Python Notes for mca i year students osmania university.docx
Physics barriers and tunneling
1. 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
3. Note
All problems today are unbounded problem, i.e. the
particle is not confined in a certain region, so:
We will not be able to do the normalization condition.
Therefore, we will not be able to solve for all unknowns.
Therefore, we will not get a characteristic equation.
Therefore, energy levels are not quantized, and all energies
are possible.
But still there are a lot of important characteristics to
understand and learn today.
hatem@oilpeakinarabic.org Dr. Hatem El-Refaei 2
5. Potential step of infinite height and infinite width
Energy
∞
∞
E
x
Since the barrier height is infinite, incident particles can’t
penetrate through it, and particles reflect back.
So, there is zero probability of finding the particle inside
the step barrier.
Here, the QM solution leads to the same classical solution.
hatem@oilpeakinarabic.org Dr. Hatem El-Refaei 4
6. Potential step of infinite height and infinite
width
Energy, ψ
∞
∞
2mE
k=
h E
x
d ψ I 2m
2 0
2
+ 2 E ψI = 0
dx h
U =∞
ψ ( x ) = Ae jkx
+ Be − jkx
E
j kx − t
E
j − kx − t ψ (x ) = 0
Ψ (x, t ) = Ae h
+ Be h
Forward Backward
propagating wave propagating wave
hatem@oilpeakinarabic.org Dr. Hatem El-Refaei 5
7. Potential step of infinite height and infinite
width
Energy, ψ
∞
∞
2mE
k= E
h
x
d ψ I 2m
2 0
2
+ 2 E ψI = 0
dx h
U =∞
ψ ( x ) = Ae jkx
+ Be − jkx
ψ (x = 0 − ) = ψ (x = 0 + ) ψ (x ) = 0
A+ B = 0
hatem@oilpeakinarabic.org Dr. Hatem El-Refaei 6
8. Potential step of infinite height and infinite
width
Energy, ψ
∞
∞
Sin(x) E
x
d ψ I 2m
2 0
2
+ 2 E ψI = 0
dx h
B = −A U =∞
ψ (x ) = Ae jkx − Ae − jkx ψ (x ) = 0
ψ (x ) = A(e jkx − e − jkx )
ψ ( x ) = 2 jA sin (kx )
hatem@oilpeakinarabic.org Dr. Hatem El-Refaei 7
9. Potential step of infinite height and infinite
width
Energy, ψ
∞
∞
Sin(x) E
x
0
B = −A
2
Re flected Amplitude B∗B
Reflectivity R= = ∗ =1
Incident Amplitude A A
All the incident particle stream is reflected back.
hatem@oilpeakinarabic.org Dr. Hatem El-Refaei 8
10. Potential step of infinite height and infinite
width
Energy, ψ
∞
∞
Sin(x) E
x
0
Since the barrier extends to infinity in the x direction, no particle can penetrate
through the whole barrier. From phenomenological understanding, as x→∞,
ψ→0.
hatem@oilpeakinarabic.org Dr. Hatem El-Refaei 9
12. Potential step of finite height and infinite
width
Energy
E<Uo
∞
E
Uo
x
0
As the step height Uo gets smaller (but still E<Uo),
the penetration of the particles inside the step
increases, but finally no particles will succeed to
travel through the whole step to x→∞.
hatem@oilpeakinarabic.org Dr. Hatem El-Refaei 11
13. Potential step of finite height and infinite
width
2mE Energy 2m(U o − E )
k= E<Uo α= >0
h h
∞
E
Uo
x
0
d 2ψ I 2m d 2ψ II 2m
2
+ 2 EψI =0 2
+ 2 (E − U o ) ψ II = 0
dx h dx h
ψ I ( x ) = Ae jkx + Be − jkx ψ II ( x ) = C e −α x + Deα x
We have 4 unknowns (A,B,C, and D) and 3 equations:
Finiteness of ψ at x=∞
Continuity of ψ at x=0
Continuity of ∇ψ at x=0
hatem@oilpeakinarabic.org Dr. Hatem El-Refaei 12
14. Potential step of finite height and infinite
width
2mE Energy 2m(U o − E )
k= E<Uo α= >0
h h
∞
E
Uo
x
0
d 2ψ I 2m d 2ψ II 2m
2
+ 2 EψI =0 2
+ 2 (E − U o ) ψ II = 0
dx h dx h
ψ I ( x ) = Ae jkx + Be − jkx ψ II ( x ) = C e −α x + Deα x
The condition that ψ(x) must be finite as x→∞, leads to D=0
hatem@oilpeakinarabic.org Dr. Hatem El-Refaei 13
15. Potential step of finite height and infinite
width
2mE Energy 2m(U o − E )
k= E<Uo α= >0
h h
∞
E
Uo
x
0
d 2ψ I 2m d 2ψ II 2m
2
+ 2 EψI =0 2
+ 2 (E − U o ) ψ II = 0
dx h dx h
ψ I ( x ) = Ae jkx + Be − jkx ψ II (x ) = C e −α x
The condition that ψ(x) must be finite as x→∞, leads to D=0
hatem@oilpeakinarabic.org Dr. Hatem El-Refaei 14
16. Potential step of finite height and infinite
width
2mE Energy 2m(U o − E )
k= E<Uo α= >0
h h
∞
E
Uo
x
0
d 2ψ I 2m d 2ψ II 2m
2
+ 2 EψI =0 2
+ 2 (E − U o ) ψ II = 0
dx h dx h
ψ I ( x ) = Ae jkx + Be − jkx ψ II (x ) = C e −α x
We have 3 unknowns (A,B,C) and 2 equations:
Continuity of ψ
Continuity of ∇ψ
Thus, the best we can get is the ratio between parameters.
hatem@oilpeakinarabic.org Dr. Hatem El-Refaei 15
17. Potential step of finite height and infinite
width
2mE Energy 2m(U o − E )
k= E<Uo α= >0
h h
∞
E
Uo
x
ψ I ( x ) = Ae jkx + Be − jkx 0
ψ II (x ) = C e −α x
Continuity of ψ Continuity of dψ/dx
dψ I dψ II
ψ I ( x = 0) = ψ II (x = 0) =
dx dx
jk ( A − B ) = −α C
x =0 x =0
A+ B = C
1 α 1 α
A= 1 + j C B = 1 − j C
2 k 2 k
hatem@oilpeakinarabic.org Dr. Hatem El-Refaei 16
18. Potential step of finite height and infinite
width
2mE Energy 2m(U o − E )
k= E<Uo α= >0
h h
E
0
1 α 1 α
A= 1+ j C B = 1− j C
2 k 2 k
α α
2 1 −
∗ j 1 + j
Re flected Amplitude BB k k
Reflectivity R= = ∗ = =1
Incident Amplitude AA α α
1 + j 1 − j
k k
hatem@oilpeakinarabic.org Dr. Hatem El-Refaei 17
19. Potential step of finite height and infinite
width
Energy
Uo
∞
C
x
0
1 α jkx 1 α − jkx
ψ I (x ) = Ae + Be
jkx − jkx
= 1 + j Ce + 1 − j Ce
2 k 2 k
e jkx + e − jkx α e jkx − e − jkx
ψ I (x ) = C
+ j C
2 k 2
α
C cos(kx ) − C sin (kx ) x≤0
ψ (x ) = k
C e −α x x ≥ 0
hatem@oilpeakinarabic.org Dr. Hatem El-Refaei 18
20. Potential step of finite height and infinite
width
Energy
Uo
∞
C
x
0
Since the barrier height is finite, particles can penetrate partially
in the vicinity of the potential step, and then they reflect back.
So, there is a finite probability of finding the particle in the
classically forbidden position.
Here, the QM solution is different from the classical one.
hatem@oilpeakinarabic.org Dr. Hatem El-Refaei 19
21. Potential step of finite height and infinite
width
Energy
E<Uo Case 3 2m(U o − E )
α= >0
Case 2 h
Case 2 ψ ( x ) = C e −α x
x
0
Case 3
As the potential barrier height increases (Uo
increases) (from case 2 to case 3), α also increases,
and thus the exponential function dies quicker inside
the barrier. Thus it becomes less probable to find the
particle inside the barrier.
hatem@oilpeakinarabic.org Dr. Hatem El-Refaei 20
22. Potential step of finite height and infinite
width
Energy
E<Uo 2m(U o − E )
α= >0
h
ψ ( x ) = C e −α x
x
0
If the barrier height (Uo) is kept constant, but the particle
energy increases (provided E<Uo), thus α decreases, and
the particle exponential function dies slower inside the
barrier. Hence, it becomes more probable to find the
particle in the vicinity of the barrier edge.
hatem@oilpeakinarabic.org Dr. Hatem El-Refaei 21
23. Tunneling through a potential
barrier
hatem@oilpeakinarabic.org Dr. Hatem El-Refaei 22
24. Potential barrier of finite height and finite
width
Energy E<Uo
Uo
E
x
A stream of particles incident on a finite width and height
potential barrier with E<Uo.
Part of the incident stream will succeed to penetrate through the
barrier and appear on the other side, this is the transmitted stream.
The other part will reflect back forming the reflected stream.
Note, a single particle doesn’t split into two.
hatem@oilpeakinarabic.org Dr. Hatem El-Refaei 23
25. Potential barrier of finite height and finite
width
Energy E<Uo
2mE Uo 2m(U o − E )
k= γ= >0
h h
0 a x
d 2ψ I 2m d 2ψ II 2m d 2ψ III 2m
+ 2 EψI = 0 + 2 (E − U o ) ψ II = 0 + 2 E ψ III = 0
dx 2 h dx 2 h dx 2 h
ψ I ( x ) = Ae jkx + Be − jkx ψ II (x ) = C eγ x + D e −γ x ψ III (x ) = Ge jkx + Fe − jkx
We have 6 unknowns (A,B,C,D,G,F) and 5 boundary conditions
Continuity of ψ and ∇ψ at x=0.
Continuity of ψ and ∇ψ at x=a.
Only a forward propagating wave on the right hand side of the barrier.
hatem@oilpeakinarabic.org Dr. Hatem El-Refaei 24
26. Potential barrier of finite height and finite
width
Energy E<Uo
Uo
0 a x
d 2ψ I 2m d 2ψ II 2m d 2ψ III 2m
+ 2 EψI = 0 + 2 (E − U o ) ψ II = 0 + 2 E ψ III = 0
dx 2 h dx 2 h dx 2 h
ψ I ( x ) = Ae jkx + Be − jkx ψ II (x ) = C eγ x + D e −γ x ψ III (x ) = Ge jkx + Fe − jkx
We have 6 unknowns (A,B,C,D,G,F) and 5 boundary conditions
Continuity of ψ and ∇ψ at x=0.
Continuity of ψ and ∇ψ at x=a.
Only a forward propagating wave on the right hand side of the barrier.
hatem@oilpeakinarabic.org Dr. Hatem El-Refaei 25
27. Potential barrier of finite height and finite
width
Energy E<Uo
Uo
0 a x
d 2ψ I 2m d 2ψ II 2m d 2ψ III 2m
+ 2 EψI = 0 + 2 (E − U o ) ψ II = 0 + 2 E ψ III = 0
dx 2 h dx 2 h dx 2 h
ψ I ( x ) = Ae jkx + Be − jkx ψ II (x ) = C eγ x + D e −γ x ψ III ( x ) = Ge jkx
We have 5 unknowns (A,B,C,D,G) and 4 boundary conditions
Continuity of ψ and ∇ψ at x=0.
Continuity of ψ and ∇ψ at x=a.
hatem@oilpeakinarabic.org Dr. Hatem El-Refaei 26
28. Potential barrier of finite height and finite
width
Energy E<Uo
Uo
0 a x
d 2ψ I 2m d 2ψ II 2m d 2ψ III 2m
+ 2 EψI = 0 + 2 (E − U o ) ψ II = 0 + 2 E ψ III = 0
dx 2 h dx 2 h dx 2 h
ψ I ( x ) = Ae jkx + Be − jkx ψ II (x ) = C eγ x + D e −γ x ψ III ( x ) = Ge jkx
We are interested in the transmission probability (T)
2 2
Transmitte d Amplitude G ∗G G
T= = ∗ =
Incident Amplitude A A A
hatem@oilpeakinarabic.org Dr. Hatem El-Refaei 27
29. Potential barrier of finite height and finite
width
Energy E<Uo
Uo
0 a x
d 2ψ I 2m d 2ψ II 2m d 2ψ III 2m
+ 2 EψI = 0 + 2 (E − U o ) ψ II = 0 + 2 E ψ III = 0
dx 2 h dx 2 h dx 2 h
ψ I ( x ) = Ae jkx + Be − jkx ψ II (x ) = C eγ x + D e −γ x ψ III ( x ) = Ge jkx
And reflection probability (R)
2 2
Re flected Amplitude B ∗B B
R = = ∗ =
Incident Amplitude A A A
hatem@oilpeakinarabic.org Dr. Hatem El-Refaei 28
30. Potential barrier of finite height and finite
width
First set of boundary conditions at x=0
dψ I dψ II
ψ I (x = 0) = ψ II (x = 0) =
dx x =0 dx x =0
A+ B = C + D jkA − jkB = γC − γD
Second set of boundary conditions at x=a
dψ II dψ III
ψ II (x = a ) = ψ III (x = a ) =
dx x =a dx x =a
C eγa + D e −γa = G eika γ C eγa − γ D e −γa = ik G eika
hatem@oilpeakinarabic.org Dr. Hatem El-Refaei 29
31. Potential barrier of finite height and finite
width
A+ B = C + D (1)
jkA − jkB = γC − γD (2)
γa −γa (3)
C e +De =Ge ika
γ C eγa − γ D e −γa = ik G eika (4)
Eq. (3) × γ + Eq. (4) to eliminate D.
γ + jk ( jk −γ )a
C= e G (5)
2γ
Eq. (3) × (-γ) + Eq. (4) to eliminate C.
γ − jk ( jk +γ )a
D= e G (6)
2γ
hatem@oilpeakinarabic.org Dr. Hatem El-Refaei 30
32. Potential barrier of finite height and finite
width
A+ B = C + D (1)
jkA − jkB = γC − γD (2)
γa −γa (3)
C e +De =Ge ika
γ C eγa − γ D e −γa = ik G eika (4)
Eq. (1) × jk + Eq. (2) to eliminate B
2 jkA = ( jk + γ )C + ( jk − γ ) D (7)
Substitute from eq. (5) and (6) into (7), we get an equation of A
and G only.
2 jkA =
1
2γ
[ ]
(γ + jk )2 e( jk −γ )a − (γ − jk )2 e( jk +γ )a G (8)
hatem@oilpeakinarabic.org Dr. Hatem El-Refaei 31
33. Transmission and Reflection Coefficients
It is an assignment to show that the transmission coefficient is
given by
G ∗G 1
T= ∗ = 2
AA 1 U
1+ o
sinh 2 (γa )
4 E (U o − E )
Also it is an assignment to show that the reflection coefficient
is given by
1 U o2
sinh 2 (γa )
B∗ B 4 E (U o − E )
R= ∗ =
AA 1 U o2
1+ sinh 2 (γa )
4 E (U o − E )
hatem@oilpeakinarabic.org Dr. Hatem El-Refaei 32
34. Transmission and Reflection Coefficients
After proving both relations, it will be clear to you that
R +T =1
Which is logical as an incident particle is either reflected or
transmitted.
You may need to use the following relations
e z − e− z
sinh ( z ) =
2
e z + e− z
cosh ( z ) =
2
cosh 2 ( z ) − sinh 2 ( z ) = 1
hatem@oilpeakinarabic.org Dr. Hatem El-Refaei 33
35. Transmission coefficient versus barrier width
An electron is tunneling through 1.5 eV barrier
0.35
Transmission Coffecient
0.3
0.25
a=0.5 nm
0.2 a=1 nm
0.15 a=2 nm
a=4 nm
0.1
0.05
0
0 0.25 0.5 0.75 1 1.25 1.5
Energy (e.V)
One notices that shrinking the barrier width by half results in a dramatic
increase in the transmission coefficient. It is not a linear relation.
Also doubling the particle energy results in exponential increase in the
transmission coefficient.
hatem@oilpeakinarabic.org Dr. Hatem El-Refaei 34
36. Reflection coefficient versus barrier width
An electron is tunneling through 1.5 eV barrier
1.2
1
Reflection Coffecient
0.8 a=0.5 nm
a=1 nm
0.6
a=2 nm
0.4 a=4 nm
0.2
0
0 0.25 0.5 0.75 1 1.25 1.5
Energy (e.V)
Notice that R=1-T
hatem@oilpeakinarabic.org Dr. Hatem El-Refaei 35
37. Plotting the wave function
Now we have found all
constants in terms of A.
G
We can plot the shape of
the wave function in all
regions.
One would expect that as the barrier gets smaller in height
and/or narrower in width more particles will be able to cross
the barrier to the other side.
This results in a higher transmission coefficient “T”, and also a
larger amplitude for the transmitted wave “G”.
Check: http://phys.educ.ksu.edu/vqm/html/qtunneling.html
hatem@oilpeakinarabic.org Dr. Hatem El-Refaei 36
38. Contradiction with Classical Mechanics
G
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
not exist in the right hand side.
hatem@oilpeakinarabic.org Dr. Hatem El-Refaei 37
39. Remember classical mechanics
Will never be able to
A man at rest here pass this point
So he can’t
exist here
hatem@oilpeakinarabic.org Dr. Hatem El-Refaei 38
40. Ain Shams University
Mathematics and Engineering Physics Department
1st Year Electrical Engineering
Lecture 11
Modern Physics and Quantum Mechanics Course
Dr. Hatem El-Refaei
42. Note
All problems today are unbounded problem, i.e. the
particle is not confined in a certain region, so:
We will not be able to do the normalization condition.
Therefore, we will not be able to solve for all unknowns.
Therefore, we will not get a characteristic equation.
Therefore, energy levels are not quantized, and all energies
are possible.
But still there are a lot of important characteristics to
understand and learn today.
Dr. Hatem El-Refaei 2
44. Potential step of infinite height and infinite width
Energy
∞
∞
E
x
Since the barrier height is infinite, incident particles can’t
penetrate through it, and particles reflect back.
So, there is zero probability of finding the particle inside
the step barrier.
Here, the QM solution leads to the same classical solution.
Dr. Hatem El-Refaei 4
45. Potential step of infinite height and infinite
width
Energy, ψ
∞
∞
2mE
k=
h E
x
d ψ I 2m
2 0
2
+ 2 E ψI = 0
dx h
U =∞
ψ ( x ) = Ae jkx
+ Be − jkx
E
j kx − t
E
j − kx − t ψ (x ) = 0
Ψ (x, t ) = Ae h
+ Be h
Forward Backward
propagating wave propagating wave
Dr. Hatem El-Refaei 5
46. Potential step of infinite height and infinite
width
Energy, ψ
∞
∞
2mE
k= E
h
x
d ψ I 2m
2 0
2
+ 2 E ψI = 0
dx h
U =∞
ψ ( x ) = Ae jkx
+ Be − jkx
ψ (x = 0 − ) = ψ (x = 0 + ) ψ (x ) = 0
A+ B = 0
Dr. Hatem El-Refaei 6
47. Potential step of infinite height and infinite
width
Energy, ψ
∞
∞
Sin(x) E
x
d ψ I 2m
2 0
2
+ 2 E ψI = 0
dx h
B = −A U =∞
ψ (x ) = Ae jkx − Ae − jkx ψ (x ) = 0
ψ (x ) = A(e jkx − e − jkx )
ψ ( x ) = 2 jA sin (kx )
Dr. Hatem El-Refaei 7
48. Potential step of infinite height and infinite
width
Energy, ψ
∞
∞
Sin(x) E
x
0
B = −A
2
Re flected Amplitude B∗B
Reflectivity R= = ∗ =1
Incident Amplitude A A
All the incident particle stream is reflected back.
Dr. Hatem El-Refaei 8
49. Potential step of infinite height and infinite
width
Energy, ψ
∞
∞
Sin(x) E
x
0
Since the barrier extends to infinity in the x direction, no particle can penetrate
through the whole barrier. From phenomenological understanding, as x→∞,
ψ→0.
Dr. Hatem El-Refaei 9
51. Potential step of finite height and infinite
width
Energy
E<Uo
∞
E
Uo
x
0
As the step height Uo gets smaller (but still E<Uo),
the penetration of the particles inside the step
increases, but finally no particles will succeed to
travel through the whole step to x→∞.
Dr. Hatem El-Refaei 11
52. Potential step of finite height and infinite
width
2mE Energy 2m(U o − E )
k= E<Uo α= >0
h h
∞
E
Uo
x
0
d 2ψ I 2m d 2ψ II 2m
2
+ 2 EψI =0 2
+ 2 (E − U o ) ψ II = 0
dx h dx h
ψ I ( x ) = Ae jkx + Be − jkx ψ II ( x ) = C e −α x + Deα x
We have 4 unknowns (A,B,C, and D) and 3 equations:
Finiteness of ψ at x=∞
Continuity of ψ at x=0
Continuity of ∇ψ at x=0
Dr. Hatem El-Refaei 12
53. Potential step of finite height and infinite
width
2mE Energy 2m(U o − E )
k= E<Uo α= >0
h h
∞
E
Uo
x
0
d 2ψ I 2m d 2ψ II 2m
2
+ 2 EψI =0 2
+ 2 (E − U o ) ψ II = 0
dx h dx h
ψ I ( x ) = Ae jkx + Be − jkx ψ II ( x ) = C e −α x + Deα x
The condition that ψ(x) must be finite as x→∞, leads to D=0
Dr. Hatem El-Refaei 13
54. Potential step of finite height and infinite
width
2mE Energy 2m(U o − E )
k= E<Uo α= >0
h h
∞
E
Uo
x
0
d 2ψ I 2m d 2ψ II 2m
2
+ 2 EψI =0 2
+ 2 (E − U o ) ψ II = 0
dx h dx h
ψ I ( x ) = Ae jkx + Be − jkx ψ II (x ) = C e −α x
The condition that ψ(x) must be finite as x→∞, leads to D=0
Dr. Hatem El-Refaei 14
55. Potential step of finite height and infinite
width
2mE Energy 2m(U o − E )
k= E<Uo α= >0
h h
∞
E
Uo
x
0
d 2ψ I 2m d 2ψ II 2m
2
+ 2 EψI =0 2
+ 2 (E − U o ) ψ II = 0
dx h dx h
ψ I ( x ) = Ae jkx + Be − jkx ψ II (x ) = C e −α x
We have 3 unknowns (A,B,C) and 2 equations:
Continuity of ψ
Continuity of ∇ψ
Thus, the best we can get is the ratio between parameters.
Dr. Hatem El-Refaei 15
56. Potential step of finite height and infinite
width
2mE Energy 2m(U o − E )
k= E<Uo α= >0
h h
∞
E
Uo
x
ψ I ( x ) = Ae jkx + Be − jkx 0
ψ II (x ) = C e −α x
Continuity of ψ Continuity of dψ/dx
dψ I dψ II
ψ I ( x = 0) = ψ II (x = 0) =
dx dx
jk ( A − B ) = −α C
x =0 x =0
A+ B = C
1 α 1 α
A= 1 + j C B = 1 − j C
2 k 2 k
Dr. Hatem El-Refaei 16
57. Potential step of finite height and infinite
width
2mE Energy 2m(U o − E )
k= E<Uo α= >0
h h
E
0
1 α 1 α
A= 1+ j C B = 1− j C
2 k 2 k
α α
2 1 −
∗ j 1 + j
Re flected Amplitude BB k k
Reflectivity R= = ∗ = =1
Incident Amplitude AA α α
1 + j 1 − j
k k
Dr. Hatem El-Refaei 17
58. Potential step of finite height and infinite
width
Energy
Uo
∞
C
x
0
1 α jkx 1 α − jkx
ψ I (x ) = Ae + Be
jkx − jkx
= 1 + j Ce + 1 − j Ce
2 k 2 k
e jkx + e − jkx α e jkx − e − jkx
ψ I (x ) = C
+ j C
2 k 2
α
C cos(kx ) − C sin (kx ) x≤0
ψ (x ) = k
C e −α x x ≥ 0
Dr. Hatem El-Refaei 18
59. Potential step of finite height and infinite
width
Energy
Uo
∞
C
x
0
Since the barrier height is finite, particles can penetrate partially
in the vicinity of the potential step, and then they reflect back.
So, there is a finite probability of finding the particle in the
classically forbidden position.
Here, the QM solution is different from the classical one.
Dr. Hatem El-Refaei 19
60. Potential step of finite height and infinite
width
Energy
E<Uo Case 3 2m(U o − E )
α= >0
Case 2 h
Case 2 ψ ( x ) = C e −α x
x
0
Case 3
As the potential barrier height increases (Uo
increases) (from case 2 to case 3), α also increases,
and thus the exponential function dies quicker inside
the barrier. Thus it becomes less probable to find the
particle inside the barrier.
Dr. Hatem El-Refaei 20
61. Potential step of finite height and infinite
width
Energy
E<Uo 2m(U o − E )
α= >0
h
ψ ( x ) = C e −α x
x
0
If the barrier height (Uo) is kept constant, but the particle
energy increases (provided E<Uo), thus α decreases, and
the particle exponential function dies slower inside the
barrier. Hence, it becomes more probable to find the
particle in the vicinity of the barrier edge.
Dr. Hatem El-Refaei 21
63. Potential barrier of finite height and finite
width
Energy E<Uo
Uo
E
x
A stream of particles incident on a finite width and height
potential barrier with E<Uo.
Part of the incident stream will succeed to penetrate through the
barrier and appear on the other side, this is the transmitted stream.
The other part will reflect back forming the reflected stream.
Note, a single particle doesn’t split into two.
Dr. Hatem El-Refaei 23
64. Potential barrier of finite height and finite
width
Energy E<Uo
2mE Uo 2m(U o − E )
k= γ= >0
h h
0 a x
d 2ψ I 2m d 2ψ II 2m d 2ψ III 2m
+ 2 EψI = 0 + 2 (E − U o ) ψ II = 0 + 2 E ψ III = 0
dx 2 h dx 2 h dx 2 h
ψ I ( x ) = Ae jkx + Be − jkx ψ II (x ) = C eγ x + D e −γ x ψ III (x ) = Ge jkx + Fe − jkx
We have 6 unknowns (A,B,C,D,G,F) and 5 boundary conditions
Continuity of ψ and ∇ψ at x=0.
Continuity of ψ and ∇ψ at x=a.
Only a forward propagating wave on the right hand side of the barrier.
Dr. Hatem El-Refaei 24
65. Potential barrier of finite height and finite
width
Energy E<Uo
Uo
0 a x
d 2ψ I 2m d 2ψ II 2m d 2ψ III 2m
+ 2 EψI = 0 + 2 (E − U o ) ψ II = 0 + 2 E ψ III = 0
dx 2 h dx 2 h dx 2 h
ψ I ( x ) = Ae jkx + Be − jkx ψ II (x ) = C eγ x + D e −γ x ψ III (x ) = Ge jkx + Fe − jkx
We have 6 unknowns (A,B,C,D,G,F) and 5 boundary conditions
Continuity of ψ and ∇ψ at x=0.
Continuity of ψ and ∇ψ at x=a.
Only a forward propagating wave on the right hand side of the barrier.
Dr. Hatem El-Refaei 25
66. Potential barrier of finite height and finite
width
Energy E<Uo
Uo
0 a x
d 2ψ I 2m d 2ψ II 2m d 2ψ III 2m
+ 2 EψI = 0 + 2 (E − U o ) ψ II = 0 + 2 E ψ III = 0
dx 2 h dx 2 h dx 2 h
ψ I ( x ) = Ae jkx + Be − jkx ψ II (x ) = C eγ x + D e −γ x ψ III ( x ) = Ge jkx
We have 5 unknowns (A,B,C,D,G) and 4 boundary conditions
Continuity of ψ and ∇ψ at x=0.
Continuity of ψ and ∇ψ at x=a.
Dr. Hatem El-Refaei 26
67. Potential barrier of finite height and finite
width
Energy E<Uo
Uo
0 a x
d 2ψ I 2m d 2ψ II 2m d 2ψ III 2m
+ 2 EψI = 0 + 2 (E − U o ) ψ II = 0 + 2 E ψ III = 0
dx 2 h dx 2 h dx 2 h
ψ I ( x ) = Ae jkx + Be − jkx ψ II (x ) = C eγ x + D e −γ x ψ III ( x ) = Ge jkx
We are interested in the transmission probability (T)
2 2
Transmitte d Amplitude G ∗G G
T= = ∗ =
Incident Amplitude A A A
Dr. Hatem El-Refaei 27
68. Potential barrier of finite height and finite
width
Energy E<Uo
Uo
0 a x
d 2ψ I 2m d 2ψ II 2m d 2ψ III 2m
+ 2 EψI = 0 + 2 (E − U o ) ψ II = 0 + 2 E ψ III = 0
dx 2 h dx 2 h dx 2 h
ψ I ( x ) = Ae jkx + Be − jkx ψ II (x ) = C eγ x + D e −γ x ψ III ( x ) = Ge jkx
And reflection probability (R)
2 2
Re flected Amplitude B ∗B B
R = = ∗ =
Incident Amplitude A A A
Dr. Hatem El-Refaei 28
69. Potential barrier of finite height and finite
width
First set of boundary conditions at x=0
dψ I dψ II
ψ I (x = 0) = ψ II (x = 0) =
dx x =0 dx x =0
A+ B = C + D jkA − jkB = γC − γD
Second set of boundary conditions at x=a
dψ II dψ III
ψ II (x = a ) = ψ III (x = a ) =
dx x =a dx x =a
C eγa + D e −γa = G eika γ C eγa − γ D e −γa = ik G eika
Dr. Hatem El-Refaei 29
70. Potential barrier of finite height and finite
width
A+ B = C + D (1)
jkA − jkB = γC − γD (2)
γa −γa (3)
C e +De =Ge ika
γa − γa
γ C e − γ D e = ik G e ika (4)
Eq. (3) × γ + Eq. (4) to eliminate D.
γ + jk ( jk −γ )a
C= e G (5)
2γ
Eq. (3) × (-γ) + Eq. (4) to eliminate C.
γ − jk ( jk +γ )a
D= e G (6)
2γ
Dr. Hatem El-Refaei 30
71. Potential barrier of finite height and finite
width
A+ B = C + D (1)
jkA − jkB = γC − γD (2)
γa −γa (3)
C e +De =Ge ika
γa − γa
γ C e − γ D e = ik G e ika (4)
Eq. (1) × jk + Eq. (2) to eliminate B
2 jkA = ( jk + γ )C + ( jk − γ ) D (7)
Substitute from eq. (5) and (6) into (7), we get an equation of A
and G only.
2 jkA =
1
2γ
[ ]
(γ + jk )2 e( jk −γ )a − (γ − jk )2 e( jk +γ )a G (8)
Dr. Hatem El-Refaei 31
72. Transmission and Reflection Coefficients
It is an assignment to show that the transmission coefficient is
given by
G ∗G 1
T= ∗ = 2
AA 1 U
1+ o
sinh 2 (γa )
4 E (U o − E )
Also it is an assignment to show that the reflection coefficient
is given by
1 U o2
sinh 2 (γa )
B∗ B 4 E (U o − E )
R= ∗ =
AA 1 U o2
1+ sinh 2 (γa )
4 E (U o − E )
Dr. Hatem El-Refaei 32
73. Transmission and Reflection Coefficients
After proving both relations, it will be clear to you that
R +T =1
Which is logical as an incident particle is either reflected or
transmitted.
You may need to use the following relations
e z − e− z
sinh ( z ) =
2
e z + e− z
cosh ( z ) =
2
cosh 2 ( z ) − sinh 2 ( z ) = 1
Dr. Hatem El-Refaei 33
74. Transmission coefficient versus barrier width
An electron is tunneling through 1.5 eV barrier
0.35
Transmission Coffecient
0.3
0.25
a=0.5 nm
0.2 a=1 nm
0.15 a=2 nm
a=4 nm
0.1
0.05
0
0 0.25 0.5 0.75 1 1.25 1.5
Energy (e.V)
One notices that shrinking the barrier width by half results in a dramatic
increase in the transmission coefficient. It is not a linear relation.
Also doubling the particle energy results in exponential increase in the
transmission coefficient.
Dr. Hatem El-Refaei 34
75. Reflection coefficient versus barrier width
An electron is tunneling through 1.5 eV barrier
1.2
1
Reflection Coffecient
0.8 a=0.5 nm
a=1 nm
0.6
a=2 nm
0.4 a=4 nm
0.2
0
0 0.25 0.5 0.75 1 1.25 1.5
Energy (e.V)
Notice that R=1-T
Dr. Hatem El-Refaei 35
76. Plotting the wave function
Now we have found all
constants in terms of A.
G
We can plot the shape of
the wave function in all
regions.
One would expect that as the barrier gets smaller in height
and/or narrower in width more particles will be able to cross
the barrier to the other side.
This results in a higher transmission coefficient “T”, and also a
larger amplitude for the transmitted wave “G”.
Check: http://phys.educ.ksu.edu/vqm/html/qtunneling.html
Dr. Hatem El-Refaei 36
77. Contradiction with Classical Mechanics
G
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
not exist in the right hand side.
Dr. Hatem El-Refaei 37
78. Remember classical mechanics
Will never be able to
A man at rest here pass this point
So he can’t
exist here
Dr. Hatem El-Refaei 38