1. Unbound States A review on calculations for the potential step.
Topics in Unbound States:
The potential step.
Two steps: The potential barrier and tunneling.
Real-life examples: Alpha decay and other applications.
A summary: Particle-wave propagation.
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2. The potential step: solve the equation
Initial condition: free particles moving from left to right.
The Schrödinger Equation:
When
When
When
Solution:
Inc.
Refl.
Trans.
Apply normalization and wave function smoothness

3. The potential step: solve the equation
Initial condition: free particles moving from left to right.
The Schrödinger Equation:
When
When
When
Solution:
Inc.
Refl.

4. The potential step: transmission and reflection
When
When
Reflection probability:
Reflection probability:
Transmission probability:
Transmission probability:
Penetration depth:

5. Examples
An electron of kinetic energy 5 eV encounters a 2 eV potential step. What is the probability that it will be reflected?
This is the case when
Straightforward, right? How about this:
An electron of kinetic energy 5 eV encounters a 2 eV potential step down. What is the probability that it will be reflected?
Not 0!

6. Two steps: The potential barrier and tunneling.
Initial condition: free particles moving from left to right.
When
When
 Tunneling
Solution:
Solution:
Inc.
Refl.
Inc.
Refl.
Trans.
Trans.
Apply normalization and wave function smoothness

7. Two steps: The potential barrier and tunneling.
When
When
 Tunneling
Results:
Results:
Resonant transmission.
Thin film optics analogy.

8. Tunneling through a wide barrier
Transmission probability is very sensitive to barrier width L and the energy E. This leads to some wonderful applications of QM.
How sensitive? An electron encounters a barrier of 5.0 eV with a width of 1.6 nm. What is the transmission probability when the electron is (a) 2.0 eV and (b) 3.0 eV?
When E = 2.0 eV When E = 3.0 eV
50% energy increase leads to 150 times more transmission!

9. Real-life examples: Alpha decay and other applications.
Who took my cheese? Who took the energy from my alphas?

10. The Tunnel Diode
Invented in 1957 by Leo Esaki (Nobel prize in 1973) et al., a tunnel diode is also called an Esaki diode. First manufactured by Sony in 1957, tunnel diodes are still produced in small volume today and used in frequency converters and detectors, and sometimes in oscillators and amplifiers as well.
A tunnel diode is made by highly doped p and n areas that form a very thin (~10 nm) depletion region. The thickness of the depletion region is controlled by external voltage, hence the probability of electrons tunneling through this barrier is also controlled.
For more reading materials about this device please check:
I-V curve of a tunnel diode with a negative resistance region

11. SQUID, Field Emission and STM
A SQUID, superconducting quantum interference device, is a very sensitive magnetometer based on superconducting loops that contain Josephson (1973 Nobel prize) Junctions, the tunneling of the Cooper pairs.
Field electron emission is emission of electrons induced by an electrostatic field, through the tunnel effect.
Scanning Tunneling Microscope.
Please research on these quantum effects and their applications in sciences and technologies. They provide good topics for semester end presentations.

12. Self study: Section 6.4 Particle-wave propagation.
Study section 6.4 and write a one page summary.
We will compare the summaries in next class.

13. 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.

14. Preview for the next class
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?

15. Homework ch6-2, due by 3/31 Problem 21 on page 224.