Physics barriers and tunneling
Mohamed_Anwarvic
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Dec 10, 2012
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Slide Content
Ain Shams University
Mathematics and Engineering Physics Department
Pre-Junior Communication Systems Engineering Students
Lecture 11
Modern Physics and Quantum Mechanics Course (EPHS 240)
9 December 2009
Dr. Hatem El-Refaei
Mathematics and Engineering Physics Department
Pre-Junior Communication Systems Engineering Students
Lecture 11
Modern Physics and Quantum Mechanics Course (EPHS 240)
9 December 2009
Dr. Hatem El-Refaei
[email protected] Dr. Hatem El-Refaei 1
Contents
MInfinite barrier
MFinite barrier
MQuantum tunnelling
Contents
MInfinite barrier
MFinite barrier
MQuantum tunnelling
[email protected] Dr. Hatem El-Refaei 2
Note
MAll problems today are unbounded problem, i.e. the
particle is not confined in a certain region, so:
MWe will not be able to do the normalization condition.
MTherefore, we will not be able to solve for all unknowns.
MTherefore, we will not get a characteristic equation.
MTherefore, energy levels are not quantized, and all energies
are possible.
MBut still there are a lot of important characteristics to
understand and learn today.
Note
MAll problems today are unbounded problem, i.e. the
particle is not confined in a certain region, so:
MWe will not be able to do the normalization condition.
MTherefore, we will not be able to solve for all unknowns.
MTherefore, we will not get a characteristic equation.
MTherefore, energy levels are not quantized, and all energies
are possible.
MBut still there are a lot of important characteristics to
understand and learn today.
[email protected] Dr. Hatem El-Refaei 3
Infinite barrier
Infinite barrier
[email protected] Dr. Hatem El-Refaei 4
Potential step of infiniteheight and infinitewidth
MSince the barrier height is infinite, incident particles can’t
penetratethroughit,andparticlesreflectback.
MSo, there is zero probability of finding the particle inside
thestepbarrier.
MHere,theQMsolutionleadstothesameclassicalsolution.
Energy
∞ ∞
x
E
Potential step of infiniteheight and infinitewidth
MSince the barrier height is infinite, incident particles can’t
penetratethroughit,andparticlesreflectback.
MSo, there is zero probability of finding the particle inside
thestepbarrier.
MHere,theQMsolutionleadstothesameclassicalsolution.
Energy
∞ ∞
x
E
[email protected] Dr. Hatem El-Refaei 5
Potential step of infiniteheight and infinite
width
()
jkxjkx
BeAex
−
+= ψ
( )
−−
−
+=Ψ
t
E
kxjt
E
kxj
BeAetx
hh
,
0
2
22
2
=+
I
IE
m
dx
dψ
ψ
h
Backward
propagating wave
Forward
propagating wave
∞=U
()0=xψ
h
mE
k
2
=
E
Energy, ψ
∞ ∞
x
0
Potential step of infiniteheight and infinite
width
()
jkxjkx
BeAex
−
+= ψ
( )
−−
−
+=Ψ
t
E
kxjt
E
kxj
BeAetx
hh
,
0
2
22
2
=+
I
IE
m
dx
dψ
ψ
h
Backward
propagating wave
Forward
propagating wave
∞=U
()0=xψ
h
mE
k
2
=
E
Energy, ψ
∞ ∞
x
0
[email protected] Dr. Hatem El-Refaei 6
Potential step of infiniteheight and infinite
width
( )( )
+−
=== 00 xxψ ψ
0=+BA
Energy, ψ
∞ ∞
x
0
∞=U
()0=xψ
E
()
jkxjkx
BeAex
−
+= ψ
0
2
22
2
=+
I
IE
m
dx
dψ
ψ
h
h
mE
k
2
=
Potential step of infiniteheight and infinite
width
( )( )
+−
=== 00 xxψ ψ
0=+BA
Energy, ψ
∞ ∞
x
0
∞=U
()0=xψ
E
()
jkxjkx
BeAex
−
+= ψ
0
2
22
2
=+
I
IE
m
dx
dψ
ψ
h
h
mE
k
2
=
[email protected] Dr. Hatem El-Refaei 7
Potential step of infiniteheight and infinite
width
AB−=
() ()kxjAx sin2=ψ
Energy, ψ
∞ ∞
x
0
∞=U
()0=xψ()
jkxjkx
AeAex
−
−= ψ
()( )
jkxjkx
eeAx
−
−= ψ
Sin(x) E
0
2
22
2
=+
I
IE
m
dx
dψ
ψ
h
Potential step of infiniteheight and infinite
width
AB−=
() ()kxjAx sin2=ψ
Energy, ψ
∞ ∞
x
0
∞=U
()0=xψ()
jkxjkx
AeAex
−
−= ψ
()( )
jkxjkx
eeAx
−
−= ψ
Sin(x) E
0
2
22
2
=+
I
IE
m
dx
dψ
ψ
h
[email protected] Dr. Hatem El-Refaei 8
Potential step of infiniteheight and infinite
width
AB−=
Reflectivity 1
Re
2
===
∗
∗
AA
BB
AmplitudeIncident
Amplitudeflected
R
All the incident particle stream is reflected back.
Energy, ψ
∞ ∞
x
0
Sin(x) E
Potential step of infiniteheight and infinite
width
AB−=
Reflectivity 1
Re
2
===
∗
∗
AA
BB
AmplitudeIncident
Amplitudeflected
R
All the incident particle stream is reflected back.
Energy, ψ
∞ ∞
x
0
Sin(x) E
[email protected] Dr. Hatem El-Refaei 9
Potential step of infiniteheight and infinite
width
MSince the barrier extends to infinity in the x direction, no particle can penetrate
through the whole barrier. From phenomenological understanding, as x→∞,
ψ→0.
Energy, ψ
∞ ∞
x
0
Sin(x) E
Potential step of infiniteheight and infinite
width
MSince the barrier extends to infinity in the x direction, no particle can penetrate
through the whole barrier. From phenomenological understanding, as x→∞,
ψ→0.
Energy, ψ
∞ ∞
x
0
Sin(x) E
[email protected] Dr. Hatem El-Refaei 10
Finite barrier
Finite barrier
[email protected] Dr. Hatem El-Refaei 11
Potential step of finite height and infinite
width
MAsthestepheightU
ogetssmaller(butstillE<U
o),
the penetration of the particles inside the step
increases, but finally no particles will succeed to
travelthroughthewholesteptox→∞.
Energy
∞
x
U
o
0
E<U
o
E
Potential step of finite height and infinite
width
MAsthestepheightU
ogetssmaller(butstillE<U
o),
the penetration of the particles inside the step
increases, but finally no particles will succeed to
travelthroughthewholesteptox→∞.
Energy
∞
x
U
o
0
E<U
o
E
[email protected] Dr. Hatem El-Refaei 12
Potential step of finite height and infinite
width
0
()
jkxjkx
I
BeAex
−
+=
ψ
0
2
22
2
=+
I
I
E
m
dx
d
ψ
ψ
h ( ) 0
2
22
2
=−+
IIo
II
UE
m
dx
d
ψ
ψ
h
()
xx
II
DeeCx
αα ψ +=
−
Energy
∞
x
U
o
MWe have 4 unknowns (A,B,C, and D) and 3 equations:
MFiniteness of ψat x=∞
MContinuity of ψat x=0
MContinuity of ∇ψat x=0
E<U
o
( )
0
2
>
−
=
h
EUm
o
α
h
mE
k
2
=
E
Potential step of finite height and infinite
width
0
()
jkxjkx
I
BeAex
−
+=
ψ
0
2
22
2
=+
I
I
E
m
dx
d
ψ
ψ
h ( ) 0
2
22
2
=−+
IIo
II
UE
m
dx
d
ψ
ψ
h
()
xx
II
DeeCx
αα ψ +=
−
Energy
∞
x
U
o
MWe have 4 unknowns (A,B,C, and D) and 3 equations:
MFiniteness of ψat x=∞
MContinuity of ψat x=0
MContinuity of ∇ψat x=0
E<U
o
( )
0
2
>
−
=
h
EUm
o
α
h
mE
k
2
=
E
[email protected] Dr. Hatem El-Refaei 13
Potential step of finite height and infinite
width
0
()
jkxjkx
I
BeAex
−
+=
ψ
0
2
22
2
=+
I
I
E
m
dx
d
ψ
ψ
h ( ) 0
2
22
2
=−+
IIo
II
UE
m
dx
d
ψ
ψ
h
()
xx
II
DeeCx
αα ψ +=
−
Energy
∞
x
U
o
E<U
o
MTheconditionthatψ(x)mustbefiniteasx→∞,leadstoD=0
( )
0
2
>
−
=
h
EUm
o
α
h
mE
k
2
=
E
Potential step of finite height and infinite
width
0
()
jkxjkx
I
BeAex
−
+=
ψ
0
2
22
2
=+
I
I
E
m
dx
d
ψ
ψ
h ( ) 0
2
22
2
=−+
IIo
II
UE
m
dx
d
ψ
ψ
h
()
xx
II
DeeCx
αα ψ +=
−
Energy
∞
x
U
o
E<U
o
MTheconditionthatψ(x)mustbefiniteasx→∞,leadstoD=0
( )
0
2
>
−
=
h
EUm
o
α
h
mE
k
2
=
E
[email protected] Dr. Hatem El-Refaei 14
Potential step of finite height and infinite
width
0
()
jkxjkx
I
BeAex
−
+=
ψ
0
2
22
2
=+
I
I
E
m
dx
d
ψ
ψ
h ( ) 0
2
22
2
=−+
IIo
II
UE
m
dx
d
ψ
ψ
h
()
x
II
eCx
αψ
−
=
Energy
∞
x
U
o
E<U
o
MTheconditionthatψ(x)mustbefiniteasx→∞,leadstoD=0
( )
0
2
>
−
=
h
EUm
o
α
h
mE
k
2
=
E
Potential step of finite height and infinite
width
0
()
jkxjkx
I
BeAex
−
+=
ψ
0
2
22
2
=+
I
I
E
m
dx
d
ψ
ψ
h ( ) 0
2
22
2
=−+
IIo
II
UE
m
dx
d
ψ
ψ
h
()
x
II
eCx
αψ
−
=
Energy
∞
x
U
o
E<U
o
MTheconditionthatψ(x)mustbefiniteasx→∞,leadstoD=0
( )
0
2
>
−
=
h
EUm
o
α
h
mE
k
2
=
E
[email protected] Dr. Hatem El-Refaei 15
Potential step of finite height and infinite
width
0
()
jkxjkx
I
BeAex
−
+=
ψ
0
2
22
2
=+
I
I
E
m
dx
d
ψ
ψ
h ( ) 0
2
22
2
=−+
IIo
II
UE
m
dx
d
ψ
ψ
h
()
x
II
eCx
αψ
−
=
Energy
∞
x
U
o
MWe have 3 unknowns (A,B,C) and 2 equations:
MContinuity of ψ
MContinuity of ∇ψ
MThus, the best we can get is the ratio between parameters.
E<U
o
( )
0
2
>
−
=
h
EUm
o
α
h
mE
k
2
=
E
Potential step of finite height and infinite
width
0
()
jkxjkx
I
BeAex
−
+=
ψ
0
2
22
2
=+
I
I
E
m
dx
d
ψ
ψ
h ( ) 0
2
22
2
=−+
IIo
II
UE
m
dx
d
ψ
ψ
h
()
x
II
eCx
αψ
−
=
Energy
∞
x
U
o
MWe have 3 unknowns (A,B,C) and 2 equations:
MContinuity of ψ
MContinuity of ∇ψ
MThus, the best we can get is the ratio between parameters.
E<U
o
( )
0
2
>
−
=
h
EUm
o
α
h
mE
k
2
=
E
[email protected] Dr. Hatem El-Refaei 16
Potential step of finite height and infinite
width
() ()00 === xx
III ψψ
Continuity ofψ
CBA=+
Continuity of dψ/dx
00 ==
=
x
II
x
Idx
d
dx
d
ψψ
( ) CBAjk α−=−
C
k
jA
+=
α
1
2
1
C
k
jB
−=
α
1
2
1
∞
x
U
o
Energy
E<U
o
( )
0
2
>
−
=
h
EUm
o
α
h
mE
k
2
=
0
E
()
jkxjkx
I
BeAex
−
+=
ψ ()
x
II
eCx
αψ
−
=
Potential step of finite height and infinite
width
() ()00 === xx
III ψψ
Continuity ofψ
CBA=+
Continuity of dψ/dx
00 ==
=
x
II
x
Idx
d
dx
d
ψψ
( ) CBAjk α−=−
C
k
jA
+=
α
1
2
1
C
k
jB
−=
α
1
2
1
∞
x
U
o
Energy
E<U
o
( )
0
2
>
−
=
h
EUm
o
α
h
mE
k
2
=
0
E
()
jkxjkx
I
BeAex
−
+=
ψ ()
x
II
eCx
αψ
−
=
[email protected] Dr. Hatem El-Refaei 17
Potential step of finite height and infinite
width
C
k
jA
+=
α
1
2
1
C
k
jB
−=
α
1
2
1
Reflectivity 1
11
11
Re
2
=
−
+
+
−
===
∗
∗
k
j
k
j
k
j
k
j
AA
BB
AmplitudeIncident
Amplitudeflected
R
αα
αα
Energy
E<U
o
( )
0
2
>
−
=
h
EUm
o
α
h
mE
k
2
=
0
E
Potential step of finite height and infinite
width
C
k
jA
+=
α
1
2
1
C
k
jB
−=
α
1
2
1
Reflectivity 1
11
11
Re
2
=
−
+
+
−
===
∗
∗
k
j
k
j
k
j
k
j
AA
BB
AmplitudeIncident
Amplitudeflected
R
αα
αα
Energy
E<U
o
( )
0
2
>
−
=
h
EUm
o
α
h
mE
k
2
=
0
E
[email protected] Dr. Hatem El-Refaei 18
Potential step of finite height and infinite
width
jkxjkx
Ce
k
jCe
k
j
−
−+
+=αα
1
2
1
1
2
1
( )
( ) ( )
≥
≤−
=
−
0
0sincos
xeC
xkxC
k
kxC
x
xα
α
ψ
()
jkxjkx
I
BeAex
−
+=
ψ
∞
x
U
o
0
C
( )
−
+
+
=
−−
22
jkxjkxjkxjkx
I
ee
C
k
j
ee
Cxα
ψ
Energy
Potential step of finite height and infinite
width
jkxjkx
Ce
k
jCe
k
j
−
−+
+=αα
1
2
1
1
2
1
( )
( ) ( )
≥
≤−
=
−
0
0sincos
xeC
xkxC
k
kxC
x
xα
α
ψ
()
jkxjkx
I
BeAex
−
+=
ψ
∞
x
U
o
0
C
( )
−
+
+
=
−−
22
jkxjkxjkxjkx
I
ee
C
k
j
ee
Cxα
ψ
Energy
[email protected] Dr. Hatem El-Refaei 19
Potential step of finite height and infinite
width
MSince the barrier height is finite, particles can penetrate partially
inthevicinityofthepotentialstep,andthentheyreflectback.
MSo, there is a finite probability of finding the particle in the
classicallyforbiddenposition.
MHere,theQMsolutionisdifferentfromtheclassicalone.
∞
x
U
o
0
C
Energy
Potential step of finite height and infinite
width
MSince the barrier height is finite, particles can penetrate partially
inthevicinityofthepotentialstep,andthentheyreflectback.
MSo, there is a finite probability of finding the particle in the
classicallyforbiddenposition.
MHere,theQMsolutionisdifferentfromtheclassicalone.
∞
x
U
o
0
C
Energy
[email protected] Dr. Hatem El-Refaei 20
Potential step of finite height and infinite
width
Case 2
Case 2
x
0
Energy
Case 3
Case 3
E<U
o
( )
0
2
>
−
=
h
EUm
o
α
MAs the potential barrier height increases (U
o
increases)(fromcase2tocase 3),αalsoincreases,
andthustheexponentialfunctiondiesquickerinside
thebarrier.Thusitbecomeslessprobabletofindthe
particleinsidethebarrier.
()
x
eCx
α
ψ
−
=
Potential step of finite height and infinite
width
Case 2
Case 2
x
0
Energy
Case 3
Case 3
E<U
o
( )
0
2
>
−
=
h
EUm
o
α
MAs the potential barrier height increases (U
o
increases)(fromcase2tocase 3),αalsoincreases,
andthustheexponentialfunctiondiesquickerinside
thebarrier.Thusitbecomeslessprobabletofindthe
particleinsidethebarrier.
()
x
eCx
α
ψ
−
=
[email protected] Dr. Hatem El-Refaei 21
Potential step of finite height and infinite
width
MIfthebarrierheight(U
o)iskeptconstant,buttheparticle
energyincreases(providedE<U
o),thusαdecreases,and
the particle exponential function dies slower inside the
barrier. Hence, it becomes more probable to find the
particleinthevicinityofthebarrieredge.
x
0
Energy
E<U
o
( )
0
2
>
−
=
h
EUm
o
α
()
x
eCx
α
ψ
−
=
Potential step of finite height and infinite
width
MIfthebarrierheight(U
o)iskeptconstant,buttheparticle
energyincreases(providedE<U
o),thusαdecreases,and
the particle exponential function dies slower inside the
barrier. Hence, it becomes more probable to find the
particleinthevicinityofthebarrieredge.
x
0
Energy
E<U
o
( )
0
2
>
−
=
h
EUm
o
α
()
x
eCx
α
ψ
−
=
[email protected] Dr. Hatem El-Refaei 23
Potential barrier of finiteheight and finite
width
MA stream of particles incident on a finite width and height
potentialbarrierwith
E<U
o.
MPart of the incident stream will succeed to penetrate through the
barrierandappearontheotherside,thisisthetransmittedstream.
Theotherpartwillreflectbackformingthereflectedstream.
MNote,asingleparticledoesn’tsplitintotwo.
x
U
o
E<U
o
E
Energy
Potential barrier of finiteheight and finite
width
MA stream of particles incident on a finite width and height
potentialbarrierwith
E<U
o.
MPart of the incident stream will succeed to penetrate through the
barrierandappearontheotherside,thisisthetransmittedstream.
Theotherpartwillreflectbackformingthereflectedstream.
MNote,asingleparticledoesn’tsplitintotwo.
x
U
o
E<U
o
E
Energy
[email protected] Dr. Hatem El-Refaei 24
Potential barrier of finiteheight and finite
width
()
jkxjkx
I
BeAex
−
+=
ψ
0
2
22
2
=+
I
IE
m
dx
d
ψ
ψ
h ( ) 0
2
22
2
=−+
IIo
IIUE
m
dx
d
ψ
ψ
h
()
xx
II
eDeCx
γγ
ψ
−
+= ()
jkxjkx
III
FeGex
−
+=
ψ
0
2
22
2
=+
III
IIIE
m
dx
d
ψ
ψ
h
Energy
x
U
o
0a
E<U
o
( )
0
2
>
−
=
h
EUm
o
γ
h
mE
k
2
=
MWe have 6 unknowns (A,B,C,D,G,F) and 5 boundary conditions
MContinuity of ψand ∇ψat x=0.
MContinuity of ψand ∇ψat x=a.
MOnly a forward propagating wave on the right hand side of the barrier.
Potential barrier of finiteheight and finite
width
()
jkxjkx
I
BeAex
−
+=
ψ
0
2
22
2
=+
I
IE
m
dx
d
ψ
ψ
h ( ) 0
2
22
2
=−+
IIo
IIUE
m
dx
d
ψ
ψ
h
()
xx
II
eDeCx
γγ
ψ
−
+= ()
jkxjkx
III
FeGex
−
+=
ψ
0
2
22
2
=+
III
IIIE
m
dx
d
ψ
ψ
h
Energy
x
U
o
0a
E<U
o
( )
0
2
>
−
=
h
EUm
o
γ
h
mE
k
2
=
MWe have 6 unknowns (A,B,C,D,G,F) and 5 boundary conditions
MContinuity of ψand ∇ψat x=0.
MContinuity of ψand ∇ψat x=a.
MOnly a forward propagating wave on the right hand side of the barrier.
[email protected] Dr. Hatem El-Refaei 25
Potential barrier of finiteheight and finite
width
()
jkxjkx
I
BeAex
−
+=
ψ
0
2
22
2
=+
I
IE
m
dx
d
ψ
ψ
h ( ) 0
2
22
2
=−+
IIo
IIUE
m
dx
d
ψ
ψ
h
()
xx
II
eDeCx
γγ
ψ
−
+= ()
jkxjkx
III
FeGex
−
+=
ψ
0
2
22
2
=+
III
IIIE
m
dx
d
ψ
ψ
h
Energy
x
U
o
0a
E<U
o
MWe have 6 unknowns (A,B,C,D,G,F) and 5 boundary conditions
MContinuity of ψand ∇ψat x=0.
MContinuity of ψand ∇ψat x=a.
MOnly a forward propagating wave on the right hand side of the barrier.
Potential barrier of finiteheight and finite
width
()
jkxjkx
I
BeAex
−
+=
ψ
0
2
22
2
=+
I
IE
m
dx
d
ψ
ψ
h ( ) 0
2
22
2
=−+
IIo
IIUE
m
dx
d
ψ
ψ
h
()
xx
II
eDeCx
γγ
ψ
−
+= ()
jkxjkx
III
FeGex
−
+=
ψ
0
2
22
2
=+
III
IIIE
m
dx
d
ψ
ψ
h
Energy
x
U
o
0a
E<U
o
MWe have 6 unknowns (A,B,C,D,G,F) and 5 boundary conditions
MContinuity of ψand ∇ψat x=0.
MContinuity of ψand ∇ψat x=a.
MOnly a forward propagating wave on the right hand side of the barrier.
[email protected] Dr. Hatem El-Refaei 26
Potential barrier of finiteheight and finite
width
()
jkxjkx
I
BeAex
−
+=
ψ
0
2
22
2
=+
I
IE
m
dx
d
ψ
ψ
h ( ) 0
2
22
2
=−+
IIo
IIUE
m
dx
d
ψ
ψ
h
()
xx
II
eDeCx
γγ
ψ
−
+= ()
jkx
III
Gex=
ψ
0
2
22
2
=+
III
IIIE
m
dx
d
ψ
ψ
h
Energy
x
U
o
0a
E<U
o
MWe have 5 unknowns (A,B,C,D,G) and 4 boundary conditions
MContinuity of ψand ∇ψat x=0.
MContinuity of ψand ∇ψat x=a.
Potential barrier of finiteheight and finite
width
()
jkxjkx
I
BeAex
−
+=
ψ
0
2
22
2
=+
I
IE
m
dx
d
ψ
ψ
h ( ) 0
2
22
2
=−+
IIo
IIUE
m
dx
d
ψ
ψ
h
()
xx
II
eDeCx
γγ
ψ
−
+= ()
jkx
III
Gex=
ψ
0
2
22
2
=+
III
IIIE
m
dx
d
ψ
ψ
h
Energy
x
U
o
0a
E<U
o
MWe have 5 unknowns (A,B,C,D,G) and 4 boundary conditions
MContinuity of ψand ∇ψat x=0.
MContinuity of ψand ∇ψat x=a.
[email protected] Dr. Hatem El-Refaei 27
Potential barrier of finiteheight and finite
width
We are interested in the transmission probability (T)
22
A
G
AA
GG
AmplitudeIncident
AmplitudedTransmitte
T ===
∗
∗
Energy
x
U
o
0a
E<U
o
()
jkxjkx
I
BeAex
−
+=
ψ
0
2
22
2
=+
I
IE
m
dx
d
ψ
ψ
h ( ) 0
2
22
2
=−+
IIo
IIUE
m
dx
d
ψ
ψ
h
()
xx
II
eDeCx
γγ
ψ
−
+= ()
jkx
III
Gex=
ψ
0
2
22
2
=+
III
IIIE
m
dx
d
ψ
ψ
h
Potential barrier of finiteheight and finite
width
We are interested in the transmission probability (T)
22
A
G
AA
GG
AmplitudeIncident
AmplitudedTransmitte
T ===
∗
∗
Energy
x
U
o
0a
E<U
o
()
jkxjkx
I
BeAex
−
+=
ψ
0
2
22
2
=+
I
IE
m
dx
d
ψ
ψ
h ( ) 0
2
22
2
=−+
IIo
IIUE
m
dx
d
ψ
ψ
h
()
xx
II
eDeCx
γγ
ψ
−
+= ()
jkx
III
Gex=
ψ
0
2
22
2
=+
III
IIIE
m
dx
d
ψ
ψ
h
[email protected] Dr. Hatem El-Refaei 28
Potential barrier of finiteheight and finite
width
And reflection probability (R)
22
Re
A
B
AA
BB
AmplitudeIncident
Amplitudeflected
R ===
∗
∗
Energy
x
U
o
0a
E<U
o
()
jkxjkx
I
BeAex
−
+=
ψ
0
2
22
2
=+
I
IE
m
dx
d
ψ
ψ
h ( ) 0
2
22
2
=−+
IIo
IIUE
m
dx
d
ψ
ψ
h
()
xx
II
eDeCx
γγ
ψ
−
+= ()
jkx
III
Gex=
ψ
0
2
22
2
=+
III
IIIE
m
dx
d
ψ
ψ
h
Potential barrier of finiteheight and finite
width
And reflection probability (R)
22
Re
A
B
AA
BB
AmplitudeIncident
Amplitudeflected
R ===
∗
∗
Energy
x
U
o
0a
E<U
o
()
jkxjkx
I
BeAex
−
+=
ψ
0
2
22
2
=+
I
IE
m
dx
d
ψ
ψ
h ( ) 0
2
22
2
=−+
IIo
IIUE
m
dx
d
ψ
ψ
h
()
xx
II
eDeCx
γγ
ψ
−
+= ()
jkx
III
Gex=
ψ
0
2
22
2
=+
III
IIIE
m
dx
d
ψ
ψ
h
[email protected] Dr. Hatem El-Refaei 29
Potential barrier of finiteheight and finite
width
MFirst set of boundary conditions at x=0
() ()00 === xx
III
ψψ
DCBA +=+
00 ==
=
x
II
x
Idx
d
dx
d
ψψ
DCjkBjkA
γγ−=−
MSecond set of boundary conditions at x=a
() ()axax
IIIII
===ψψ
ax
III
ax
IIdx
d
dx
d
==
=
ψψ
ikaaa
eGeDeC =+
−γγ ikaaa
eGikeDeC =−
−γγ
γγ
Potential barrier of finiteheight and finite
width
MFirst set of boundary conditions at x=0
() ()00 === xx
III
ψψ
DCBA +=+
00 ==
=
x
II
x
Idx
d
dx
d
ψψ
DCjkBjkA
γγ−=−
MSecond set of boundary conditions at x=a
() ()axax
IIIII
===ψψ
ax
III
ax
IIdx
d
dx
d
==
=
ψψ
ikaaa
eGeDeC =+
−γγ ikaaa
eGikeDeC =−
−γγ
γγ
[email protected] Dr. Hatem El-Refaei 30
Potential barrier of finiteheight and finite
width
DCBA +=+
(1)
DCjkBjkA γγ−=−
(2)
ikaaa
eGeDeC =+
−γγ
(3)
ikaaa
eGikeDeC =−
−γγ
γγ
(4)
MEq. (3) ×γ+ Eq. (4) to eliminate D.
MEq. (3)×(-γ) + Eq. (4) to eliminate C.
( )
Ge
jk
C
ajkγ
γ
γ
−+
=
2
(5)
( )
Ge
jk
D
ajkγ
γ
γ
+−
=
2
(6)
Potential barrier of finiteheight and finite
width
DCBA +=+
(1)
DCjkBjkA γγ−=−
(2)
ikaaa
eGeDeC =+
−γγ
(3)
ikaaa
eGikeDeC =−
−γγ
γγ
(4)
MEq. (3) ×γ+ Eq. (4) to eliminate D.
MEq. (3)×(-γ) + Eq. (4) to eliminate C.
( )
Ge
jk
C
ajkγ
γ
γ
−+
=
2
(5)
( )
Ge
jk
D
ajkγ
γ
γ
+−
=
2
(6)
[email protected] Dr. Hatem El-Refaei 31
Potential barrier of finiteheight and finite
width
MEq. (1) × jk+ Eq. (2) to eliminate B
( )( )DjkCjkjkA γγ −++=2 (7)
MSubstitute from eq. (5) and (6) into (7), we get an equation of A
and G only.
( )
( )
( )
( )[ ]GejkejkjkA
ajkajkγγ
γγ
γ
+−
−−+=
22
2
1
2
(8)
DCBA +=+
(1)
DCjkBjkA γγ−=−
(2)
ikaaa
eGeDeC =+
−γγ
(3)
ikaaa
eGikeDeC =−
−γγ
γγ
(4)
Potential barrier of finiteheight and finite
width
MEq. (1) × jk+ Eq. (2) to eliminate B
( )( )DjkCjkjkA γγ −++=2 (7)
MSubstitute from eq. (5) and (6) into (7), we get an equation of A
and G only.
( )
( )
( )
( )[ ]GejkejkjkA
ajkajkγγ
γγ
γ
+−
−−+=
22
2
1
2
(8)
DCBA +=+
(1)
DCjkBjkA γγ−=−
(2)
ikaaa
eGeDeC =+
−γγ
(3)
ikaaa
eGikeDeC =−
−γγ
γγ
(4)
[email protected] Dr. Hatem El-Refaei 32
Transmission and Reflection Coefficients
MIt is an assignment to show that the transmission coefficient is
givenby
( )
( )a
EUE
UAA
GG
T
o
o
γ
2
2
sinh
4
1
1
1
−
+
==
∗
∗
MAlso it is an assignment to show that the reflection coefficient
isgivenby
( )
( )
( )
( )a
EUE
U
a
EUE
U
AA
BB
R
o
o
o
o
γ
γ
2
2
2
2
sinh
4
1
1
sinh
4
1
−
+
−
==
∗
∗
Transmission and Reflection Coefficients
MIt is an assignment to show that the transmission coefficient is
givenby
( )
( )a
EUE
UAA
GG
T
o
o
γ
2
2
sinh
4
1
1
1
−
+
==
∗
∗
MAlso it is an assignment to show that the reflection coefficient
isgivenby
( )
( )
( )
( )a
EUE
U
a
EUE
U
AA
BB
R
o
o
o
o
γ
γ
2
2
2
2
sinh
4
1
1
sinh
4
1
−
+
−
==
∗
∗
[email protected] Dr. Hatem El-Refaei 33
Transmission and Reflection Coefficients
MAfter proving both relations, it will be clear to you that
1=+TR
MWhich is logical as an incident particle is either reflected or
transmitted.
MYou may need to use the following relations
( )
2
sinh
zz
ee
z
−
−
=
( )
2
cosh
zz
ee
z
−
+
=
() ()1sinhcosh
22
=− zz
Transmission and Reflection Coefficients
MAfter proving both relations, it will be clear to you that
1=+TR
MWhich is logical as an incident particle is either reflected or
transmitted.
MYou may need to use the following relations
( )
2
sinh
zz
ee
z
−
−
=
( )
2
cosh
zz
ee
z
−
+
=
() ()1sinhcosh
22
=− zz
[email protected] Dr. Hatem El-Refaei 34
Transmission coefficient versus barrier width
EOne notices that shrinking the barrier width by half resultsin a dramatic
increaseinthetransmissioncoefficient.Itisnota linearrelation.
EAlso doubling the particle energy results in exponential increase in the
transmissioncoefficient.
An electron is tunneling through 1.5 eV barrier
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0 0.25 0.5 0.75 1 1.25 1.5
Energy (e.V)
Transmission Coffecient
a=0.5 nm
a=1 nm
a=2 nm
a=4 nm
Transmission coefficient versus barrier width
EOne notices that shrinking the barrier width by half resultsin a dramatic
increaseinthetransmissioncoefficient.Itisnota linearrelation.
EAlso doubling the particle energy results in exponential increase in the
transmissioncoefficient.
An electron is tunneling through 1.5 eV barrier
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0 0.25 0.5 0.75 1 1.25 1.5
Energy (e.V)
Transmission Coffecient
a=0.5 nm
a=1 nm
a=2 nm
a=4 nm
[email protected] Dr. Hatem El-Refaei 35
Reflection coefficient versus barrier width
An electron is tunneling through 1.5 eV barrier
0
0.2
0.4
0.6
0.8
1
1.2
0 0.25 0.5 0.75 1 1.25 1.5
Energy (e.V)
Reflection Coffecient
a=0.5 nm
a=1 nm
a=2 nm
a=4 nm
ENoticethatR=1-T
Reflection coefficient versus barrier width
An electron is tunneling through 1.5 eV barrier
0
0.2
0.4
0.6
0.8
1
1.2
0 0.25 0.5 0.75 1 1.25 1.5
Energy (e.V)
Reflection Coffecient
a=0.5 nm
a=1 nm
a=2 nm
a=4 nm
ENoticethatR=1-T
[email protected] Dr. Hatem El-Refaei 36
Plotting the wave function
MNow we have found all
constantsintermsofA.
MWe can plot the shape of
the wave function in all
regions.
MOne would expect that as the barrier gets smaller in height
and/or narrower in width more particles will be able to cross
thebarriertotheotherside.
MThisresultsinahighertransmissioncoefficient“T”,andalsoa
largeramplitudeforthetransmittedwave“G”.
MCheck:http://phys.educ.ksu.edu/vqm/html/qtunneling.html
G
Plotting the wave function
MNow we have found all
constantsintermsofA.
MWe can plot the shape of
the wave function in all
regions.
MOne would expect that as the barrier gets smaller in height
and/or narrower in width more particles will be able to cross
thebarriertotheotherside.
MThisresultsinahighertransmissioncoefficient“T”,andalsoa
largeramplitudeforthetransmittedwave“G”.
MCheck:http://phys.educ.ksu.edu/vqm/html/qtunneling.html
G
[email protected] Dr. Hatem El-Refaei 37
Contradiction with Classical Mechanics
MThis results are in contradiction with the classical mechanics
which predicts that is the particle’s energy is lower than the
barrier height, the particle overcome the barrier and thus can
notexistintherighthandside.
G
Contradiction with Classical Mechanics
MThis results are in contradiction with the classical mechanics
which predicts that is the particle’s energy is lower than the
barrier height, the particle overcome the barrier and thus can
notexistintherighthandside.
G
[email protected] Dr. Hatem El-Refaei 38
Remember classical mechanics
A man at rest here
Will never be able to
pass this point
So he can’t
exist here
Remember classical mechanics
A man at rest here
Will never be able to
pass this point
So he can’t
exist here
Ain Shams University
Mathematics and Engineering Physics Department
1
st
Year Electrical Engineering
Lecture 11
Modern Physics and Quantum Mechanics Course
Dr. Hatem El-Refaei
Mathematics and Engineering Physics Department
1
st
Year Electrical Engineering
Lecture 11
Modern Physics and Quantum Mechanics Course
Dr. Hatem El-Refaei
Dr. Hatem El-Refaei1
Contents
M
Infinite barrier
M
Finite barrier
M
Quantum tunnelling
Contents
M
Infinite barrier
M
Finite barrier
M
Quantum tunnelling
Dr. Hatem El-Refaei2
Note
M
All problems today are unbounded problem, i.e. the
particle is not confined in a certain region, so:
M
We will not be able to do the normalization condition.
M
Therefore, we will not be able to solve for all unknowns.
M
Therefore, we will not get a characteristic equation.
M
Therefore, energy levels are not quantized, and all energies
are possible.
M
But still there are a lot of important characterist ics to
understand and learn today.
Note
M
All problems today are unbounded problem, i.e. the
particle is not confined in a certain region, so:
M
We will not be able to do the normalization condition.
M
Therefore, we will not be able to solve for all unknowns.
M
Therefore, we will not get a characteristic equation.
M
Therefore, energy levels are not quantized, and all energies
are possible.
M
But still there are a lot of important characterist ics to
understand and learn today.
Dr. Hatem El-Refaei3
Infinite barrier
Infinite barrier
Dr. Hatem El-Refaei4
Potential step of
infinite
height and
infinite
width
M
Since the barrier height is infinite, incident particles can’t
penetratethroughit,andparticlesreflectback. M
So, there is zero probability of finding the particle inside
thestepbarrier. M
Here,theQMsolutionleadstothesameclassicalsolution.
Energy
∞
∞
x
E
Potential step of
infinite
height and
infinite
width
M
Since the barrier height is infinite, incident particles can’t
penetratethroughit,andparticlesreflectback. M
So, there is zero probability of finding the particle inside
thestepbarrier. M
Here,theQMsolutionleadstothesameclassicalsolution.
Energy
∞
∞
x
E
Dr. Hatem El-Refaei5
Potential step of
infinite
height and
infinite
width
(
)
jkx jkx
Be Ae x
−
+ =
ψ
( )
− −
−
+ = Ψ
t
E
kx j t
E
kxj
Be Ae tx
h h
,
0
2
2 2
2
= +
I
I
E
m
dx
d
ψ
ψ
h
Backward
propagating wave
Forward
propagating wave
∞
=
U
(
)
0
=
x
ψ
h
mE
k
2
=
E
Energy, ψ
∞
∞
x
0
Potential step of
infinite
height and
infinite
width
(
)
jkx jkx
Be Ae x
−
+ =
ψ
( )
− −
−
+ = Ψ
t
E
kx j t
E
kxj
Be Ae tx
h h
,
0
2
2 2
2
= +
I
I
E
m
dx
d
ψ
ψ
h
Backward
propagating wave
Forward
propagating wave
∞
=
U
(
)
0
=
x
ψ
h
mE
k
2
=
E
Energy, ψ
∞
∞
x
0
Dr. Hatem El-Refaei6
Potential step of
infinite
height and
infinite
width
(
)
(
)
+ −
= = =0 0x x
ψ
ψ
0
=
+
B A
Energy, ψ
∞
∞
x
0
∞
=
U
(
)
0
=
x
ψ
E
(
)
jkx jkx
Be Ae x
−
+ =
ψ
0
2
2 2
2
= +
I
I
E
m
dx
d
ψ
ψ
h
h
mE
k
2
=
Potential step of
infinite
height and
infinite
width
(
)
(
)
+ −
= = =0 0x x
ψ
ψ
0
=
+
B A
Energy, ψ
∞
∞
x
0
∞
=
U
(
)
0
=
x
ψ
E
(
)
jkx jkx
Be Ae x
−
+ =
ψ
0
2
2 2
2
= +
I
I
E
m
dx
d
ψ
ψ
h
h
mE
k
2
=
Dr. Hatem El-Refaei7
Potential step of
infinite
height and
infinite
width
A
B
−
=
(
)
(
)
kx jA xsin 2
=
ψ
Energy, ψ
∞
∞
x
0
∞
=
U
(
)
0
=
x
ψ
(
)
jkx jkx
Ae Ae x
−
− =
ψ
(
)
(
)
jkx jkx
e eA x
−
− =
ψ
Sin(x)
E 0
2
2 2
2
= +
I
I
E
m
dx
d
ψ
ψ
h
Potential step of
infinite
height and
infinite
width
A
B
−
=
(
)
(
)
kx jA xsin 2
=
ψ
Energy, ψ
∞
∞
x
0
∞
=
U
(
)
0
=
x
ψ
(
)
jkx jkx
Ae Ae x
−
− =
ψ
(
)
(
)
jkx jkx
e eA x
−
− =
ψ
Sin(x)
E 0
2
2 2
2
= +
I
I
E
m
dx
d
ψ
ψ
h
Dr. Hatem El-Refaei8
Potential step of
infinite
height and
infinite
width
A
B
−
=
Reflectivity
1
Re
2
= = =
∗
∗
AA
BB
Amplitude Incident
Amplitude flected
R
All the incident particle stream is reflected back.
Energy, ψ
∞
∞
x
0
Sin(x)
E
Potential step of
infinite
height and
infinite
width
A
B
−
=
Reflectivity
1
Re
2
= = =
∗
∗
AA
BB
Amplitude Incident
Amplitude flected
R
All the incident particle stream is reflected back.
Energy, ψ
∞
∞
x
0
Sin(x)
E
Dr. Hatem El-Refaei9
Potential step of
infinite
height and
infinite
width
M
Since the barrier extends to infinity in the x direction, no p article can penetrate
through the whole barrier. From phenomenological understa nding, as x→∞,
ψ→0.
Energy, ψ
∞
∞
x
0
Sin(x)
E
Potential step of
infinite
height and
infinite
width
M
Since the barrier extends to infinity in the x direction, no p article can penetrate
through the whole barrier. From phenomenological understa nding, as x→∞,
ψ→0.
Energy, ψ
∞
∞
x
0
Sin(x)
E
Dr. Hatem El-Refaei 10
Finite barrier
Finite barrier
Dr. Hatem El-Refaei 11
Potential step of
finite
height and
infinite
width
M
Asthestepheight
U
o
getssmaller(butstill
E<U
o
),
the penetration of the particles inside the step
increases, but finally no particles will succeed to
travelthroughthewholesteptox→∞.
Energy
∞
x
U
o
0
E<U
o
E
Potential step of
finite
height and
infinite
width
M
Asthestepheight
U
o
getssmaller(butstill
E<U
o
),
the penetration of the particles inside the step
increases, but finally no particles will succeed to
travelthroughthewholesteptox→∞.
Energy
∞
x
U
o
0
E<U
o
E
Dr. Hatem El-Refaei 12
Potential step of
finite
height and
infinite
width
0
(
)
jkx jkx
I
Be Ae x
−
+ =
ψ
0
2
2 2
2
= +
I
I
E
m
dx
d
ψ
ψ
h
( )
0
2
2 2
2
= − +
II o
II
UE
m
dx
d
ψ
ψ
h
(
)
x x
II
De eC x
α α
ψ
+ =
−
Energy
∞
x
U
o
M
We have 4 unknowns (A,B,C, and D) and 3 equations:
M
Finiteness of ψat x=∞
M
Continuity of ψat x=0
M
Continuity of ∇ψat x=0
E<U
o
(
)
0
2
>
−
=
h
E Um
o
α
h
mE
k
2
=
E
Potential step of
finite
height and
infinite
width
0
(
)
jkx jkx
I
Be Ae x
−
+ =
ψ
0
2
2 2
2
= +
I
I
E
m
dx
d
ψ
ψ
h
( )
0
2
2 2
2
= − +
II o
II
UE
m
dx
d
ψ
ψ
h
(
)
x x
II
De eC x
α α
ψ
+ =
−
Energy
∞
x
U
o
M
We have 4 unknowns (A,B,C, and D) and 3 equations:
M
Finiteness of ψat x=∞
M
Continuity of ψat x=0
M
Continuity of ∇ψat x=0
E<U
o
(
)
0
2
>
−
=
h
E Um
o
α
h
mE
k
2
=
E
Dr. Hatem El-Refaei 13
Potential step of
finite
height and
infinite
width
0
(
)
jkx jkx
I
Be Ae x
−
+ =
ψ
0
2
2 2
2
= +
I
I
E
m
dx
d
ψ
ψ
h
( )
0
2
2 2
2
= − +
II o
II
UE
m
dx
d
ψ
ψ
h
(
)
x x
II
De eC x
α α
ψ
+ =
−
Energy
∞
x
U
o
E<U
o
M
Theconditionthatψ(x)mustbefiniteasx→∞,leadstoD=0
(
)
0
2
>
−
=
h
E Um
o
α
h
mE
k
2
=
E
Potential step of
finite
height and
infinite
width
0
(
)
jkx jkx
I
Be Ae x
−
+ =
ψ
0
2
2 2
2
= +
I
I
E
m
dx
d
ψ
ψ
h
( )
0
2
2 2
2
= − +
II o
II
UE
m
dx
d
ψ
ψ
h
(
)
x x
II
De eC x
α α
ψ
+ =
−
Energy
∞
x
U
o
E<U
o
M
Theconditionthatψ(x)mustbefiniteasx→∞,leadstoD=0
(
)
0
2
>
−
=
h
E Um
o
α
h
mE
k
2
=
E
Dr. Hatem El-Refaei 14
Potential step of
finite
height and
infinite
width
0
(
)
jkx jkx
I
Be Ae x
−
+ =
ψ
0
2
2 2
2
= +
I
I
E
m
dx
d
ψ
ψ
h
( )
0
2
2 2
2
= − +
II o
II
UE
m
dx
d
ψ
ψ
h
(
)
x
II
eC x
α
ψ
−
=
Energy
∞
x
U
o
E<U
o
M
Theconditionthatψ(x)mustbefiniteasx→∞,leadstoD=0
(
)
0
2
>
−
=
h
E Um
o
α
h
mE
k
2
=
E
Potential step of
finite
height and
infinite
width
0
(
)
jkx jkx
I
Be Ae x
−
+ =
ψ
0
2
2 2
2
= +
I
I
E
m
dx
d
ψ
ψ
h
( )
0
2
2 2
2
= − +
II o
II
UE
m
dx
d
ψ
ψ
h
(
)
x
II
eC x
α
ψ
−
=
Energy
∞
x
U
o
E<U
o
M
Theconditionthatψ(x)mustbefiniteasx→∞,leadstoD=0
(
)
0
2
>
−
=
h
E Um
o
α
h
mE
k
2
=
E
Dr. Hatem El-Refaei 15
Potential step of
finite
height and
infinite
width
0
(
)
jkx jkx
I
Be Ae x
−
+ =
ψ
0
2
2 2
2
= +
I
I
E
m
dx
d
ψ
ψ
h
( )
0
2
2 2
2
= − +
II o
II
UE
m
dx
d
ψ
ψ
h
(
)
x
II
eC x
α
ψ
−
=
Energy
∞
x
U
o
M
We have 3 unknowns (A,B,C) and 2 equations:
M
Continuity of ψ
M
Continuity of ∇ψ
M
Thus, the best we can get is the ratio between parameters.
E<U
o
(
)
0
2
>
−
=
h
E Um
o
α
h
mE
k
2
=
E
Potential step of
finite
height and
infinite
width
0
(
)
jkx jkx
I
Be Ae x
−
+ =
ψ
0
2
2 2
2
= +
I
I
E
m
dx
d
ψ
ψ
h
( )
0
2
2 2
2
= − +
II o
II
UE
m
dx
d
ψ
ψ
h
(
)
x
II
eC x
α
ψ
−
=
Energy
∞
x
U
o
M
We have 3 unknowns (A,B,C) and 2 equations:
M
Continuity of ψ
M
Continuity of ∇ψ
M
Thus, the best we can get is the ratio between parameters.
E<U
o
(
)
0
2
>
−
=
h
E Um
o
α
h
mE
k
2
=
E
Dr. Hatem El-Refaei 16
Potential step of
finite
height and
infinite
width
(
)
(
)
0 0
=
=
=
x x
II I
ψ
ψ
Continuity ofψ
C B A
=
+
Continuity of dψ/dx
0 0= =
=
x
II
x
I
dx
d
dx
dψ ψ
(
)
C BAjk
α
−
=
−
C
k
j A
+ =
α
1
2
1
C
k
j B
− =
α
1
2
1
∞
x
U
o
Energy
E<U
o
(
)
0
2
>
−
=
h
E Um
o
α
h
mE
k
2
=
0
E
(
)
jkx jkx
I
Be Ae x
−
+ =
ψ
(
)
x
II
eC x
α
ψ
−
=
Potential step of
finite
height and
infinite
width
(
)
(
)
0 0
=
=
=
x x
II I
ψ
ψ
Continuity ofψ
C B A
=
+
Continuity of dψ/dx
0 0= =
=
x
II
x
I
dx
d
dx
dψ ψ
(
)
C BAjk
α
−
=
−
C
k
j A
+ =
α
1
2
1
C
k
j B
− =
α
1
2
1
∞
x
U
o
Energy
E<U
o
(
)
0
2
>
−
=
h
E Um
o
α
h
mE
k
2
=
0
E
(
)
jkx jkx
I
Be Ae x
−
+ =
ψ
(
)
x
II
eC x
α
ψ
−
=
Dr. Hatem El-Refaei 17
Potential step of
finite
height and
infinite
width
C
k
j A
+ =
α
1
2
1
C
k
j B
− =
α
1
2
1
Reflectivity
1
1 1
1 1
Re
2
=
−
+
+
−
= = =
∗
∗
k
j
k
j
k
j
k
j
AA
BB
Amplitude Incident
Amplitude flected
Rα α
α α
Energy
E<U
o
(
)
0
2
>
−
=
h
E Um
o
α
h
mE
k
2
=
0
E
Potential step of
finite
height and
infinite
width
C
k
j A
+ =
α
1
2
1
C
k
j B
− =
α
1
2
1
Reflectivity
1
1 1
1 1
Re
2
=
−
+
+
−
= = =
∗
∗
k
j
k
j
k
j
k
j
AA
BB
Amplitude Incident
Amplitude flected
Rα α
α α
Energy
E<U
o
(
)
0
2
>
−
=
h
E Um
o
α
h
mE
k
2
=
0
E
Dr. Hatem El-Refaei 18
Potential step of
finite
height and
infinite
width
jkx jkx
Ce
k
j Ce
k
j
−
− +
+ =
α α
1
2
1
1
2
1
( )
( )
( )
≥
≤ −
=
−
0
0 sin cos
x eC
x kx C
k
kx C
x
x
α
α
ψ
(
)
jkx jkx
I
Be Ae x
−
+ =
ψ
∞
x
U
o
0
C
( )
−
+
+
=
− −
2 2
jkx jkx jkx jkx
I
e e
C
k
j
e e
C x
α
ψ
Energy
Potential step of
finite
height and
infinite
width
jkx jkx
Ce
k
j Ce
k
j
−
− +
+ =
α α
1
2
1
1
2
1
( )
( )
( )
≥
≤ −
=
−
0
0 sin cos
x eC
x kx C
k
kx C
x
x
α
α
ψ
(
)
jkx jkx
I
Be Ae x
−
+ =
ψ
∞
x
U
o
0
C
( )
−
+
+
=
− −
2 2
jkx jkx jkx jkx
I
e e
C
k
j
e e
C x
α
ψ
Energy
Dr. Hatem El-Refaei 19
Potential step of
finite
height and
infinite
width
M
Since the barrier height is finite, particles can penetrate parti ally
inthevicinityofthepotentialstep,andthentheyreflectback. M
So, there is a finite probability of finding the particle in the
classicallyforbiddenposition. M
Here,theQMsolutionisdifferentfromtheclassicalone.
∞
x
U
o
0
C
Energy
Potential step of
finite
height and
infinite
width
M
Since the barrier height is finite, particles can penetrate parti ally
inthevicinityofthepotentialstep,andthentheyreflectback. M
So, there is a finite probability of finding the particle in the
classicallyforbiddenposition. M
Here,theQMsolutionisdifferentfromtheclassicalone.
∞
x
U
o
0
C
Energy
Dr. Hatem El-Refaei 20
Potential step of
finite
height and
infinite
width
Case 2
Case 2
x
0
Energy
Case 3
Case 3
E<U
o
(
)
0
2
>
−
=
h
E Um
o
α
M
As the potential barrier height increases (U
o
increases)(fromcase2tocase 3),αalsoincreases,
andthustheexponentialfunctiondiesquickerinside
thebarrier.Thusitbecomeslessprobabletofindthe
particleinsidethebarrier.
(
)
x
eC x
α
ψ
−
=
Potential step of
finite
height and
infinite
width
Case 2
Case 2
x
0
Energy
Case 3
Case 3
E<U
o
(
)
0
2
>
−
=
h
E Um
o
α
M
As the potential barrier height increases (U
o
increases)(fromcase2tocase 3),αalsoincreases,
andthustheexponentialfunctiondiesquickerinside
thebarrier.Thusitbecomeslessprobabletofindthe
particleinsidethebarrier.
(
)
x
eC x
α
ψ
−
=
Dr. Hatem El-Refaei 21
Potential step of
finite
height and
infinite
width
M
Ifthebarrierheight(U
o
)iskeptconstant,buttheparticle
energyincreases(providedE<U
o
),thusαdecreases,and
the particle exponential function dies slower inside the
barrier. Hence, it becomes more probable to find the
particleinthevicinityofthebarrieredge.
x
0
Energy
E<U
o
(
)
0
2
>
−
=
h
E Um
o
α(
)
x
eC x
α
ψ
−
=
Potential step of
finite
height and
infinite
width
M
Ifthebarrierheight(U
o
)iskeptconstant,buttheparticle
energyincreases(providedE<U
o
),thusαdecreases,and
the particle exponential function dies slower inside the
barrier. Hence, it becomes more probable to find the
particleinthevicinityofthebarrieredge.
x
0
Energy
E<U
o
(
)
0
2
>
−
=
h
E Um
o
α(
)
x
eC x
α
ψ
−
=
Dr. Hatem El-Refaei 22
Tunneling through a potential
barrier
Tunneling through a potential
barrier
Dr. Hatem El-Refaei 23
Potential barrier of
finite
height and
finite
width
M
A stream of particles incident on a finite width and height
potentialbarrierwith
E<U
o.
M
Part of the incident stream will succeed to penetrate through the
barrierandappearontheotherside,thisisthetransmittedstream.
Theotherpartwillreflectbackformingthereflectedstream.
M
Note,asingleparticledoesn’tsplitintotwo.
x
U
o
E<U
o
E
Energy
Potential barrier of
finite
height and
finite
width
M
A stream of particles incident on a finite width and height
potentialbarrierwith
E<U
o.
M
Part of the incident stream will succeed to penetrate through the
barrierandappearontheotherside,thisisthetransmittedstream.
Theotherpartwillreflectbackformingthereflectedstream.
M
Note,asingleparticledoesn’tsplitintotwo.
x
U
o
E<U
o
E
Energy
Dr. Hatem El-Refaei 24
Potential barrier of
finite
height and
finite
width
(
)
jkx jkx
I
Be Ae x
−
+ =
ψ
0
2
2 2
2
= +
I
I
E
m
dx
d
ψ
ψ
h
( )
0
2
2 2
2
= − +
II o
II
UE
m
dx
d
ψ
ψ
h
(
)
x x
II
eD eC x
γ γ
ψ
−
+ =
(
)
jkx jkx
III
Fe Ge x
−
+ =
ψ
0
2
2 2
2
= +
III
III
E
m
dx
d
ψ
ψ
h
Energy
x
U
o
0a
E<U
o
(
)
0
2
>
−
=
h
E Um
o
γ
h
mE
k
2
=
M
We have 6 unknowns (A,B,C,D,G,F) and 5 boundary conditions
M
Continuity of ψand ∇ψat x=0.
M
Continuity of ψand ∇ψat x=a.
M
Only a forward propagating wave on the right hand s ide of the barrier.
Potential barrier of
finite
height and
finite
width
(
)
jkx jkx
I
Be Ae x
−
+ =
ψ
0
2
2 2
2
= +
I
I
E
m
dx
d
ψ
ψ
h
( )
0
2
2 2
2
= − +
II o
II
UE
m
dx
d
ψ
ψ
h
(
)
x x
II
eD eC x
γ γ
ψ
−
+ =
(
)
jkx jkx
III
Fe Ge x
−
+ =
ψ
0
2
2 2
2
= +
III
III
E
m
dx
d
ψ
ψ
h
Energy
x
U
o
0a
E<U
o
(
)
0
2
>
−
=
h
E Um
o
γ
h
mE
k
2
=
M
We have 6 unknowns (A,B,C,D,G,F) and 5 boundary conditions
M
Continuity of ψand ∇ψat x=0.
M
Continuity of ψand ∇ψat x=a.
M
Only a forward propagating wave on the right hand s ide of the barrier.
Dr. Hatem El-Refaei 25
Potential barrier of
finite
height and
finite
width
(
)
jkx jkx
I
Be Ae x
−
+ =
ψ
0
2
2 2
2
= +
I
I
E
m
dx
d
ψ
ψ
h
( )
0
2
2 2
2
= − +
II o
II
UE
m
dx
d
ψ
ψ
h
(
)
x x
II
eD eC x
γ γ
ψ
−
+ =
(
)
jkx jkx
III
Fe Ge x
−
+ =
ψ
0
2
2 2
2
= +
III
III
E
m
dx
d
ψ
ψ
h
Energy
x
U
o
0a
E<U
o
M
We have 6 unknowns (A,B,C,D,G,F) and 5 boundary conditions
M
Continuity of ψand ∇ψat x=0.
M
Continuity of ψand ∇ψat x=a.
M
Only a forward propagating wave on the right hand s ide of the barrier.
Potential barrier of
finite
height and
finite
width
(
)
jkx jkx
I
Be Ae x
−
+ =
ψ
0
2
2 2
2
= +
I
I
E
m
dx
d
ψ
ψ
h
( )
0
2
2 2
2
= − +
II o
II
UE
m
dx
d
ψ
ψ
h
(
)
x x
II
eD eC x
γ γ
ψ
−
+ =
(
)
jkx jkx
III
Fe Ge x
−
+ =
ψ
0
2
2 2
2
= +
III
III
E
m
dx
d
ψ
ψ
h
Energy
x
U
o
0a
E<U
o
M
We have 6 unknowns (A,B,C,D,G,F) and 5 boundary conditions
M
Continuity of ψand ∇ψat x=0.
M
Continuity of ψand ∇ψat x=a.
M
Only a forward propagating wave on the right hand s ide of the barrier.
Dr. Hatem El-Refaei 26
Potential barrier of
finite
height and
finite
width
(
)
jkx jkx
I
Be Ae x
−
+ =
ψ
0
2
2 2
2
= +
I
I
E
m
dx
d
ψ
ψ
h
( )
0
2
2 2
2
= − +
II o
II
UE
m
dx
d
ψ
ψ
h
(
)
x x
II
eD eC x
γ γ
ψ
−
+ =
(
)
jkx
III
Ge x=
ψ
0
2
2 2
2
= +
III
III
E
m
dx
d
ψ
ψ
h
Energy
x
U
o
0a
E<U
o
M
We have 5 unknowns (A,B,C,D,G) and 4 boundary conditions
M
Continuity of ψand ∇ψat x=0.
M
Continuity of ψand ∇ψat x=a.
Potential barrier of
finite
height and
finite
width
(
)
jkx jkx
I
Be Ae x
−
+ =
ψ
0
2
2 2
2
= +
I
I
E
m
dx
d
ψ
ψ
h
( )
0
2
2 2
2
= − +
II o
II
UE
m
dx
d
ψ
ψ
h
(
)
x x
II
eD eC x
γ γ
ψ
−
+ =
(
)
jkx
III
Ge x=
ψ
0
2
2 2
2
= +
III
III
E
m
dx
d
ψ
ψ
h
Energy
x
U
o
0a
E<U
o
M
We have 5 unknowns (A,B,C,D,G) and 4 boundary conditions
M
Continuity of ψand ∇ψat x=0.
M
Continuity of ψand ∇ψat x=a.
Dr. Hatem El-Refaei 27
Potential barrier of
finite
height and
finite
width
We are interested in the transmission probability (T)
2 2
A
G
AA
GG
Amplitude Incident
Amplitude d Transmitte
T= = =
∗
∗
Energy
x
U
o
0a
E<U
o
(
)
jkx jkx
I
Be Ae x
−
+ =
ψ
0
2
2 2
2
= +
I
I
E
m
dx
d
ψ
ψ
h
( )
0
2
2 2
2
= − +
II o
II
UE
m
dx
d
ψ
ψ
h
(
)
x x
II
eD eC x
γ γ
ψ
−
+ =
(
)
jkx
III
Ge x=
ψ
0
2
2 2
2
= +
III
III
E
m
dx
d
ψ
ψ
h
Potential barrier of
finite
height and
finite
width
We are interested in the transmission probability (T)
2 2
A
G
AA
GG
Amplitude Incident
Amplitude d Transmitte
T= = =
∗
∗
Energy
x
U
o
0a
E<U
o
(
)
jkx jkx
I
Be Ae x
−
+ =
ψ
0
2
2 2
2
= +
I
I
E
m
dx
d
ψ
ψ
h
( )
0
2
2 2
2
= − +
II o
II
UE
m
dx
d
ψ
ψ
h
(
)
x x
II
eD eC x
γ γ
ψ
−
+ =
(
)
jkx
III
Ge x=
ψ
0
2
2 2
2
= +
III
III
E
m
dx
d
ψ
ψ
h
Dr. Hatem El-Refaei 28
Potential barrier of
finite
height and
finite
width
And reflection probability (R)
2 2
Re
A
B
AA
BB
Amplitude Incident
Amplitude flected
R= = =
∗
∗
Energy
x
U
o
0a
E<U
o
(
)
jkx jkx
I
Be Ae x
−
+ =
ψ
0
2
2 2
2
= +
I
I
E
m
dx
d
ψ
ψ
h
( )
0
2
2 2
2
= − +
II o
II
UE
m
dx
d
ψ
ψ
h
(
)
x x
II
eD eC x
γ γ
ψ
−
+ =
(
)
jkx
III
Ge x=
ψ
0
2
2 2
2
= +
III
III
E
m
dx
d
ψ
ψ
h
Potential barrier of
finite
height and
finite
width
And reflection probability (R)
2 2
Re
A
B
AA
BB
Amplitude Incident
Amplitude flected
R= = =
∗
∗
Energy
x
U
o
0a
E<U
o
(
)
jkx jkx
I
Be Ae x
−
+ =
ψ
0
2
2 2
2
= +
I
I
E
m
dx
d
ψ
ψ
h
( )
0
2
2 2
2
= − +
II o
II
UE
m
dx
d
ψ
ψ
h
(
)
x x
II
eD eC x
γ γ
ψ
−
+ =
(
)
jkx
III
Ge x=
ψ
0
2
2 2
2
= +
III
III
E
m
dx
d
ψ
ψ
h
Dr. Hatem El-Refaei 29
Potential barrier of
finite
height and
finite
width
M
First set of boundary conditions at x=0
(
)
(
)
0 0
=
=
=
x x
II I
ψ
ψ
D C BA
+
=
+
0 0= =
=
x
II
x
I
dx
d
dx
dψ ψ
D C jkB jkA
γ
γ
−
=
−
M
Second set of boundary conditions at x=a
(
)
(
)
ax ax
III II
=
=
=
ψ
ψ
ax
III
ax
II
dx
d
dx
d
= =
=
ψ ψ
ika a a
eG eD eC= +
−
γ γ
ika a a
eGik eD eC= −
−
γ γ
γ γ
Potential barrier of
finite
height and
finite
width
M
First set of boundary conditions at x=0
(
)
(
)
0 0
=
=
=
x x
II I
ψ
ψ
D C BA
+
=
+
0 0= =
=
x
II
x
I
dx
d
dx
dψ ψ
D C jkB jkA
γ
γ
−
=
−
M
Second set of boundary conditions at x=a
(
)
(
)
ax ax
III II
=
=
=
ψ
ψ
ax
III
ax
II
dx
d
dx
d
= =
=
ψ ψ
ika a a
eG eD eC= +
−
γ γ
ika a a
eGik eD eC= −
−
γ γ
γ γ
Dr. Hatem El-Refaei 30
Potential barrier of
finite
height and
finite
width
D C BA
+
=
+
(1)
D C jkB jkA
γ
γ
−
=
−
(2)
ika a a
eG eD eC= +
−
γ γ
(3)
ika a a
eGik eD eC= −
−
γ γ
γ γ
(4)
M
Eq. (3) ×γ+ Eq. (4) to eliminate D.
M
Eq. (3)×(-γ) + Eq. (4) to eliminate C.
( )
G e
jk
C
a jk
γ
γ
γ
−
+
=
2
(5)
( )
G e
jk
D
a jk
γ
γ
γ
+
−
=
2
(6)
Potential barrier of
finite
height and
finite
width
D C BA
+
=
+
(1)
D C jkB jkA
γ
γ
−
=
−
(2)
ika a a
eG eD eC= +
−
γ γ
(3)
ika a a
eGik eD eC= −
−
γ γ
γ γ
(4)
M
Eq. (3) ×γ+ Eq. (4) to eliminate D.
M
Eq. (3)×(-γ) + Eq. (4) to eliminate C.
( )
G e
jk
C
a jk
γ
γ
γ
−
+
=
2
(5)
( )
G e
jk
D
a jk
γ
γ
γ
+
−
=
2
(6)
Dr. Hatem El-Refaei 31
Potential barrier of
finite
height and
finite
width
M
Eq. (1) ×
jk
+ Eq. (2) to eliminate B
(
)
(
)
D jk C jk jkA
γ
γ
−
+
+
=
2
(7)
M
Substitute from eq. (5) and (6) into (7), we get an equation of A
and G only.
( )
( )
( )
( )
[
]
G e jk e jk jkA
a jk a jk
γ γ
γ γ
γ
+ −
− − + =
2 2
2
1
2
(8)
D C BA
+
=
+
(1)
D C jkB jkA
γ
γ
−
=
−
(2)
ika a a
eG eD eC= +
−
γ γ
(3)
ika a a
eGik eD eC= −
−
γ γ
γ γ
(4)
Potential barrier of
finite
height and
finite
width
M
Eq. (1) ×
jk
+ Eq. (2) to eliminate B
(
)
(
)
D jk C jk jkA
γ
γ
−
+
+
=
2
(7)
M
Substitute from eq. (5) and (6) into (7), we get an equation of A
and G only.
( )
( )
( )
( )
[
]
G e jk e jk jkA
a jk a jk
γ γ
γ γ
γ
+ −
− − + =
2 2
2
1
2
(8)
D C BA
+
=
+
(1)
D C jkB jkA
γ
γ
−
=
−
(2)
ika a a
eG eD eC= +
−
γ γ
(3)
ika a a
eGik eD eC= −
−
γ γ
γ γ
(4)
Dr. Hatem El-Refaei 32
Transmission and Reflection Coefficients
M
It is an assignment to show that the transmission coefficient is
givenby
( )
( )
a
E UE
U AA
GG
T
o
o
γ
2
2
sinh
4
1
1
1
−
+
= =
∗
∗
M
Also it is an assignment to show that the reflection coefficient
isgivenby
( )
( )
( )
( )
a
E UE
U
a
E UE
U
AA
BB
R
o
o
o
o
γ
γ
2
2
2
2
sinh
4
1
1
sinh
4
1
−
+
−
= =
∗
∗
Transmission and Reflection Coefficients
M
It is an assignment to show that the transmission coefficient is
givenby
( )
( )
a
E UE
U AA
GG
T
o
o
γ
2
2
sinh
4
1
1
1
−
+
= =
∗
∗
M
Also it is an assignment to show that the reflection coefficient
isgivenby
( )
( )
( )
( )
a
E UE
U
a
E UE
U
AA
BB
R
o
o
o
o
γ
γ
2
2
2
2
sinh
4
1
1
sinh
4
1
−
+
−
= =
∗
∗
Dr. Hatem El-Refaei 33
Transmission and Reflection Coefficients
M
After proving both relations, it will be clear to you that
1
=
+
T
R
M
Which is logical as an incident particle is either reflected or
transmitted. M
You may need to use the following relations
( )
2
sinh
z z
e e
z
−
−
=
( )
2
cosh
z z
e e
z
−
+
=
(
)
(
)
1 sinh cosh
2 2
= −z z
Transmission and Reflection Coefficients
M
After proving both relations, it will be clear to you that
1
=
+
T
R
M
Which is logical as an incident particle is either reflected or
transmitted. M
You may need to use the following relations
( )
2
sinh
z z
e e
z
−
−
=
( )
2
cosh
z z
e e
z
−
+
=
(
)
(
)
1 sinh cosh
2 2
= −z z
Dr. Hatem El-Refaei 34
Transmission coefficient versus barrier width
E
One notices that shrinking the barrier width by half results in a dramatic
increaseinthetransmissioncoefficient.Itisnota linear relation.
E
Also doubling the particle energy results in exponential in crease in the
transmissioncoefficient.
An electron is tunneling through 1.5 eV barrier
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0 0.25 0.5 0.75 1 1.25 1.5
Energy (e.V)
Transmission Coffecient
a=0.5 nm a=1 nm a=2 nm a=4 nm
Transmission coefficient versus barrier width
E
One notices that shrinking the barrier width by half results in a dramatic
increaseinthetransmissioncoefficient.Itisnota linear relation.
E
Also doubling the particle energy results in exponential in crease in the
transmissioncoefficient.
An electron is tunneling through 1.5 eV barrier
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0 0.25 0.5 0.75 1 1.25 1.5
Energy (e.V)
Transmission Coffecient
a=0.5 nm a=1 nm a=2 nm a=4 nm
Dr. Hatem El-Refaei 35
Reflection coefficient versus barrier width
An electron is tunneling through 1.5 eV barrier
0
0.2
0.4
0.6
0.8
1
1.2
0 0.25 0.5 0.75 1 1.25 1.5
Energy (e.V)
Reflection Coffecient
a=0.5 nm a=1 nm a=2 nm a=4 nm
E
NoticethatR=1-T
Reflection coefficient versus barrier width
An electron is tunneling through 1.5 eV barrier
0
0.2
0.4
0.6
0.8
1
1.2
0 0.25 0.5 0.75 1 1.25 1.5
Energy (e.V)
Reflection Coffecient
a=0.5 nm a=1 nm a=2 nm a=4 nm
E
NoticethatR=1-T
Dr. Hatem El-Refaei 36
Plotting the wave function
M
Now we have found all
constantsintermsofA. M
We can plot the shape of
the wave function in all
regions.
M
One would expect that as the barrier gets smaller in height
and/or narrower in width more particles will be able to cross
thebarriertotheotherside.
M
Thisresultsinahighertransmissioncoefficient“T”,andalsoa
largeramplitudeforthetransmittedwave“G”. M
Check:
http://phys.educ.ksu.edu/vqm/html/qtunneling.html
G
Plotting the wave function
M
Now we have found all
constantsintermsofA. M
We can plot the shape of
the wave function in all
regions.
M
One would expect that as the barrier gets smaller in height
and/or narrower in width more particles will be able to cross
thebarriertotheotherside.
M
Thisresultsinahighertransmissioncoefficient“T”,andalsoa
largeramplitudeforthetransmittedwave“G”. M
Check:
http://phys.educ.ksu.edu/vqm/html/qtunneling.html
G
Dr. Hatem El-Refaei 37
Contradiction with Classical Mechanics
M
This results are in contradiction with the classical mechanics
which predicts that is the particle’s energy is lower than the
barrier height, the particle overcome the barrier and thus can
notexistintherighthandside.
G
Contradiction with Classical Mechanics
M
This results are in contradiction with the classical mechanics
which predicts that is the particle’s energy is lower than the
barrier height, the particle overcome the barrier and thus can
notexistintherighthandside.
G
Dr. Hatem El-Refaei 38
Remember classical mechanics
A man at rest here
Will never be able to
pass this point
So he can’t
exist here
Remember classical mechanics
A man at rest here
Will never be able to
pass this point
So he can’t
exist here
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