ACM30210 Foundations of Quantum Mechanics
Academic Year 2020/2021
This module introduces Quantum Mechanics in its modern mathematical setting. Several canonical, exactly-solvable models are studied, including one-dimensional piecewise constant potentials, Dirac potentials, the harmonic oscillator, and the Hydrogen atom. Three calculational techniques are introduced: time-independent perturbation theory, variational methods, and numerical (spectral) methods.
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 [Piecewise constant one-dimensional potentials] Bound and unbound states, wells and barriers, scattering, transmission coefficients, tunneling, [The harmonic oscillator] Solution by power series, Hermite polynomials, creation and annihilation operators, coherent states, [The Hydrogen atom] Solution by separation of variables, 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, [Approximation methods] Time-independent perturbation theory: the non-degenerate case, variational methods for estimating the ground-state energy
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 piecewise constant potential wells and barriers, Dirac potentials, and the Harmonic oscillator;
4. Perform calculations based on Hermite polynomials, including the characterization 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. Perform time-independent non-degenerate perturbation theory up to and including the second order
Student Effort Hours:
Lectures |
36 |
Specified Learning Activities |
24 |
Autonomous Student Learning |
40 |
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
Continuous Assessment: Assignments |
Varies over the Trimester |
n/a |
Other |
No |
70 |
Class Test: Class Test |
Unspecified |
n/a |
Other |
No |
30 |
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 |