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 2014/5

Outline Description of Module

  • To provide foundations of the description of matter by wave mechanics, in particular through the Schrödinger equation and the interpretation and use of the wave function.
  • To introduce more formal aspects of wave mechanics.
  • To use worked examples and model systems to develop understanding of the meaning of wave functions, eigenvalues, eigenfunctions and operators.
  • To apply quantum mechanics to describing the hydrogen atom.

On completion of the module a student should be able to

  • Recall and use basic quantum-mechanical concepts, including Schrödinger’s time-independent and time-dependent wave equations, expectation values, operators, and the uncertainty principle.
  • Find normalised energy eigenfunctions and eigenvalues in some simple 1-D potentials.
  • Describe scattering from step potentials.
  • Describe quantum-mechanical tunnelling.
  • Be able to describe mathematically the time development of quantum states.
  • Appreciate how angular momentum appears in quantum mechanics. Solve problems relating to this and quantum states of the hydrogen atom.

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 80%.  Coursework 20%.  [Examination duration: 2 hours].

Assessment Breakdown

Type % Title Duration(hrs)
Exam - Autumn Semester 80 Introductory Quantum Mechanics 2
Written Assessment 20 Introductory Quantum Mechanics N/A

Syllabus content

Introductory quantum concepts: Brief historical survey. Difficulties of classical physics. The Rutherford-Bohr atom and quantisation of energy. Black-body radiation and Planck’s hypothesis. Photoelectric effect.

Compton effect. Wave-particle duality.

Foundations of wave mechanics: Schrödinger’s equation and the wave function; physical interpretation.

Probability density, normalisation. Postulates of quantum mechanics. Observables as operators. Expectation values and observables.

Schrödinger’s time-independent wave equation: The Hamiltonian. Eigenvalue equations. Stationary states and time-independent probability distributions. Example of a 1-D region of constant potential; method of solution; boundary conditions.

1-D potential wells: The infinite square well (interior and exterior solutions; boundary conditions; orthogonality; parity). Energy eigenvalues and eigenfunctions and their physical meaning. Uncertainty principle. The infinite square well (numerical solution methods). The harmonic oscillator potential. Finite polynomials and Hermite’s equation. Physical relevance of the solutions.

Scattering potentials: Probability flux density, continuity. Step potentials and barrier potentials. Reflection and transmission coefficient. Tunnelling. Examples such as alpha decay and scanning tunnelling microscope.
Quantum measurement: Wave function superposition. General quantum state. Complete sets and orthogonal functions. Wave function collapse and measurements. Compatibility of operators and the commutation relations. Commutation with Hamiltonian. Waves, wave packets and the uncertainty principle revisited. Summary of postulates.

2-D and 3-D systems: Operators and wave equations in 3-D. Symmetry and separation of variables. Schrödinger’s equation in 3-D Cartesian and in spherical polar coordinates. Worked example: particle in a 2-D box. Energy levels and degeneracy.

Angular momentum: Cartesian representation of angular momentum operators; commutation relations; polar representation of angular momentum operators; eigenfunctions and eigenvalues. Example: rotational energy levels of a diatomic molecule.

The Hydrogen atom: Separation of variables. Angular equation. Radial solutions (Laguerre’s equation). Radial probability densities. H-atom energy levels. The four quantum numbers. “Centrifugal barrier”. Spherical harmonics. Full solution to the H-atom and its physical significance.

Essential Reading and Resource List

Please see Background Reading List for an indicative list.

Background Reading and Resource List

Introduction to Quantum Mechanics, A C Phillips (Wiley).
Principles of Physics (Extended Version), Halliday, Resnick and Walker (Wiley).
Quantum Mechanics, A Rae (Taylor and Francis 2007)
Concepts of Modern Physics, 5th Edn., A Beiser (McGraw-Hill).


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