Download N96770 Statistical Mechanics class on 11/29/2002

N96770
微奈米統計力學
上課地點 : 國立成功大學工程科學系越生講堂
(41X01教室)
2002.11.29
N96770 微奈米統計力學
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OUTLINES

Fermi-Dirac & Bose-Einstein Gases

Microcanonical Ensemble

Grand Canonical Ensemble
Reference:
K. Huang, Statistical Mechanics, John Wiley & Sons, Inc., 1987.
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Quick Review
| 
is a vector and a state of a system.
| q
is an eigenvector of the position operators of all
particles in a system.
 q |    (q)
is the wave function of the system in the state | .
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At any instant of time the wave function of a truly isolated
system can be expressed as a complete orthonormal set of
stationary wave functions { n }
   cn  n
n
cn : a complex number and a function of time
n : a set of quantum numbers
| cn |2 : the probability associated with n
orthonormal
A subset of a vector space V {v1,…vk}, with the inner product <,>, is called
orthonormal if <vi,vj> = 0 when i ≠ j. That is, the vectors are mutually
perpendicular. Moreover, they are all required to have length one: |vi| = 1 .
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Ideal Gases
Two types of a system composed of N identical particles:
Fermi-Dirac system
The wave functions are antisymmetric under an
interchange of any pair of particle coordinates.
Particles with such characteristics are called fermions.
Examples: electrons, protons.
Bose-Einstein system
The wave functions are symmetric under an interchange of
any pair of particle coordinates.
Particles with such characteristics are called bosons.
Examples: deuterons (2H), photons.
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Microcanonical Ensemble
N(E) : the number of states of a system having an energy
eigenvalue that is between E and E+E.
A state of an ideal system can be specified by a set of
occupation numbers {np} so that there are np particles
having the momentum p in the state.
E   pnp
total energy
N   np
total number
of particles
p
p
np = 0, 1, 2, … for bosons
np = 0, 1
for fermions
 p  p 2 2m level (energy eigenvalue)
p  nh L
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L  V 1/ 3
h : Planck’s constant
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The levels p become continuous as the
system volume V→∞.
g4
The spectrum can be divided into groups of
levels containing g1, g2, g3, g4,… subcells.
g3
Each group is called a cell and has an
average energy i.
g2
The occupation number ni is the sum
of np over all levels in the i-th cell.
W{ni} is the number of states
corresponding to the set of
occupation number {ni}.
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cell
g1
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N ( E )  W {ni }
{ni }
W {ni }   wi
i
wi : The number of ways in which ni particles can be
assigned to the i-th cell.
For Fermions
The number of particles in each of the gi subcell of the i-th
cell is either 0 or 1.
 gi 
gi !


wi    
 ni  ni !( g i  ni )!
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gi !
W {ni }  
i ni !( g i  ni )!
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For Bosons
Each of the gi subcell of the i-th cell can be occupied by any
number of particles.
wi 
Entropy :
(ni  g i  1)!
ni !( g i  1)!
W {ni }  
i
(ni  g i  1)!
ni !( g i  1)!



S  k B ln N ( E )  k B ln   W {ni }
 {ni }

It can be shown that N ( E )  W {ni }
{ni }: the set of occupation numbers that maximizes W {ni }
 S  k B ln W {ni }
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ni 
ni 
where
gi
e
 ( i   )
1
(for bosons)
1
(for fermions)
gi
e
 ( i   )
  1 (k BT )
 : chemical potential
kB : Boltzmann’s constant
It can be shown that (by using Stirling’s approximation)
  ( i   )

S  k B  g i   ( i   )
 ln 1  e   ( i   ) 
1
e

i
(for bosons)
  ( i   )

S  k B  g i   ( i   )
 ln 1  e   ( i   ) 
1
e

i
(for fermions)


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

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Grand Canonical Ensemble
Partition function for ideal gases
z (V , T )   g{n p }e
 E {n p }
{n p }
E{n p }    p n p
where
p
the occupation numbers {np} are subject to the condition :
n
p
N
p
the number of states corresponding to {np} is
g{n p }  1
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for bosons and fermions
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Consider the grand partition function Z,

Z (  , V , T )   e N z (V , T )
N 0

  e
N
e
  pn p
p
N  0 {n p }


   e
N  0 {n p } p

   e
n0

  ( p   ) n p
 e
  (  0   ) n0

  (  1   ) n1
n1

  e   (  0   )
 n0



n0

  (1   )
e
 
  n1


n1


  ( p   ) n 
   e

p  n

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

N96770 微奈米統計力學



n = 0, 1, 2, … for bosons
n = 0, 1
for fermions
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Z (  ,V , T )  
1
1 e
  ( p   )
Z (  , V , T )  1  e
  ( p   )
p
(for bosons)
(for fermions)
p
PV
 ln Z (  ,V , T )
k BT
Equations of state :

PV
  (   )
  ln 1  e p
k BT
p

PV
  (   )
  ln 1  e p
k BT
p
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

N96770 微奈米統計力學
(for bosons)
(for fermions)
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Now let V → ∞, then the possible
values of p become continuous.

      dp
p
0
Equations of state for ideal Fermi-Dirac gases
P
4
 3
k BT h

N 4
 3
V h

p
2

ln 1  e
  ( p 2 2m )
dp
0
 e
p2
( p 2 2 m  )
0
1
dp
Equations of state for ideal Bose-Einstein gases
P
4
 3
k BT
h

p
2

ln 1  e
  ( p 2 2 m  )
0


1
dp  ln 1  e 
V


N 4
p2
e 
 3   ( p 2 2 m  )
dp 
V h 0e
V 1  e 
1

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
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  2 2 mkBT
Let
  e 
and
Then equations of state for ideal Fermi-Dirac gases become
P
1
 3 f 5 / 2 ( )
k BT 
N
1
 3 f 3 / 2 ( )
V 
where
f 5 / 2 ( ) 
4

x


0
2

ln 1  e
 x2

(1) j 1  j
dx  
5/ 2
j
j 1



(1) j 1  j
f 3 / 2 ( )  
f 5 / 2 ( )  

j 3/ 2
j 1
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And equations of state for ideal Bose-Einstein gases become
P
1
1
 3 g 5 / 2 ( )  ln 1   
k BT 
V
N 1

 3 g 3 / 2 ( ) 
V 
V 1   
where
g 5 / 2 ( ) 
4

x


0
2

ln 1  e
 x2
dx   j

j 1
j
5/ 2


j
g 3 / 2 ( )  
g 5 / 2 ( )   3 / 2

j 1 j
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