Transcript Precursors to Modern Physics

Unbound States
1.
2.
3.
A review on calculations for the potential
step.
Quiz 10.23
Topics in Unbound States:


today


The potential step.
Two steps: The potential barrier and
tunneling.
Real-life examples: Alpha decay and other
applications.
A summary: Particle-wave propagation.
The potential step: solve the equation
Initial condition: free
particles moving
from left to right.
E
KE  E  U 0
U  x  0  0
x
x0
The Schrödinger Equation: 
2
2m
d 2  x 
dx 2
 U  x   x   E  x 
When U  0
d 2  x 
dx 2

2mE
2
U  x  0  U0
KE  E
When U  U 0
  x   k   x 
2
d 2  x 
dx 2

2m  E  U 0 
2
  x   k' 2   x 
When E  U 0
Solution:
  x   Aeikx  Beikx
Inc.
Refl.
Apply normalization and wave
function smoothness
  x   Ceik' x
Trans.
The potential step: solve the equation
Initial condition: free
particles moving
from left to right.
E
KE  E
U  x  0  0
KE  E  U0  0
x
x0
The Schrödinger Equation: 
2
2m
d 2  x 
dx 2
 U  x   x   E  x 
When U  0
d 2  x 
dx 2

2mE
2
U  x  0  U0
When U  U 0
  x   k   x 
2
d 2  x 
dx 2

2m U 0  E 
2
  x    2  x 
 U0
When E 
Solution:
  x   Aeikx  Beikx
Inc.
Refl.
  x   Ce x
The potential step: transmission and reflection
E
U0
U0
E
x0
x0
x
When E  U 0
Reflection
probability:

R

Transmission T  4
probability:
When E  U 0
E  E U0
E  E U0

x


E  E U0 
E  E U0
Reflection
probability:
2
2
B* B
R  * 1
AA
Transmission T  1  R  0
probability:

2
Penetration
depth:

1


2m U 0  E 
Two steps: The potential barrier and
tunneling.
E
Initial condition: free
particles moving
from left to right.
U  0  x  L   U0
U  x  L  0
U  x  0  0
x0
When E  U 0
When E  U 0
Solution:
x  0:   x   Aeikx  Beikx
Inc.
xL
x
 Tunneling
Solution:
x  0:   x   Aeikx  Beikx
Refl.
Inc.
Refl.
0  x  L:   x   Ceik' x  Deik' x
0  x  L:   x   Ce x  De x
x  L:   x   Feikx
x  L:   x   Feikx
Trans.
Trans.
Apply normalization and wave
function smoothness
Two steps: The potential barrier and
tunneling.
When E  U 0
When E  U 0
Results:
Results:
R
sin 2
sin 2
T
sin
2


 Tunneling


2m  E  U 0  L
  4  E U  E U
2m  E  U 0  L
0
0
 1
4  E U 0  E U 0  1
2m  E  U 0  L
when:
  4  E U  E U
2m  E  U 0 
R0
0
T
0
 1
n2 2 2
L  n or E  U 0 
2mL2
Resonant transmission.
Thin film optics analogy.
R
sinh 2
sinh 2
sinh 2




2m  E  U 0  L
2m  E  U 0  L
  4  E U  E U
0
0
 1
0
 1
4  E U 0  E U 0  1
2m  E  U 0  L
  4  E U  E U
0
Tunneling through a wide barrier
Wide barrier:
L

Tunneling:
L 
2m  E  U0 
E 
E  2 L
T  16 1   e
U0  U0 
L
1
2 m E U 0 
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 and other
applications.
Who took my cheese? Who took the energy from
my alphas?
Scanning Tunneling Microscope
Please read about the tunneling diode,
field emission and the SQUIDS yourself.
We will discuss about the STM here.
A summary: Particle-wave propagation.
Review questions

Please review the solutions to the Schrödinger
equation with the step and two steps condition
and make sure that you feel comfortable with
the results.
Preview for the next class (10/28)

Text to be read:


Please skim from 7.1 to 7.8. If you have difficulty in
understanding the materials, see the slides by next Monday.
Questions:



What is the fundamental change to move the Schrödinger
equation from 1-D to 3-D?
What is the quantization condition for the z component of
angular momentum?
According to QM, can you have a visual presentation for the
electron’s whereabouts in a hydrogen atom?
Homework 9, due by 10/30
1.
2.
3.
Problem 21 on page 224.
Problem 24 on page 225.
Problem 32 on page 225.