified by quantum number (n,ms)
with magnetic quantum number ms =  ½ for electron spin s = 1/2 (ms = ½ are
degenerate in energy)
(ex 1) lowest energy level with quantum state (n,ms) = (31/2,½) (two states)
e(111) = (ħ2/2m)(p/L)2(12 + 12 + 12) = 3.(ħ2/2m)(p/L)2
(ex 2) level with quantum state (n,ms) = (61/2,½) (six states)
e(211) = e(121) = e(112) = 6.(ħ2/2m)(p/L)2
Fermi energy (費米能量) eF is the energy of the highest filled states at T = 0 K
eF = (ħ2/2m)(pnF/L)2
(kF = 2p/lF = 2p/(2L/nF) = pnF/L)
here nF is the radius of a sphere in n-space that separated filled and empty orbitals
(7-1-1)
for the free electron gas with single electron quantum states (orbitals) (n,ms)
nF is determined by
N = SmsSn<nF f(en,t,m) (f(en,t,m) = 1 for n < nF at T = 0 K)
= 2.(1/8).(4pnF3/3) (Sms = 2s + 1 = 2 for electron spin s = ½)
= pnF3/3
(1/8 = (1/2)3 for positive octant of sphere in n-space)
Fermi energy eF in electron concentration n  N/V with associates Fermi temperature
TF (費米溫度)
eF = (ħ2/2m)(3p2N/V)2/3 (do not confuse n = N/V with quantum number (n,ms))
= m(0 K)  kTF
Total energy of system in ground state (T = 0 K)
U(0) = Sms.Sn<nF en
= 2.(1/8).(4p)0nFdn.n2.en
= (p3/2m)(ħ/L)20nFdn.n4
= (3/5)NeF
 (1/V)2/3
at constant N, U(0 K) = U0 gives a repulsive
contribution (increases with decreasing volume V)
(7-2)
(Density of states)(狀態密度)
Thermal average of X expressed as an integer over the state energy e
<X> = Sn,ms.f(en,t,m).X(n,ms)
(quantum states (n,ms) with energy en)
= dN(e).f(e,t,m).X(e)
(dN(e): number of states (orbitals))
= de.D(e).f(e,t,m).X(e)
where dN(e)  D(e)de is number of states of energy between e and e + de
D(e) is called density of states (density of orbitals)
Number of free electron states of energy less or equal to e
N(e) = 2.(1/8)(4pn3/3) = (V/3p2)(2m/ħ2)3/2.e3/2
take logarithm
log N = (3/2).log e + constant
take differential
dN/N = (3/2)(de/e)
density of states (狀態密度)
D(e)  dN(e)/de
= 3N(e)/2e
= (V/2p2)(2m/ħ2)3/2.e1/2
 e1/2
multiply by the Fermi-Dirac distribution
function f(e), D(e)f(e) is the density of occupied states (dashed line T ≠ 0 K)
(7-3)(12/18)
total number of electrons in the system at temperature T
N(T) = 0de.D(e).f(e,t,m) = N = constant
total energy (total kinetic energy of free electrons) at T
U(T) = 0de.e.D(e).f(e,t,m)
For the system in ground state (T = 0 K) all orbitals are filled up to Fermi energy eF
total number of electrons at 0 K
N(0) = 0eFde.D(e) = N
total energy of system at 0 K
U(0) = 0eFde.e.D(e)
(Heat capacity of electron gas)
(電子氣體之熱容量)
Increase of total energy of the Fermi gas of
electrons from 0 K to T
DU = U(T) – U(0)
= [0eF + eF]de.e.D(e).f(e) - 0eFde.e.D(e)
since total number of electrons N is constant
N.eF = [0eF + eF]de.eF.D(e).f(e) = 0eFde.eF.D(e)
rewrite DU = U(T) – U(0) – [N(T).eF - N(0).eF]
DU = 0eFde.(eF – e)[1 - f(e)].D(e) + eFde.(e – eF).f(e).D(e)
first term: energy needed to bring the electrons to eF from e < eF
second term: energy needed to bring the electrons from eF to e > eF
(7-4)
Constant volume heat capacity of electron gas (電子氣體熱容量) CV(electron) ≡ Ce
is differentiating DU with respect to temperature, the only temperature-dependent
term is f(e,t.m)
Ce = dU/dT = d(DU)/dT (V = constant)
= 0de.(e - eF)(df/dT)D(e)
since temperature of interest for simple metals
T << TF ~ 104 K (degenerate Fermi gas)
derivative df/dT is large only at e ~ eF
use D(e) ~ D(eF) at eF
Ce ~ D(eF)0de.(e – eF)(df/dT)
Fermi level m(t) ~ m(0)  eF for t << eF
replace m by eF
df/dT = k.[(e - eF)/t2].exp[(e – eF)/t]/{exp[(e – eF)/t] + 1}2
set x  (e – eF)/t
Ce = k2T.D(eF).-eF/tdx.x2.exp(x)/[exp(x) + 1]2
replace the lower limit by – since t << eF
and use -dx.x2.exp(x)/[exp(x) + 1]2 = p2/3
Ce = (p2/3).D(eF).k2T
use D(eF) = 3N/2eF = 3N/2kTF
Ce = (p2Nk/2TF).T
= gT  T
(7-5)(12/23)
(Fermi gas in metals)(金屬之費米氣體)
The alkali metals (鹼金屬) (Li, Na, K, Rb, Cs) and noble metals (貴金屬) (Cu, Ag, Au)
have one valence electron per atom, and valence electron becomes conduction electron
in metals
The conduction electrons in simple metal can be
treated as a free electron gas with Fermi energy
eF, Fermi temperature TF and Fermi velocity vF
eF = (ħ2/2m)(3p2n)2/3
= (1/2)mvF2
= kTF
where n = N/V is conduction electron concentration
(ex 1) Li (2s) n = 4.70 x 1022 cm-3, eF = 4.72 eV
vF = 1.29 x 106 m/s << c, TF = 5.48 x 104 K >> T
(ex 2) Cu (3d104s) n = 8.45 x 1022 cm-3, eF = 7.00 eV
vF = 1.57 x 106 m/s, TF = 8.12 x 104 K
Since T = 300 K << TF, the free electron gas in metal is a degenerate Fermi gas
(7-6)
Calculated free electron Fermi surface parameters (自由電子費米表面參數)
for some simple s/p metals (s, d10s, s2, d10s2, s2p, s2p2)
conduction electron concentration n = N/V [= 1/(4pr03/3), with rs  rn  r0/aH]
(7-7)
Measured constant volume heat capacity at low temperature for simple metals
CV = CV(electron) + CV(phonon)
= gT + AT3 (second term is the Debye T3 law for phonons in Chap 5)
(ex) Molar heat capacity C = Cp ~ CV of potassium (K) (鉀) plotted as C/T
versus T2
for T < 0.6 K << qD (= 91 K) << TF (= 2.5 x 104 K)
C/T = 2.08 + 2.57 T2
with observed intercept = g
g(exp) = 2.08 mJ/(mol.K2)
compared with calculated g
from free electron gas
g(calc) = p2R/2TF
= 1.69 mJ/mol-K2
ratio
g(exp)/g(cal) = 1.23 ~ 1
electron gas in K metal is close to a degenerate
Fermi gas but with a larger thermal effective
mass (熱有效質量) due to interactions
mth = 1.23 m
(7-8)
(White dwarf stars)(白矮星)
White dwarfs (degenerate dwarfs 簡併矮星) gave masses comparable to sun, but with
radius comparable to earth, atoms (mostly C and O) under high density are ionized into
free electrons and nuclei. Treat free electrons as a degenerate Fermi gas (T << TF)
(ex) Sirius B (a white dwarf) (天狼星伴星)(faint dot to lower left)
and companion star Sirius A (天狼星) (Hubble space telescope)
mass M ~ 2 x 1030 kg (~1 solar mass)
radius R ~ 2 x 107 m (~3 earth radius)
density d ~ 7 x 107 kg/m3
high electron concentration n ~ 1 x 1030 cm-3
Fermi energy eF ~ 3 x 105 eV
Fermi temperature TF ~ 3 x 109 K >> T(interior) ~ 107 K
relativistic effect significant, but not dominant (see exercise)
rest mass energy of electron mc2 ~ eF
(note) White dwarf in a binary system can accrete materials from
companion star, increasing mass and density. As mass approaches
Chandrasekhar limit of ~1.4 solar mass, this could lead to explosive
ignition of fusion into a type Ia supernova (超新星)
(7-9)
(SN1572, remnant of type Ia supernova)
2. Bose Gas and Einstein Condensation
(玻色氣體和愛因斯坦凝結)
Bose-Einstein condensation (BEC) (玻色-愛因思坦凝結): ground-orbital effect (基態軌
道效應) for a system of non-interacting free bosons (Bose gas)
(Chemical potential near absolute zero)(接近絕對零度之化學位)
Bose-Einstein distribution function for Bose gas
f(e,t,m) = 1/{exp[(e – m)t] – 1} = 1/(x-1 – 1)
let the ground state energy at zero e0 (= e(111)) = 0
occupancy of ground state (orbital) at temperature T
f(e0=0,t,m) = 1/[exp(-m/t) - 1]
when T 0, ground state occupancy equals to total number of particles of system
limt0f(0,t,m) = N
= limt01/[exp(-m/t) – 1]
~ 1/[1 – (m/t) – 1]
(exp(-m/t) ~ 1 – m/t + … for m/t < 1)
= -t/m
chemical potential near absolute zero
m(t,V,N) = -t/N < e0 = 0
(ex) for N = 1022 at T = 1 K, m(1 K) = -(1.38 x 10-23 J/K).(1 K)/1022 ~ -1.4 x 10-45 J
(7-10)
(ex) spacing of lowest and second lowest states of boson gas
energy levels of a free boson (N = 1) in a cube of volume V = L3
en = (ħ2/2M)(p/L)2(nx2 + ny2 + nz2)
= (ħ2/2M)(p/L)2n2
single boson quantum state with quantum number n = (nx,ny,nz) for spin = 0
the lowest excitation energy (let ground state energy e0 = e(111) = 0)
De = e(211) – e(111)
= 3.(ħ2/2M)(p/L)2
(ex) for 4He boson M(4He) = 4.00 u = 4 x 1.66 x 10-27 kg and V = L3 = 1 cm3
De = 2.48 x 10-37 J (or in temperature unit De/k = 1.80 x 10-14 K)
for N ~ 1022 atoms (n ~ 1022 cm-3) at T = 1 mK
m(1 mK) = -1.4 x 10-48 J << De, much closer to ground orbital e(111) = 0
occupancy of the first excited state e(211)
f(e(211)=De,t,m) = 1/exp[(De/t – m/t) – 1]
~ 1/[exp(De/t) - 1]
(ex) for 4He boson at T = 1 mK, fraction of N particles in this orbital is very small
f(De,1 mK)/N ~ 5 x 1010/1022 ~ 5 x 10-12 << f(e0=0,1 mK)/N ~ 1
(7-11)
(Orbital occupancy versus temperature)(隨溫度軌道佔據)
Number of free boson (spin = 0) states of energy less or equal to e for volume V
N(e) = (1/8)(4pn3/3)
(Sms = 1 for boson spin s = 0)
2
2
3/2
3/2
= (V/6p )(2M/ħ ) .e
density of states (orbitals)
D(e)  dN(e)/de
= (V/4p2)(2M/ħ2)3/2.e1/2
Total number of atom at temperature T
N = Sn.f(en,t)
= f(0,t) + 0 de.D(e).f(e,t)
= N0(t) + Ne(t)
N0: number of atoms in the ground
orbital e = 0 (condensed phase 凝結相)
Ne: number of atoms in the excited
orbitals (normal phase 正常相) since
D(e = 0) = 0 at ground orbital
(ex) plot of Bose-Einstein distribution
function f(e,t) for two temperatures
(7-12)
number of atoms at ground orbital e = 0
N0(t) = f(0,t)
= 1/[exp(0 - m)/t – 1]
= 1/[l-1 – 1]
number of particles in all excited orbitals
Ne(t) = (V/4p2)(2M/ħ2)3/2.0.de.e1/2/[l-1.exp(e/t) – 1]
= (V/4p2)(2M/ħ2)3/2.t3/2.0.dx.x1/2/[l-1.exp(x) – 1] (x  e/t)
At sufficiently low temperature
N0 >> 1
or l-1 = exp(-m/t) ~ exp(1/N0) ~ 1
set l = 1 in integer of Ne
0.dx.x1/2/[exp(x) – 1] = Ss=10.dx.x1/2e-sx
(let y = sx)

-3/2

1/2
= (Ss=1 .s ).0 .dy.y e-y = (Ss=1.s-3/2).G(-1/2)
= (2.612)(p1/2/2)
thus
Ne = (1.306.V/4)(2Mt/pħ2)3/2
= 2.612.nQV
or ne/nQ = neldB3 = 2.612 (ne = Ne/V, ldB = 1/nQ1/3 is thermal de Broglie wavelength)
fraction of atoms in excited orbitals when N0 >> 1
Ne/N = 2.612.nQ/n
(7-13)
(Einstein condensation temperature)(愛因思坦凝結溫度)
Einstein condensation temperature: temperature for which the number of atoms in
excited states is equal to total number of atoms
Ne(TE) = N (macroscopic quantum phenomena 巨觀量子現象 below TE)
N0(TE) ~ 0 for T > TE (macroscopic number only below TE)
TE  (2pħ2/M)(N/2.612V)2/3
(ex) liquid 4He (液氦四) treated as a noninteracting bose gas
M(4He) = 4.00 u = 4.00 x 1.66 x 10-27 kg
n = N/V = d/M = (0.125 g/cm3)/M = 1.88 x 1022 cm-3
TE = 3.1 K
particle number in excited orbitals at T
Ne/N ~ (t/tE)3/2
particle number in ground orbital at T
N0 = N – Ne
= N[1 – (t/tE)3/2]
particles N0 in the ground orbital below TE form
the condensed phase (凝結相): Bose-Einstein
condensation (BEC 玻色-愛因思坦凝結)
particles Ne: normal phase (正常相)
(7-14)(12/25)
(Liquid 4He)(液氦四)
Heat capacity C = Cp ~ CV of liquid 4He shows
a liquid phase transition from normal liquid
(正常液體) (liquid He I) to superfluid phase
(超流體) (liquid He II) at 2.17 K (l point)
l-line separates LHe I and LHe II
triple point (l point) at 2.17 K where LHe I,
LHe II and vapor coexists
another triple point at 1.74 K where solid (bcc)
is in equilibrium with LHe I and LHe II
liquid 4He behaves like is a weakly interacting
Bose gas with large quantum zero-point motion
1. large molar volume (大摩爾體積)
Vm(observed) = NA/n = 27.6 cm3/mol
> Vm(calculated) ~ 9 cm3/mol
2. low critical point (低臨界點)
Tc = 5.20 K (low binding energy)
3. stable liquid phase from 4.22 K to 0 K
at 1 atm (穩定液相), solid phase only
for p > 25 atm (see Chap 12 cryogenics)
(7-15)
(Superfluid phase He II)(超流相He II)
Rate of mass flow (質量流率) (in g/s)
under gravity through a fine hole for
L4He and L3He
sudden onset of high fluidity or superfluidity
(超流性) in LHe II phase at 2.17 K (1937)
Superfluidity in LHe II can be regarded as
a consequence of Bose-Einstein condensation
(BEC) in the weakly interacting system with a
condensation of a substantial fraction of 4He atoms into ground orbital
Two-fluid model (二流體模型): LHe II is a mixture of normal fluid component
(正常流體分量) in thermally excited orbitals and superfluid component (超流體
分量) in the ground orbital
for superfluid component (超流體分量)
zero viscosity h = 0 (零黏滯度) (transverse momentum flux Jpxz = -h.dvx/dz = 0)
zero entropy S = 0 (零熵(亂度)) (in the ground orbital with g = 1, S = k.log 1 = 0)
infinite thermal conductivity K =  (無限大熱傳導率) (see Chap 14 transport processes)
(energy flux density Juz = -K.dT/dz, no temperature gradient dT/dz = 0 in superfluid)
(7-16)(12/30)
Fountain effect (噴泉效應) (1938)
thermo-mechanical effect (熱機效應)
when radiation warms a superfluid,
expansion push up the free surface
of liquid forming a fountain
manifestation of two-fluid model of LHe II
Creeping effect (爬行效應)
LHe II will creep up the side of open
container to form a layer, 30 nm thick liquid
film (Rollin film) on surface (will creep out
of a open container and forming a drop below)
(7-17)
P. L. Kapitsa (1894-1984)
(discovery of superfluid in 1937)
1978 Nobel physics prize, “for his
basic inventions and discoveries
in the area of low temperature
physics”
(Quasiparticles and superfluidity)(似粒子和超流性)
The development of superfluid properties is not an automatic consequence of the
Bose-Einstein condensation of noninteracting Bose gas, interactions are needed
The superfluid component of LHe II behaves like a vacuum, N0 atoms are
condensed into the ground orbital and have no excitation energy (e0 = 0)
For a system of weakly interacting particles, low-lying excited states (低階激發態)
are called elementary excitations (基本激發) and in their particle aspect the states
are called quasiparticles (似粒子)
Liquid 4He is a quantum liquid (量子液體), not non-interacting boson gas
Necessary condition for superfluidity (超流性必要條件): phonon-like nature of
elementary excitations (基本激發有聲子特性)
Consider a slow neutron (慢中子) with mass M0 falling with velocity V
down a column of L4He at rest at T = 0 K (all atoms in ground orbital)
in order to generate an elementary excitation of energy ek with
momentum ħk, conservation of energy and momentum require
(1/2)M0V2 = (1/2)M0V’2 + ek
M0V = M0V’ + ħk
(7-18)
eliminate scattered neutron velocity V’ to obtain
ħV.k – (1/2M0)ħ2k2 = ek
lowest (Landau) critical velocity will occur when
direction of quasiparticle k is parallel to V
Vc = minimum of [ek + (1/2M0)ħ2k2]/ħk]
~ min [ek/ħk] (for slow neutron)
V < Vc: no excitation h = 0 (superfluid)
(1) for free excited atoms ek = ħ2k2/2M
Vc = min [ħk/2M] = 0 (no superfluidity)
(2) for low energy phonon ek = ħw = ħvsk
Vc = min [ħvsk/ħk] = vs (sound velocity)
(ex) Energy ek versus wavevector k of
elementary excitations in liquid He-II at
1.12 K from slow neutron inelastic scattering
solid line: Landau critical velocity
Vc = D/ħk0 ~ 50 m/s > 0 (roton 旋子, quantized vortex lines)
broken line: longitudinal phonon with sound vs = 237 m/s
(7-19)
Lev D. Laudau (1908-1968)
1962 Nobel physics prize
“for his pioneering theories of condensed
matter, especially liquid helium”
(Superfluidity in liquid 3He)(液氦三超流性)
3He
is a fermion with nuclear spin I = ½
3He
remains liquid for pressure less than
~34 atm (~3.4 MPa)
(note)
1 Pa = 1 N/m2 = 9.87 x 10-6 atm
1 atm = 1.01 MPa
L3He shows a phase transition from
normal liquid to superfluid phase
(liquid A or liquid B) for T < 2.5 mK
(1971)
(note)
to liquid B at 1.9 mK (1 atm)
(see Chap 12 Cryogenics for low
temperature cooling method)
Normal liquid 3He behaves like a weakly
interacting Fermi gas
(7-20)
In the superfluid phase, two 3He fermions form a loosely boson pair (Cooper pair,
mediated by spin fluctuation) with radius much larger than the average interatomic
spacing (pair radius r >> d)
nuclear spin I = 1, 2I + 1 = 3, mI = 0, ±1
orbital angular momentum L = 1 (p-wave pairing)
phase A: mixture of mI = 1 and -1
(in applied magnetic field Ba, the phase
divides into two components of opposite
nuclear magnetic moments, A and A1)
phase B: mixture of mI = 0, 1, -1
Fermionic condensation (費米子凝結):
BCS-like pairing + BEC
(Bardeen-Cooper-Schrieffer theory on
electron pairing in superconductors)
D. M. Lee, D. D. Osheroff, R. C. Richardson
1996 Nobel physics prize
“for the discovery of superfluidity in helium-3”
A. J. Leggett
2003 Noble physics prize, “for the theory of superfluids”
(7-21)
(Bose-Einstein condensation of dilute gas of alkali atoms)
(鹼金屬原子稀釋氣體之玻色-愛因思坦凝結)
Bose-Einstein condensation (BEC) of a
dilute boson gas (vapor) of 2000 87Rb
atoms (e + p + n = even, boson)
cooled by combination of laser cooling
and magnetic evaporation cooling
particle velocity distribution shows a
sharp peak centered around zero
speed v = 0 (condensate atoms in
the ground state)
TE = 170 nK
(left: T ~ 400 nK, center: T ~ 200 nK, right: T ~ 50 nK) (in 3D artificial color)
for the dilute gas at T, room-mean-square velocity vrms = (3kT/M)1/2 (see Chap 14)
particle concentration n = N/V ~ 1010 cm-3 due to small trapped volume V
n ~ nQ at ultra-low temperature
E. A. Cornell, W. Ketterle, C. E. Wieman, 2001 Nobel physics prize
“for the achievement of Bose-Einstein condensation in dilute gases of alkali atoms”
(7-22)
Magneto-optical trap (MOT)(磁光陷阱) (see Chap 12 Cryogenics for detail)
Doppler laser cooling (都普勒雷射冷卻)
1. three pairs of two circularly-polarized
counter-propagating laser beams with
a red-detuned frequency (紅調變頻率)
2. two solenoid magnets with currents in
opposite directions generate a spherical
quadrupole magnetic field (球四極磁場)
a position-dependent force always
pushes atoms toward trap center
Tmin ~ 100 mK
Magnetic evaporating cooling (磁蒸發冷卻)
hot atoms escaped from magnetic trap
Tmin ~ 50 nK
Steven Chu (朱棣文), C. Cohen-Tannoudji, W. D. Philips
1997 Nobel physics prize
“for development the methods to cool and trap atoms with laser light”
(7-23)
(Report and Exercise 7) (due day: 1/13/2010) (Final exam: 1/13)
1. Energy of relativistic Fermi gas (相對論費米氣體之能量). (TP Problem 7-2)
For an electron gas in the extreme relativistic limit with electron energy e ~ pc,
where p is momentum
(a) Show that Fermi energy of gas with concentration n = N/V with cube V = L3
eF = ħpc(3n/p)1/3
(b) Show that total energy of ground state of gas
U0 = (3/4)NeF
2. Liquid 3He (L3He) as a Fermi gas (液氦三當成費米氣體). (TP Problem 7-5)
Treat L3He as a gas of noninteracting fermions with nuclear spin I = ½.
(a) Calculate Fermi parameters eF, TF and vF at 0 K
(b) Calculate constant volume heat capacity CV at low temperature T << TF and
compare with experimental value CV = 2.89 NkT for T < 0.1 K
(7-24)
3. Energy, heat capacity, and entropy of degenerate boson gas.
(簡併玻色氣體能量,熱容量和熵(亂度)). (TP Problem 7-8)
Find expressions in temperature region T < TE for energy U(t), heat capacity CV(t),
and entropy s(t) of a gas of N nointeracting bosons of spin zero in volume V.
4. Boson gas in one dimension (一維玻色氣體). (TP Problem 7-9)
Calculate the integral Ne(t) for a one-dimensional gas of noninteracting bosons,
and show that the integral does not converge (a boson ground state condensate
does not form in one dimension).
(7-25)