Lecture on Quantum tunneling .pdf

A lecture on Quantum tunneling

1. Quantum Tunneling
One of the mysteries of quantum mechanics
W. Ubachs – Lectures MNW-Quant-Tunneling

2. Finite Potential Well
)( )(l l
I
I
I
II
ψ
ψ
=
)( )(l l
dx
d
dx
d
III II
ψ
ψ
=
)0( )0(
II
I
ψ
ψ
=
A finite potential well has a potential of zero between x= 0and x= ,
but outside that range the potential is a constant U
0
.
l
The potential outside the well is no
longer zero; it fall s off exponentially.
Solve in regions I, II, and III
and use for boundary conditions
Continuity:
)0( )0(
dx
d
dx
d
II I
ψ
ψ
=
Bound states: E < E
0
Continuum states: E > E
0

3. Finite Potential Well
(
)
2
0 2
2
h
E Um
G
−
=
(
)
0
2
2
0
2
2
=
⎥
⎦
⎤
⎢
⎣
⎡−
−
ψ
ψ
h
E Um
dx
d
Gx Gx
III I
De Ce
−
+ =
,
ψ
IfE < U
0
with
in the “forbidden regions”
General solution:
Region I
0<x
hence
0
=
D
and similarly for C
Gx
I
Ce=
ψ
should match
kx B kx A
II
cos sin
+
=
ψ
Finite value at
0
=
x
exponentially decayinginto the finite walls

4. Finite Potential Well
These graphs show the wave functions and probability
distributions for the first three energy states.
Nonclassical effects
Partile can exist in the forbidden region

5. Finite Potential Well
()
0
2UEm
h
p
h
−
= =
λ
(
)
0
2
2
0
2
2
=
⎥
⎦
⎤
⎢
⎣
⎡−
+
ψ
ψ
h
UEm
dx
d
0
2
2 2
2
=
⎥
⎦
⎤
⎢
⎣
⎡
+
ψ
ψ
hmE
dx
d
IfE > U
0
In regions I and III
free particle condition
In region II
In both cases oscillating free partcile
wave function:
I,III:
II:
mE
h
p
h
2
= =
λ
0
2
0
2
2 2
1
U
m
p
U mv E+ = + =

6. Tunneling Through a Barrier
l
>
x
0
<
x
mE
h
p
h
2
= =
λ
(
)
0
2
2
0
2
2
=
⎥
⎦
⎤
⎢
⎣
⎡−
−
ψ
ψ
h
E Um
dx
d
0
2
2 2
2
=
⎥
⎦
⎤
⎢
⎣
⎡
+
ψ
ψ
hmE
dx
d
Also in region
In regionoscillating wave
mE
h
p
h
2
= =
λ
Wave with same wavelength
In the barrier:
Gx Gx
b
De Ce
−
+ =
ψ
Approximation: assume that the decaying
function is dominant
Gx
b
De
−
=
ψ
() ()
(
)
l
l
G
Gx
e
D
De
x
x
T
2
2
2
2
2
0
−
−
= =
=
=
=
ψ
ψ
Transmission:

7. Tunneling Through a Barrier
(
)
2
0
2
h
E Um
G
−
=
The probability that a particle tunnels through a barrier
can be expressed as a transmission coefficient, T, and
a reflection coefficient, R(where T+ R= 1). If Tis small,
The smaller Eis with respect to U
0
, the smaller the
probability that the particle will tunnel through the
barrier.
Gl
e
T
2
−
≈