Atilla Ozgur Cakmak, PhD Nanophotonics Atilla Ozgur Cakmak, PhD

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

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

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.

Uncertainty Principle

Uncertainty Principle

Uncertainty Principle

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.

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

Harmonic Oscillator Classical treatment Image credits: https://www.learner.org/courses/physics/visual/visual.html?shortname=simple_harmonic_oscillator

Harmonic Oscillator

The quantum mechanical treatment Harmonic Oscillator The quantum mechanical treatment

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

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

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

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

Harmonic Oscillator

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

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

Harmonic Oscillator Image credits: http://astro.dur.ac.uk/~done/foundations2A.html

Harmonic Oscillator

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

Harmonic Oscillator (solution) Generate φ1 from φ0.

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

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

Harmonic Oscillator 2nd 1st V 0th