Quantum Free electron Theory Engineering Mathematics Notes | EduRev

Created by: Vinod Sharma

Engineering Mathematics : Quantum Free electron Theory Engineering Mathematics Notes | EduRev

 Page 1


Quantum free electron theory(Sommerfield)
• Particles of micro dimension like  the electrons are studied under 
quantum physics
• moving  electrons  inside a solid material can be associated with 
waves with a wave function  ?(x)  in  one dimension (?(r)  in 3D)
• Hence  its behaviors can be studied with the Schrödinger's 
equation (1D)
For a free particle  V=0, hence the equation reduces to
E
With   
0 ) ( ) (
2 ) (
2 2
2
x V E
m
x
x
?
, 0 ) (
) (
2
2
2
x k
x
x
2
,
2
,
2
2 2
2
2
h
m
k
E or
mE
k
?
?
?
K      
E
f
Page 2


Quantum free electron theory(Sommerfield)
• Particles of micro dimension like  the electrons are studied under 
quantum physics
• moving  electrons  inside a solid material can be associated with 
waves with a wave function  ?(x)  in  one dimension (?(r)  in 3D)
• Hence  its behaviors can be studied with the Schrödinger's 
equation (1D)
For a free particle  V=0, hence the equation reduces to
E
With   
0 ) ( ) (
2 ) (
2 2
2
x V E
m
x
x
?
, 0 ) (
) (
2
2
2
x k
x
x
2
,
2
,
2
2 2
2
2
h
m
k
E or
mE
k
?
?
?
K      
E
f
Sommerfield’s model
• As the freely moving electrons can not escape the surface of the 
material, they may be treated as  particles  confined (trapped)in a 
box
• Hence,  V(x) =0,    for    0<x<L                                          . e-
= 8, for    x=0  &  x=L           V(x)
0                     L       x
Sch’s  equation , 
Solution of the equation  can be obtained as 
?(x) = A sin kx + B cos kx
From boundary conditions , at x=o & x=L , ?(x)=0,   
we can get  B=0 and  k= ± np / L ,
Putting the normalization condition  we get  A=
0 ) (
2 ) (
2 2
2
x
mE
x
x
?
P= 0
?= 0
P= 0
?= 0
L
2
Page 3


Quantum free electron theory(Sommerfield)
• Particles of micro dimension like  the electrons are studied under 
quantum physics
• moving  electrons  inside a solid material can be associated with 
waves with a wave function  ?(x)  in  one dimension (?(r)  in 3D)
• Hence  its behaviors can be studied with the Schrödinger's 
equation (1D)
For a free particle  V=0, hence the equation reduces to
E
With   
0 ) ( ) (
2 ) (
2 2
2
x V E
m
x
x
?
, 0 ) (
) (
2
2
2
x k
x
x
2
,
2
,
2
2 2
2
2
h
m
k
E or
mE
k
?
?
?
K      
E
f
Sommerfield’s model
• As the freely moving electrons can not escape the surface of the 
material, they may be treated as  particles  confined (trapped)in a 
box
• Hence,  V(x) =0,    for    0<x<L                                          . e-
= 8, for    x=0  &  x=L           V(x)
0                     L       x
Sch’s  equation , 
Solution of the equation  can be obtained as 
?(x) = A sin kx + B cos kx
From boundary conditions , at x=o & x=L , ?(x)=0,   
we can get  B=0 and  k= ± np / L ,
Putting the normalization condition  we get  A=
0 ) (
2 ) (
2 2
2
x
mE
x
x
?
P= 0
?= 0
P= 0
?= 0
L
2
• Substituting  all the  values
• ?
n
(x) = (v 2/L) sin  npx /L
• &  E
n
(x) = h
2
k
2
/ 2m = h
2
p
2
n
2
/ 2m L
2
= h
2
n
2
/ 8mL
2
This shows that energy of the electrons inside the material is quantized and hence 
is discrete 
?3                                               n=3                                                     E3
?2                                               n=2                                                     E2                   
?1                                                 n=1                                                    E1
In  3D,  ?
n
(r) = (v 8/L
3
) sin  n
x
px /L  sin n
y
py /L  sin n
z
pz /L 
& E
n
(r) = h
2
k
2
/ 2m = (n
x
2
+n
y
2
+n
z
2
) h
2
p
2
/ 2m L
2 
Page 4


Quantum free electron theory(Sommerfield)
• Particles of micro dimension like  the electrons are studied under 
quantum physics
• moving  electrons  inside a solid material can be associated with 
waves with a wave function  ?(x)  in  one dimension (?(r)  in 3D)
• Hence  its behaviors can be studied with the Schrödinger's 
equation (1D)
For a free particle  V=0, hence the equation reduces to
E
With   
0 ) ( ) (
2 ) (
2 2
2
x V E
m
x
x
?
, 0 ) (
) (
2
2
2
x k
x
x
2
,
2
,
2
2 2
2
2
h
m
k
E or
mE
k
?
?
?
K      
E
f
Sommerfield’s model
• As the freely moving electrons can not escape the surface of the 
material, they may be treated as  particles  confined (trapped)in a 
box
• Hence,  V(x) =0,    for    0<x<L                                          . e-
= 8, for    x=0  &  x=L           V(x)
0                     L       x
Sch’s  equation , 
Solution of the equation  can be obtained as 
?(x) = A sin kx + B cos kx
From boundary conditions , at x=o & x=L , ?(x)=0,   
we can get  B=0 and  k= ± np / L ,
Putting the normalization condition  we get  A=
0 ) (
2 ) (
2 2
2
x
mE
x
x
?
P= 0
?= 0
P= 0
?= 0
L
2
• Substituting  all the  values
• ?
n
(x) = (v 2/L) sin  npx /L
• &  E
n
(x) = h
2
k
2
/ 2m = h
2
p
2
n
2
/ 2m L
2
= h
2
n
2
/ 8mL
2
This shows that energy of the electrons inside the material is quantized and hence 
is discrete 
?3                                               n=3                                                     E3
?2                                               n=2                                                     E2                   
?1                                                 n=1                                                    E1
In  3D,  ?
n
(r) = (v 8/L
3
) sin  n
x
px /L  sin n
y
py /L  sin n
z
pz /L 
& E
n
(r) = h
2
k
2
/ 2m = (n
x
2
+n
y
2
+n
z
2
) h
2
p
2
/ 2m L
2 
Fermi Level and Fermi Energy:
• Electrons are fermions or Fermi particles, which obey  Pauli’s 
exclusion principle
• At   0K  temperature the highest filled energy level is called the 
“Fermi Level”  & the energy possessed by the electrons in that 
level is “Fermi Energy”
• E
f
= h
2
k
f
2
/ 2m   or   k
f 
= (2mE
f
/ h
2
)
1/2
• In 3D electrons will fill up the k-space with one state 
accommodating   2  electrons each ( ??)
• If   N  is the total no. of electrons  &  is large , electrons will occupy 
a sphere  of  radius  k
f
, then  highest occupied state 
n
f
= N/2
From uncertainty principle                                                        electron states
?x ?p =h  or, ?x h?k = h  
Or,  L (h/2p) ?k = h  or, ?k = 2p / L
k
x
k
y
k
f
Page 5


Quantum free electron theory(Sommerfield)
• Particles of micro dimension like  the electrons are studied under 
quantum physics
• moving  electrons  inside a solid material can be associated with 
waves with a wave function  ?(x)  in  one dimension (?(r)  in 3D)
• Hence  its behaviors can be studied with the Schrödinger's 
equation (1D)
For a free particle  V=0, hence the equation reduces to
E
With   
0 ) ( ) (
2 ) (
2 2
2
x V E
m
x
x
?
, 0 ) (
) (
2
2
2
x k
x
x
2
,
2
,
2
2 2
2
2
h
m
k
E or
mE
k
?
?
?
K      
E
f
Sommerfield’s model
• As the freely moving electrons can not escape the surface of the 
material, they may be treated as  particles  confined (trapped)in a 
box
• Hence,  V(x) =0,    for    0<x<L                                          . e-
= 8, for    x=0  &  x=L           V(x)
0                     L       x
Sch’s  equation , 
Solution of the equation  can be obtained as 
?(x) = A sin kx + B cos kx
From boundary conditions , at x=o & x=L , ?(x)=0,   
we can get  B=0 and  k= ± np / L ,
Putting the normalization condition  we get  A=
0 ) (
2 ) (
2 2
2
x
mE
x
x
?
P= 0
?= 0
P= 0
?= 0
L
2
• Substituting  all the  values
• ?
n
(x) = (v 2/L) sin  npx /L
• &  E
n
(x) = h
2
k
2
/ 2m = h
2
p
2
n
2
/ 2m L
2
= h
2
n
2
/ 8mL
2
This shows that energy of the electrons inside the material is quantized and hence 
is discrete 
?3                                               n=3                                                     E3
?2                                               n=2                                                     E2                   
?1                                                 n=1                                                    E1
In  3D,  ?
n
(r) = (v 8/L
3
) sin  n
x
px /L  sin n
y
py /L  sin n
z
pz /L 
& E
n
(r) = h
2
k
2
/ 2m = (n
x
2
+n
y
2
+n
z
2
) h
2
p
2
/ 2m L
2 
Fermi Level and Fermi Energy:
• Electrons are fermions or Fermi particles, which obey  Pauli’s 
exclusion principle
• At   0K  temperature the highest filled energy level is called the 
“Fermi Level”  & the energy possessed by the electrons in that 
level is “Fermi Energy”
• E
f
= h
2
k
f
2
/ 2m   or   k
f 
= (2mE
f
/ h
2
)
1/2
• In 3D electrons will fill up the k-space with one state 
accommodating   2  electrons each ( ??)
• If   N  is the total no. of electrons  &  is large , electrons will occupy 
a sphere  of  radius  k
f
, then  highest occupied state 
n
f
= N/2
From uncertainty principle                                                        electron states
?x ?p =h  or, ?x h?k = h  
Or,  L (h/2p) ?k = h  or, ?k = 2p / L
k
x
k
y
k
f
• state in k-space of radius k
f
=
• =  K
f
3
L
3
/ 6p
2
• Hence no. of electrons  ‘N’ = 2 X ( K
f
3
L
3
/ 6p
2
)
•
• So, k
f
= {3p
2
(N/ V) }
1/3
• where  n = electron density = N/V = 1/ V (k
f
3
V / 3p
2
)
• Or,  n= 
3
3
2
3
4
L
k
f
N = k
f
3
V / 3p
2
k
f
= (3p
2
n) 
1/3
2
3
2
3
2 2
2
3
1
f
E
m
?
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