1. Atilla Ozgur Cakmak, PhD
Nanophotonics
Atilla Ozgur Cakmak, PhD

2. Lecture 5: Electron in complex potentials-Part1
Unit 1
Lecture 5: Electron in complex potentials-Part1

3. Outline Uncertainty Principle Harmonic Oscillator Molecular Vibrations
Classical treatment
The quantum mechanical treatment

4. A couple of words…
We have covered the most fundamental scenarios in the previous lecture and the electron confinement due to these potentials. Now, we can continue with more advanced potentials.
Suggested reading: David J. Griffiths, Introduction to Quantum Mechanics, 2nd edition, 2nd and 4th Chapters.

5. Uncertainty Principle

6. Uncertainty Principle

7. Uncertainty Principle

8. Uncertainty Principle
Large well
Narrow well
A narrow well confines the particle to a very small space, however the momentum is spread out in k-space.
Physically showing the outcome of the uncertainty principle.

9. Harmonic Oscillator
Harmonic Oscillator is an important model that serves as the basis for the treatment of vibrations in molecules.
Molecular Vibrations

10. Harmonic Oscillator Classical treatment Image credits:

11. Harmonic Oscillator

12. The quantum mechanical treatment
Harmonic Oscillator
The quantum mechanical treatment

13. Harmonic Oscillator (problem)
Find the Hamiltonian of the Harmonic Oscillator problem in terms of the newly defined operators, a and a†.
a) H=ћω (a +a†)
b) H=ћω (a +a†)/2
c) H=ћω (aa† +a†a)/2
d) H=ћω (a†a† +aa)/2

14. Harmonic Oscillator (solution)
Find the Hamiltonian of the Harmonic Oscillator problem in terms of the newly defined operators, a and a†.

15. Harmonic Oscillator (problem)
Find [a,a†].
a) 0
b) ћω/2
c) 1
d) ћω

16. Harmonic Oscillator (solution)
Find [a,a†].

17. Harmonic Oscillator

18. Harmonic Oscillator (problem)
If φ0 is the ground state with a φ0 =0. Find a†aa† φ0
a) 0
b) φ0
c) 1
d) φ1

19. Harmonic Oscillator (solution)
If φ0 is the ground state with a φ0 =0. Find a†aa† φ0

20. Harmonic Oscillator Image credits:

21. Harmonic Oscillator

22. Harmonic Oscillator (problem)
Generate φ1 from φ0. a) b) c) d)

23. Harmonic Oscillator (solution)
Generate φ1 from φ0.

24. Harmonic Oscillator (problem)
Calculate the following for a particle in the ground state:
a) ћω/2, ћω/2
b) 3ћω/2, 5ћω/2
c) 3ћω/2, 0
d) 3ћω/2, ћω/2

25. Harmonic Oscillator (solution)
Calculate the following for a particle in the ground state:

26. Harmonic Oscillator
2nd
1st V
0th