Unbound States
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Transcript of Unbound States
Unbound StatesUnbound States
1.1. A review on calculations for the A review on calculations for the potential step.potential step.
2.2. Quiz 10.23Quiz 10.233.3. Topics in Unbound States:Topics in Unbound States:
The potential step. The potential step. Two steps: The potential barrier Two steps: The potential barrier
and tunneling. and tunneling. Real-life examples: Alpha decay Real-life examples: Alpha decay
and other applications.and other applications. A summary: Particle-wave A summary: Particle-wave
propagation.propagation.
today
The potential step: solve the The potential step: solve the equationequationKE E
00U x U
0x
0KE E U E
0 0U x
Initial condition: free particles moving from left to right.
x
0U UWhen
22
22
d xU x x E x
m dx
The Schrödinger Equation:
When 0U
2
22 2
2d x mEx k x
dx
20 2
2 2
2d x m E Ux k' x
dx
ikx ikxx Ae Be Solution:
Inc.
Refl.
Trans.
Apply normalization and wave function smoothness
0E UWhen
ik' xx Ce
The potential step: solve the The potential step: solve the equationequationKE E
00U x U
0x
0 0KE E U
E
0 0U x
Initial condition: free particles moving from left to right.
x
22
22
d xU x x E x
m dx
The Schrödinger Equation:
0U UWhen
2
0 22 2
2d x m U Ex x
dx
ikx ikxx Ae Be Solution:
Inc.
Refl.
When 0U
2
22 2
2d x mEx k x
dx
0E UWhen
xx Ce
0E U
The potential step: transmission The potential step: transmission and reflectionand reflection
0x x
Transmission probability:
Reflection probability:
0
2
0
4E E U
TE E U
2
0
2
0
E E UR
E E U
0E UWhen0E UWhen
Transmission probability:
1 0T R
Reflection probability:
1*
*
B BR
A A
Penetration depth: 0
1
2m U E
E
0x x
E0U 0U
Two steps: The potential barrier Two steps: The potential barrier and tunneling.and tunneling.
00U x L U
0x
E
0 0U x
Initial condition: free particles moving from left to right.
xx L
0U x L
0E UWhen 0E UWhen Tunneling
0: ikx ikxx x Ae Be Solution:
Inc.
Refl. 0 : ik' x ik' xx L x Ce De
: ikxx L x Fe Trans.
0: ikx ikxx x Ae Be Solution:
Inc.
Refl. 0 : x xx L x Ce De
: ikxx L x Fe Trans.Apply normalization and
wave function smoothness
Two steps: The potential barrier Two steps: The potential barrier and tunneling.and tunneling.
0E UWhen 0E UWhen Tunneling
Results:
Results:
20
20 0 0
sin 2
sin 2 4 1
m E U LR
m E U L E U E U
0 0
20 0 0
4 1
sin 2 4 1
E U E UT
m E U L E U E U
2 2 20
0 2
2when: or
2
m E U nL n E U
mL
0R
Resonant transmission.Thin film optics analogy.
20
20 0 0
sinh 2
sinh 2 4 1
m E U LR
m E U L E U E U
0 0
20 0 0
4 1
sinh 2 4 1
E U E UT
m E U L E U E U
Tunneling through a wide Tunneling through a wide barrierbarrier
Wide barrier: 02
1m E UL
L L
02 2
0 0
16 1L m E UE E
T eU U
Tunneling:
Transmission probability is very sensitive to barrier width L and the energy E. This leads to some wonderful applications of QM.
Real-life examples: Alpha decay Real-life examples: Alpha decay and other applications.and other applications.
Who took my cheese? Who took the energy from my alphas?
Scanning Tunneling Scanning Tunneling MicroscopeMicroscope
Please read about the tunneling diode, field emission and the SQUIDS yourself.We will discuss about the STM here.
A summary: Particle-wave A summary: Particle-wave propagation.propagation.
Review questionsReview questions
Please review the solutions to the Please review the solutions to the Schrödinger equation with the step Schrödinger equation with the step and two steps condition and make and two steps condition and make sure that you feel comfortable with sure that you feel comfortable with the results.the results.
Preview for the next class Preview for the next class (10/28)(10/28)
Text to be read:Text to be read: Please skim from 7.1 to 7.8. If you have Please skim from 7.1 to 7.8. If you have
difficulty in understanding the materials, see difficulty in understanding the materials, see the slides by next Monday.the slides by next Monday.
Questions:Questions: What is the fundamental change to move the What is the fundamental change to move the
Schrödinger equation from 1-D to 3-D? Schrödinger equation from 1-D to 3-D? What is the quantization condition for the What is the quantization condition for the zz
component of angular momentum? component of angular momentum? According to QM, can you have a visual According to QM, can you have a visual
presentation for the electron’s whereabouts in presentation for the electron’s whereabouts in a hydrogen atom? a hydrogen atom?
Homework 9, due by 10/30Homework 9, due by 10/30
1.1. Problem 21 on page 224.Problem 21 on page 224.
2.2. Problem 24 on page 225.Problem 24 on page 225.
3.3. Problem 32 on page 225.Problem 32 on page 225.