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ACM30210

Academic Year 2022/2023

Foundations of Quantum Mechanics (ACM30210)

Subject:
Applied & Computational Maths
College:
Science
School:
Mathematics & Statistics
Level:
3 (Degree)
Credits:
5
Module Coordinator:
Assoc Professor Miguel Bustamante
Trimester:
Spring
Mode of Delivery:
On Campus
Internship Module:
No
How will I be graded?
Letter grades

Curricular information is subject to change.

This module introduces Quantum Mechanics in its modern mathematical setting. Several canonical, exactly-solvable models are studied, including the harmonic oscillator, and the Hydrogen atom.

The postulates of Quantum Mechanics, [Mathematical background] Complex vector spaces and scalar products, linear forms and duality, the natural scalar product derived from linear forms, Hilbert spaces, linear operators, commutation relations, expectation values, uncertainty, [Time evolution and the Schrodinger equation] Derivation of the Schrodinger equation for time-independent Hamiltonians, the position and momentum representations, the probability current, the free particle [The Hydrogen atom] Quantization of energy and angular momentum, general treatment of central potentials in terms of spherical harmonics, [Angular momentum] Motivation: angular momentum in the hydrogen atom, as derived from spherical harmonics, angular momentum in the abstract setting, intrinsic angular momentum, addition of angular momenta, Clebsch-Gordan coefficients, [Piecewise constant one-dimensional potentials] Bound and unbound states, wells and barriers, scattering, transmission coefficients, tunneling, [The harmonic oscillator] Creation and annihilation operators, coherent states, [Approximation methods] Time-independent perturbation theory: the non-degenerate case, variational methods for estimating the ground-state energy [Introduction to Quantum Information] Qubits and quantum logic gates

Further topics may include: Spin coherent states, how to build a microwave laser, the Dyson series for time-evolution for time-dependent Hamiltonians, one-dimensional Dirac potentials, time-independent perturbation theory for degenerate eigenstates, the fine structure of Hydrogen, numerical (spectral) methods for solving the Schrodinger equation

About this Module

Learning Outcomes:

On completion of this module students should be able to

1. Perform standard linear-algebra calculations as they relate to the mathematical foundations of Quantum Mechanics;
2. Solve standard problems for systems with finite-dimensional Hilbert spaces, e.g. the two-level system
3. Solve standard one-dimensional models including the Harmonic oscillator;
4. Use of creation and annihilation operators, including the characterisation of coherent states;
5. Compute expectation values for appropriate observables for the Hydrogen atom;
6. Explain the quantum theory of angular momentum and compute expectation values for appropriate observables. These computations will involve both the matrix representation of intrinsic angular momentum, and the spherical-harmonic representation of orbital angular momentum;
7. Add independent angular momenta in the quantum-mechanical fashion;
8. Understand the foundations of quantum logic

Student Effort Hours:
Student Effort Type Hours
Lectures

31
Specified Learning Activities

24
Autonomous Student Learning

45
Total

100

Approaches to Teaching and Learning:
Lectures and problem-based learning

Requirements, Exclusions and Recommendations
Learning Recommendations:

Students should have followed
ACM30010 Analytical Mechanics
or equivalent.


Module Requisites and Incompatibles
Not applicable to this module.
 

Assessment Strategy Invalid Option
Description Timing Open Book Exam Component Scale Must Pass Component % of Final Grade In Module Component Repeat Offered
Continuous Assessment: Take-home assignments and in-class exams Varies over the Trimester n/a Standard conversion grade scale 40% No
40
 No
Examination: 2 hour End of Trimester Exam 2 hour End of Trimester Exam No Standard conversion grade scale 40% No
60
 No

Carry forward of passed components Invalid Option
No
 

Resit In Terminal Exam
Autumn Yes - 2 Hour
Please see Student Jargon Buster for more information about remediation types and timing. 

Feedback Strategy/Strategies

• Group/class feedback, post-assessment

How will my Feedback be Delivered?

Not yet recorded.

Name Role
Professor Adrian Ottewill Lecturer / Co-Lecturer
Dr Manya Sahni Lecturer / Co-Lecturer
Manya Sahni Lecturer / Co-Lecturer
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