PX2132: Introductory Quantum Mechanics
| School | Cardiff School of Physics & Astronomy |
| Department Code | PHYSX |
| Module Code | PX2132 |
| External Subject Code | 100425 |
| Number of Credits | 10 |
| Level | L5 |
| Language of Delivery | English |
| Module Leader | Dr Martin Elliott |
| Semester | Autumn Semester |
| Academic Year | 2013/4 |
Outline Description of Module
To provide a description of matter and radiation by wave mechanics, in particular through the Schrödinger equation and the interpretation and use of its solutions.
To introduce formal aspects of wave mechanics, and to provide worked examples to increase awareness of the meaning of wavefunctions, eigenvalues, eigenfunctions and operators.
To apply quantum mechanics to a range of model systems.
On completion of the module a student should be able to
Recall and use basic quantum-mechanical definitions, such as Schrödinger’s time-independent and time-dependent wave equations, expectation values, energy and momentum operators.
Derive, from first principles, normalised energy eigenfunctions and eigenvalues in 1-D potentials.
Describe scattering from step potentials and explain the principles of quantum-mechanical tunnelling.
Show an appreciation of the time development of quantum states and uncertainty.
Solve problems relating to the quantum states of the hydrogen atom, showing some understanding of angular momentum.
How the module will be delivered
Lectures 22 x 1 hr, marked exercises.
Skills that will be practised and developed
Mathematics. Problem solving. Investigative skills. Analytical skills.
How the module will be assessed
Examination and Continuous Assessment.
Assessment Breakdown
| Type | % | Title | Duration(hrs) |
|---|---|---|---|
| Exam - Autumn Semester | 80 | Introductory Quantum Mechanics | 2 |
| Written Assessment | 20 | Introductory Quantum Mechanics | N/A |
Syllabus content
Quantum Concepts: The failures of classical physics and Planck's hypothesis. Wave-particle duality - photons and matter waves. Compton effect. The Rutherford-Bohr atom and quantisation of energy.
Introductory Wave Mechanics: Schrödinger's equation and wave functions and their physical interpretation. Probability density. Normalisation. Expectation values and observables. Operators. Schrödinger's time-independent wave equation. Energy eigenvalues and eigenfunctions. Eigenvalue equations. Correspondence principle and complementarity.
1-D Potential Wells: The infinite square well. Energy-level diagrams. Orthogonality. Parity. The finite square well (including graphical solution methods). The harmonic oscillator potential. Finite polynomials and Hermite's equation. Physical relevance of the solutions. (This section is completed on a work-sheet in the PC Laboratory.)
Scattering Potentials: Probability flux density. Step potentials and barrier potentials. Reflection and transmission coefficient. Tunnelling. Examples such as alpha decay and scanning tunnelling microscope.
Free Particles and the time development of quantum states: Uncertainty revisited. Wave groups. Wave function superposition. Wave function collapse and measurements. Compatibility of operators.
2-D and 3-D Systems: Operators and wave equations in 3-D. Symmetry and separation of variables. Worked example: particle in a 2-D box. Energy levels and degeneracy.
The Hydrogen Atom: Radial solutions (Laguerre's equation). Radial probability densities. H-atom energy levels. Separation of variables. The four quantum numbers. “Centrifugal barrier”. Spherical harmonics. Angular momentum. Space quantisation. Full solution to the H-atom and its physical significance.
Essential Reading and Resource List
Principles of Physics (Extended Version), Halliday, Resnick and Walker (Wiley).
Introduction to Quantum Mechanics, A C Phillips (Wiley).
Introductory Quantum Mechanics, R C Greenhow (Adam Hilger). Now out of print, but copies in the library.
Concepts of Modern Physics, 5th Edn., A Beiser (McGraw-Hill).