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Lecture15 – Chapter. 6
Unbound States
Outline:
•! The Potential Step
•! The Potential Barrier & Tunneling
•! Alpha Decay & Other Applications
•! Particle-Wave Propagation
1)'(
)'(
)'(
'2''4
)'(
)'(
)'(
'4
?
2
2
2
22
2
2
2
=+
+=
=+
!++=
=+
!+
+=+
=+
kk
kk
kk
kkkkkk
kk
kk
kk
kkRT
RT
T + R = 1
C = [2k/(k+k’)]A
B = [(k-k’_/(k+k’)]A
Like in
Optics
Finally, we express the Prob. in terms of U0 & E using the definitions of k, k’
5 eV
2 eV
3 eV
Potential Wall
E > U
Case (2) 6.2 The Potential Barrier
&
Tunneling
•!Tunneling is one of the most important & startling"
ideas in QM. "
•!The simplest solution is a potential barrier, a PE "
jumps that is only temporary."
•! If a particle#s E < Barrier#s height, (i.e. E < U0) "
it should not get through - classically"
Potential Barrier
R(L)
T(L)
(!)
Resonant Transmission
Is there any “L” that would give
T = 1, R = 0
? Like i
n OPTIC
S
Resonant Transmission
Is there any “L” that would give
T = 1, R = 0
? Like i
n OPTIC
S
= 0
Resonant Transmission
T = 1, R = 0
for:
Potential Wall (KE<U) – “Potential Step”
< 0
The solution is not oscillatory !
(exponentially damped wave function)
C = 0
> 0
100% Reflected“Penetration
depth”Section 5.6
Reflection (R) &transmission (T)
probabilities for a potential step
If E < U0! wave is totally reflected.
When E > U0! R falls rapidly with increasing E
Region
II
Would a
particle
penetrate
all the way
through?
Barrier Penetration – “Tunneling”
Barrier Penetration – “Tunneling” Barrier Penetration – “Tunneling”
Reflection (R)
&transmission (T) probabilities for a potential
barrier
1st
Transmission
resonance
tunneling