Quantum Mechanics I

Yachay Tech University - 2024

About this repository:

This is a collection of lecture notes and programming exercises carried out as part of the Quantum Mechanics I course at Yachay Tech University.

Lecturer:

Wladimir E. Banda Barragán

Course description:

This course provides an introduction to the formal mathematical treatment of Quantum Mechanics. The course introduces the Schrodinger Equation and its solutions for different potentials, emphasising on its statistical interpretation and its importance for the description of experiments at quantum scales. Topics range from wave functions, the time-independent Schrodinger equation, through Hilbert spaces and the mathematical formalism of quantum mechanics, to the description of the hydrogen atom and two-particle systems. The course includes examples of different applications of quantum mechanics, including writing Hamiltonians for different physical systems and extracting information about them.

Course objectives:

• Understand the fundamental ideas and experiments that led to the formulation of quantum mechanics.

• Learn the mathematical skills and formalism needed to solve Schrödinger’s equation and interpret its solutions.

• Study the Hamiltonians of quantum systems in 1D and 3D for different potentials and coordinates, and provide a detailed quantum description of the hydrogen atom.

• Use quantum mechanics to analyse real microscopic phenomena and interpret experimental data.

Syllabus:

1. The Schrödinger equation

• Review of quantum experiments and mathematical tools.

• The wave function and the Schrödinger equation.

• Statistical interpretation of the wave function and probability.

• Normalisation, momentum, and the uncertainty principle.

2. Quantum Mechanics in 1D

• Stationary states and the time-independent Schrödinger equation.

• Free particles and wave packets.

• Finite, Infinite potential wells, and the harmonic oscillator.

• Delta-function potentials, tunnelling and scattering states.

3. Mathematical formalism of Quantum Mechanics

• Linear algebra, Hermitian operators, and Hilbert space.

• Eigenfunctions, eigenvectors, and eigenvalues for discrete and continuous spectra.

• Dirac notation and the Generalised statistical interpretation.

• Operators of position and momentum and the uncertainty principle.

4. Quantum Mechanics in 3D

• Schrodinger Equations in Spherical Coordinates.

• Coulomb potential and quantum description of the Hydrogen atom.

• Angular momentum and spin.

• Larmor precession and the Stern- Gerlach experiment.

5. Two-Particle Systems and quantum applications

• Identical particles and introduction to two-particle systems.

• Exchange interactions and covalent bonds.

• Atoms and the periodic table.

• Applications of quantum mechanics.

Full Course Syllabus and Programme:

The full course syllabus and programme can be found here:

https://github.com/wbandabarragan/quantum-mechanics-1/blob/main/QM1_Syllabus_Ing_IISEM2024.pdf

https://github.com/wbandabarragan/quantum-mechanics-1/blob/main/QM1_Program_Ing_IISEM2024.pdf

References:

1. David J. Griffiths, Introduction to Quantum Mechanics, 3rd Edition, 2018.

2. John Townsend, A Modern Approach to Quantum Mechanics, 2nd Edition, 2012.

3. Feynman, Leighton, Sands, The Feynmann Lectures on Physics (Volume III), Online (https://www.feynmanlectures.caltech.edu/).

Evaluation:

• Formative evaluation (Homework): 20%

• Laboratory (Quizzes): 20%

• Midterm Exam: 30%

• Final Exam: 30%

Both the midterm and final exams have two components, one is carried out in class, one is carried out at home. Please do your best in every assignment.

Calendar:

The assignment deadlines and exam dates will be discussed and agreed upon in class. Once fixed, all deadlines are hard deadlines.

Weekly tutoring schedule:

If you have questions on the material, you can find me in the office:

• On Mondays: 14h00 – 15h00

• On Tuesdays: 12h00 – 13h00

On academic integrity:

• Students are responsible for ensuring the academic integrity of their submitted assignments and exams.

• Cheating in exams, plagiarising, and copying code or solutions from the Internet, from AI platforms (like chatGPT), from other students, or from previous years' solutions are all breaches of academic integrity.

• Academic misconduct will be penalised according to the University’s regulations. Any assignments that infringe academic integrity (even partially) will receive zero marks.

Late assignment policy:

Late assignments accompanied by appropriate justification (e.g. a medical certificate, etc.) will receive no penalisation. Late assignments without appropriate justification will receive a penalisation of -1% per late hour.

Acknowledgements:

I would like to thank my former students and teaching assistants: Melanie Cedeño, Daniel Arias (https://github.com/VariableCefeida), Juan Castro (https://github.com/JuanAndCast) and Daniel Villarruel (https://github.com/DanV-Y) for their valuable support and help in tutoring this course in previous semesters.

Archived courses:

Please visit the repository below to see the materials developed in previous semesters:

https://github.com/wbandabarragan/quantum-mechanics-1-arxiv