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Chapter 10: Quantum physics
47
10.7
The tunnel effect
The wavefunction of a particle in an
∞
high potential step from
x
=0
to
x
=
a
is given by
ψ
(
x
)=
a

1
/
2
sin(
kx
)
. The energylevels are given by
E
n
=
n
2
h
2
/
8
a
2
m
.
If the wavefunction with energy
W
meets a potential well of
W
0
>W
the wavefunction will, unlike the
classical case, be nonzero within the potential well. If 1, 2 and 3 are the areas in front, within and behind the
potential well, holds:
ψ
1
=
A
e
ikx
+
B
e

ikx
,
ψ
2
=
C
e
ik
±
x
+
D
e

ik
±
x
,
ψ
3
=
A
±
e
ikx
with
k
±
2
=2
m
(
W

W
0
)
/
¯
h
2
and
k
2
=2
mW
. Using the boundary conditions requiring continuity:
ψ
=
continuous and
∂ψ
/
∂
x
=
continuous at
x
=0
and
x
=
a
gives
B
,
C
and
D
and
A
±
expressed in
A
. The
amplitude
T
of the transmitted wave is deFned by
T
=

A
±

2
/

A

2
. If
W>W
0
and
2
a
=
n
λ
±
=2
π
n/k
±
holds:
T
=1
.
10.8
The harmonic oscillator
±or a harmonic oscillator holds:
U
=
1
2
bx
2
and
ω
2
0
=
b/m
. The Hamiltonian
H
is then given by:
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This note was uploaded on 01/30/2011 for the course PHYSICS 208 taught by Professor Ye during the Spring '10 term at Blinn College.
 Spring '10
 Ye
 Energy, Quantum Physics

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