2002.11.29 N96770 微奈米統計力學 1

上課地點 : 國立成功大學工程科學系越生講堂

(41X01 教室 )

N96770微奈米統計力學

2002.11.29 N96770 微奈米統計力學 2

Fermi-Dirac & Bose-Einstein Gases

Microcanonical Ensemble

Reference:

K. Huang, Statistical Mechanics, John Wiley & Sons, Inc., 1987.

OUTLINES

Grand Canonical Ensemble

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is an eigenvector of the position operators of all particles in a system.

is a vector and a state of a system.|

q|

Quick Review

)(| qq

is the wave function of the system in the state .|

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orthonormalA 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 .

: a complex number and a function of timenc

n : a set of quantum numbers2|| nc : the probability associated with n

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

nnc

}{ n

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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.

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: electrons, protons.

Examples: deuterons (2H), photons.

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the number of states of a system having an energy eigenvalue that is between E and E+E.

N(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.

p

ppnE

p

pnN

total energy

total number of particles

mpp 22 level (energy eigenvalue)

Lnhp 3/1VL

np = 0, 1, 2, … for bosons

np = 0, 1 for fermions

Microcanonical Ensemble

h : Planck’s constant

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g1

g2

g3

g4

cell

Each group is called a cell and has an average energy i.

The spectrum can be divided into groups of levels containing g1, g2, g3, g4,… subcells.

The levels p become continuous as the system volume V→∞.

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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i

ii wnW }{

The number of ways in which ni particles can be assigned to the i-th cell.

wi :

}{

}{)(in

inWEN

For Fermions

The number of particles in each of the gi subcell of the i-th cell is either 0 or 1.

)!(!

!

iii

i

i

ii ngn

g

n

gw

i iii

ii ngn

gnW

)!(!

!}{

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For Bosons

Each of the gi subcell of the i-th cell can be occupied by any number of particles.

)!1(!

)!1(

ii

iii gn

gnw

i ii

iii gn

gnnW

)!1(!

)!1(}{

Entropy :

}{

}{ln)(lnin

iBB nWkENkS

}{)( inWEN

}{ln iB nWkS

It can be shown that

: the set of occupation numbers that maximizes

}{ in }{ inW

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1)( ie

gn i

i (for bosons)

1)( ie

gn i

i (for fermions)

where )(1 TkB

: chemical potential

It can be shown that (by using Stirling’s approximation)

i

iiB

i

ie

egkS )(

)( 1ln1

)(

(for bosons)

i

iiB

i

ie

egkS )(

)( 1ln1

)(

(for fermions)

kB : Boltzmann’s constant

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Grand Canonical Ensemble

Partition function for ideal gases

}{

}{}{),(p

p

n

nEp engTVz

where p

ppp nnE }{

the occupation numbers {np} are subject to the condition :

Nnp

p

the number of states corresponding to {np} is

1}{ png for bosons and fermions

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Consider the grand partition function Z,

p n

n

n

n

n

n

n n

nn

N n

n

p

N n

nN

N

N

p

p

pp

p

ppp

e

ee

ee

e

ee

TVzeTVZ

)(

)()(

)()(

0 }{

)(

0 }{

0

1

11

0

00

0 1

1100

),(),,(

n = 0, 1, 2, … for bosons

n = 0, 1 for fermions

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ppe

TVZ )(1

1),,(

p

peTVZ )(1),,(

(for bosons)

(for fermions)

Equations of state : ),,(ln TVZTk

PV

B

pB

peTk

PV )(1ln

pB

peTk

PV )(1ln

(for bosons)

(for fermions)

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Now let V → ∞, then the possible values of p become continuous.

0

dpp

Equations of state for ideal Fermi-Dirac gases

0

)2(23

2

1ln4

dpephTk

P mp

B

0)2(

2

31

42 dp

e

p

hV

Nmp

Equations of state for ideal Bose-Einstein gases

e

Vdpep

hTk

P mp

B

1ln1

1ln4

0

)2(23

2

eV

edp

e

p

hV

Nmp

11

4

0)2(

2

3 2

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Let TmkB22 and e

Then equations of state for ideal Fermi-Dirac gases become

)(1

2/53f

Tk

P

B

)(1

2/33f

V

N

where

1

2/5

1

0

22/5

)1(1ln

4)(

2

j

jjx

jdxexf

1

2/3

1

2/52/3

)1()()(

j

jj

jff

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And equations of state for ideal Bose-Einstein gases become

1ln1

)(1

2/53 Vg

Tk

P

B

1

)(1

2/33 Vg

V

N

where

12/5

0

22/5

2

1ln4

)(j

jx

jdxexg

1

2/32/52/3 )()(j

j

jgg