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* Bc.S, Faculty of science Egypt south valley university, Application Mathematics

The Hamilton operator give2

2

pV

m! "

) )

The p is the operator of moment give

p i! %h 2

2

2H V

m! & % "

) )h

The Schrdinger equation give by H # =E # , E is the scalar quantity

corresponding to the energy of the system for ( , )q t# is the system and if apply

the more ( , )j

q t# one more subscripts .generally then where

j j'# ! (#

1-The Hamilton in one dimensional2 2

2

2

1

2 2kx

m x

)! & "

)

h

A simple harmonic oscillator and we can find by solve the equation

1

2E n*

+ ,! "- .

/ 0h

Where

1/ 2k

m*

+ ,

! - ./ 0

Consider the energy states of a particle in the 3-dimensional infinite we

1 22

2 2 2

, , 28x y zn n n x y z

hn n n

ma3 ! " "

, ,x y zn n n Space with coordination give

1 22

2 2 2 2

2

8x y z

man n n R

h

3" " ! !

We treat R a continuous variable in the volume of one in sphere of radiuses R3/ 2

23

2

1 4 8( )

8 3 6

maR

h

4 4 35 3

+ ,! ! - .

/ 0

The number of sates between 3 and

( , ) ( ) ( )* 3 3 5 3 3 5 3 6 ! " 6 & 3/ 2

21/ 2 2

2

8( , ) ( )

4

maO

h

4* 3 3 3 3 3

+ ,6 ! 6 " 6- .

/ 0

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For N-particle system2

2 2 2

218

N

xj yj zj

j

hE n n n

ma !! " "7

The Hamilton can be written for a system distribution and individual energyaj

3 where the superscript denotes the particle subscript the state

Canonical partition function become:-/

( , , ) jE KT

j

Q N V T e&

! 7

.......a b cj i j kE 3 3 3! " " " ( ....) / // /

, , ,...

( , , ) ........a b c ba ci j k ji k

KT KTKT KT

i j k i j i

Q N V T e e e e3 3 3 3 3 3& " " " && &! !7 7 7 7

( , , ) ...........a b cQ N V T q q q! The last equation very important result show that if we can write the N-

particle Hamilton as a sum of independent terms the useful application of

separation individual in equation is to the molecular partition function and

the equation show that molecular Hamilton can be approximation the

degrees of freedom of the molecule

.......m ol tran rota vibration electroinq q q q q! /transj kT

trans

j

q e 3&! 7 If N-body energy

.... .......ijk l i j k l E 3 3 3 3 ! " " " " The Q(N,V,T) give portion function

( .......) /( , , ) i j k l

kT

ijkl

Q N V T e3 3 3 3 & " " " "

! 7 Fermi-Dirac And Bose Einstein statistics

There are two cases to consider the evaluation the equation conical

partion estimation . the resultant distribution function in the case of Fermions

in called Fermi-Dirac and Bosons is called Bose- Einstein statistics the energy

( , )j

E N V the system containing N-molecules andk

3 is the molecular quantum

state when the system itself is in the quantum state with energyjE the set { kn } is

the entire system total energy

j k k

k

E n 3! 7

k

k

N n! 7

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We can write Q(V,T,N) as conical ensemble as

( , , )k k

j k

nE

j j

Q V T N e e8 3

8&

& 7! !7 7

This restriction turns but the be mathematics the grand canonical ensemble 9

give

0

( , , ) ( , , ) k

N

V T Q N V T e8:3

:;

!

9 ! 7 We can write the grand conical ensemble

0

( , , )k k k

k k

n n

N

V T e8 3

:

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Grand conical ensemble

( , , ) ln(1 )k

k

V T e 83< >7

1

k

kk

e

e

83

83< 7

So N-particle is given by

1

k

kk

eN

e

83

83

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22 2

2z z z

an k k n

a

4

4! ? !

22 2 2 2 2 2 2

2( )x y z x y zk k k k n n n

a

4! " " ! " "

The moment p is given

2

h hkP

< 4! !

So the total energy is given by2 2

2 2 2

, , 2( )

2 8x y zn n n x y z

P hn n n

m ma3 ! ! " "

The basic equation associative with the two fundamental distribution laws is

grand conical ensemble1(1 )k

k

e 839 ! >=

k

k

N n! 7

1

k

kk en

e

83

83

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1-Weakly Degenerate ideal Fermi Dirac gas :

We are derive series of fermions in a region where

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By the integration

1/ 2 1

3/ 200

1( 1)y l l ly

l

y e e dy<

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And give that3 2

2 3/ 2

( )

2a

B!

And can find3

a

31 2 2

3 3/ 2 3/ 2

20

2 2

a a aa & " !

3 3

3 3/ 2

1 1( )

4 2a

C D! & BE FG H

3 3 2 3 3

3 / 2 3 / 2

1 1 1( ) ( )

2 4 2

< @ @ @ C D

! B " B " & BE FG H

Now we want get the ratioP

KTby integration

(ln(1 )k

k

PV KT e83

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3/ 2 1

2 5/ 21

2 ( 1)l l

l

P m KT

KT h l

4

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0

3/ 2 1/ 2

2

1 22

1 1

k

k

m ed

V h e

83

833 3

<

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All other thermodynamic function for a weakly degenerate ideal gas of Bosons

following in similar way in thermodynamics function useful only for small

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At room temperature the quantity0

/K: is called Fermi temperature and id

denote by FT . Fermi temperature are typically of the order of thus and degrees

Kelvin

0

3/ 2

3/ 2

02 0

24

mE V d

h

:

4 3 3+ ,

! - ./ 0 $

0

3/ 2

3/ 2

0 02

2 24 |

5

mE V

h

:4 3 3

+ ,! - .

/ 0

0 0

2

5E N!

Where we have written0

E to emphasize that this T=0K result the equation give

in zero temperature distribution but not difficult to calculate correction to be

theses zero temperature and we are expansion in power of parameter

0/KTN :)!! . now we can write all thermodynamic quantities N,E, P,.. can be

written by

0( ) ( )I f h d3 3 3

;

! $

Where I=N give h by3/ 2

3/ 2

22( ) 4 mh Vh

3 4 3+ ,! - ./ 0

And we derive the E and N by the I by integral by part give

00

( ) ( ) | ( ) ( )I f H f h d3 3 3 3 3 ;

; O! &$ Now we can fin the I give

0( ) ( )I f h d3 3 3

;O! &$

Where

0( ) ( )h d

3

3 3 3! $ We using the fact ( ) 0f 3 ! and ( )f 3O is nonzero only for some small region

around 3 :! and we can find ( )H 3 by Taylor around 3 :!

22

2

1( ) ( ) ( ) ( ) .............

2

dH d H H H

d d3 :

3 :

3 : 3 : 3 : 3 3

!!

+ ,+ ,! " & " & "- .- .

/ 0 / 0

2

0 12

1( ) ( ) .......2

dH d H I H H L Ld d

3 :3 :

3 :3 3

!!

+ ,+ ,! ! " " "- .- ./ 0 / 0

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Where

0( ) ( )jj f d3 : 3 3

;O! & &$

The first integral1

1L !

And we can find1 2, ,....L L we may replace the lower limited &; so we can writhe

the probability function like

( )

( ) 2( )

(1 )

e

e

8 3 :

8 3 :

83

8

&

&O !

" But ( )8 3 :& ! and /d dx3 8!

20

1, 0,1, 2,......

(1 )

j x

j j x

x eL dx j

e8

;

! & !"$

Now we can find 1 0L ! 2 2

2 2 2 20

1

(1 ) 3

x

x

x eL dx

e

4

8 8

;

! & !"$

Now we can write the I by2 2

2

2( ) ( )

6

d HI H KT

d3 :

4:

3!

+ ,! " - .

/ 0

Now we can defined 0( ) ( )H h d

3

3 3 3!

$ and we calculate N in this case3/ 21/ 2

2

2( ) 4

mh V

h3 4 3

+ ,! - .

/ 0

3/ 2

3/ 2

2

3/ 221/ 2

2 2

8 2( )

3

2| 2

mH V

h

d H mV

d h3 :

4: :

4 :

3

&!

+ ,! - .

/ 0

+ ,! - .

/ 0

By replace I by N and give N now3/ 2 2

3/ 2 2

2

8 21 ( )

3 8

mN V

h

4 4: 8: &

C D+ ,! "- . E F

/ 0 G H

By equation

2/3 3/ 22

0

3

2 8

h N

m V:

4

+ , + ,! - . - .

/ 0 / 0

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2 /32

2

0

2 /32

20

22

0

1 ( )8

1 ( )8

1 ( )12

4: : 8:

4: : 8:

4: : 8:

&

&

&

&

C D! "E F

G H

C D! "E FG H

C D! &E F

G H

This equation show that : change slowly with temperature and is

approximately0

: throughout the entire solid state range of a metal

Now we can calculate E by the I looks like3/ 2

3/ 2

2

2( ) 4

mh V

h3 4 3

+ ,! - .

/ 0

so we can write3/ 2

5/ 2

2

3/ 22 1/ 2

2 2

4 2( )

5

26

mH V

h

d H m Vd h

43 3

4 33

+ ,! - .

/ 0

+ ,! - ./ 0

In the finally we get on E by series3/ 2 2

5/ 2 2

2

5/ 22

2

00

8 2 51 ( )

5 8

5

1 ( )8

mE V

h

E E

4 4: 8:

: 4

8::

&

&

C D+ ,! "- . E F

/ 0 G H

+ , C D

! "- . E FG H/ 0

By the ratio we fin E

22

0

51 ( )

12E E

48: &

C D! "E F

G H

The last equation give the energy in power series

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4-A strongly Degenerate ideal Bose-Einstein :we consider the situation when

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* Bc S Faculty of science Egypt south valley university Application Mathematics

In this equation where V M ; give 1

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