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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
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3)12(0101
2
3
260303,
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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
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pH
prob
...]332
11
2
3[0...3
2
3
3
11
2
310
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8
11]
3
4/38/9[)(
)2
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2/32/3
0 0
220
30
2
10
2
0 0
2
0
2/32/3
2
++−=+−−=
+−
+=
=+−=∆⇒
+=−
+−
=+−
+=∆
=+=+=
∑
∑
≠
≠
+++
λωλω
ωλω
ψ
ωλωω
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ω
ωλωλ
hh
hh
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n n
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Q
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p
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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
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