22

2

.,

)(.2.)2

(

int2

,1)exp(

)()(..

2),exp(

1

2:1

1

∆−∆⇒

−

==⇒

=⇒

=+

==

=

statesallowedL

kcontainsspacekinklength

spinincludingstatesallowedcontainsItspacekinL

elementlengthconsider

egersnwhereL

nk

ikL

xLxcbperiodicuse

m

kEikx

L

m

pHsystemelectronfreeDConsider

prob

h

π

π

ψψ

ψ

2/12

2/12

2/12

2

21)(

2

2

122

)(,,

22*)(

.int

.,

−

−−

==⇒

==∂∑∂=+==

=−=Σ=

∆−∆⇒

Em

Eglengthunitperstatesofdensity

EEmL

EEmL

EE

EEEeneandEenestatesallowedofNo

mELspacekinlength

LEEenergyuptostatesallowedofNo

Eenergywithpoenergyconstconsider

statesallowedL

kcontainsspacekinklength

h

hh

h

π

δπ

δπ

δδ

ππ

π

3)12(0101

2

3

260303,

,00,00:,0

.'

11,1:Reminder

)(2

1

)(2

1)()

2()()(

2:

)(,2

1

2

)2

2

2/3

3

2

222233

2/3

32/3

332/3

32/3222

=+=+=

===

==

++=−=

+++++++=

+=+=⇒+=

=++=

+++

+

+

+

++++++++

+++

ωλωλ

ωλωλ

ωλ

ωλω

ωωλω

ωωλω

hh

hh

h

hh

hh

h

hh

aaaaaW

aWthus

aaanoteWnneed

opthisbystategroundtoconnectedarestateswhichseesLet

nnnannna

aaaaaaaaaaaaaaaa

aaaam

mWaa

mXuse

Xm

WWXmm

pH

prob

...]332

11

2

3[0...3

2

3

3

11

2

310

...0

0

8

11]

3

4/38/9[)(

)2

1(:,

0301...

000

.02

3

2)12(0101

2/32/3

0 0

220

30

2

10

2

0 0

2

0

2/32/3

2

++−=+−−=

+−

+=

=+−=∆⇒

+=−

+−

=+−

+=∆

=+=+=

∑

∑

≠

≠

+++

λωλω

ωλω

ψ

ωλωω

ωλ

ω

ωλωλ

hh

hh

hhh

h

h

hh

nEE

Wn

E

nEuseEE

W

EE

W

EE

WnWE

areelementsmatrixotherall

aaaaaW

n n

nn n

Q

xdxgum

mhQ

Qqm

m

p

hdqdpQ

QfindsLet

Hdqdph

TVNQ

qmmpqpH

prob

N

NN

N

i

N

ii

=⇒

=−=

=

=−−=

−=

+=

∫

∫ ∫

∫

∑

∞

∞−

∞

∞−

∞

∞−

=

)(

1

1]2/exp[2

1:sin

1221

})2

exp()2

exp(1

{

'

)exp(1

),,(

)2

12/(),(

3

22

2/12/1

21

1

222

22

1

2

ωβ

σπσωββ

πωβπ

ωββ

β

ω

h

h

NkT

UCNkTQU

kTNk

T

FSentropy

V

FPpressure

kTkT

N

Fpotentialchemical

kTNkTQkTFenergyfreeHelmholtz

VV

TN

TN

TN

=

∂∂=⇒=

∂∂−=

+=

∂∂−=

=

∂∂−=

=

∂∂=

=−=

ln

]1)[ln(,

0,

)ln(,

)ln(ln,

)(

,

,

,

β

ω

ωµ

ω

ωβ

h

h

h

h