ACM30210 Foundations of Quantum Mechanics
Academic Year 2022/2023
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
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Curricular information is subject to change
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:
| 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
| Examination: 2 hour End of Trimester Exam |
2 hour End of Trimester Exam |
No |
Standard conversion grade scale 40% |
No |
60 |
| Continuous Assessment: Take-home assignments and in-class exams |
Varies over the Trimester |
n/a |
Standard conversion grade scale 40% |
No |
40 |
Carry forward of passed components
No
Feedback Strategy/Strategies
• Group/class feedback, post-assessment
How will my Feedback be Delivered?
Not yet recorded.
| Professor Adrian Ottewill |
Lecturer / Co-Lecturer |
| Dr Manya Sahni |
Lecturer / Co-Lecturer |
| Manya Sahni |
Lecturer / Co-Lecturer |