The Department of Physics

Course Description

This is a calculus-based advanced level physics course covering the basic theory of quantum mechanics. The course is compulsory for the major in Physics. Its purpose is to describe the behaviour of microscopic systems in physics such as atoms and particles in confined space. The mathematical formalism is developed to describe these microscopic systems. Students are given problems to solve for homework and these are then discussed during the tutorial sessions. Independent work by students on these problems is critical in developing problem-solving skills as well as increasing their understanding of the applications of the theory. Assessment and evaluation is done in the form of in-course tests and a Final examination.

CONTENT

• The origins of quantum physics: Review of Blackbody radiation, the Photoelectric effect and the Compton Effect. Wave properties of material particles and electron diffraction. The Bohr atom.
• The Schrödinger equation: Wave-particle duality: radiation as particles and electrons as waves. Development of a wave equation for a free particle and for a particle moving in a potential. The time dependent and time-independent Schrödinger equations. The wave function and Born’s probability interpretation of the wave function. Heisenberg’s Uncertainty Principle. The momentum and energy operators.
• One-dimensional problems: The free particle. Solutions to the Schrödinger equation for the infinite potential well. Stationary states of the infinite well. The potential barrier and quantum tunnelling. The harmonic oscillator. Applications.
• Three-dimensional problems: Wave functions of the infinite cubical well. Degeneracy of the energy levels. Wave functions of the hydrogen atom and degeneracy of the spectrum.
• Eigenfunctions, eigenvalues and operators: The eigenfunctions, eigenvalues and Hamiltonian operator of the Schrödinger equation. Normalization and completeness of the eigenfunctions. Eigenvalues and measurement. The superposition principle and generalized time-dependent wave functions. Properties of wave functions. Expectation values of position and momentum.
• Orbital and spin angular momentum: Representation of orbital angular momentum in quantum mechanics. Eigenfunctions of L2 and Lz . Orbital magnetic moment in terms of orbital angular momentum. The Stern-Gerlack experiment and the spin hypothesis. Theory of spin 1/2 and the Pauli matrices. Spin magnetic moment of the electron in terms of spin angular momentum. Applications.

GOALS/AIMS

• To enable students to develop a good understanding of the quantum mechanics required for more advanced studies of Physics.
• To produce graduates with good critical thinking and problem solving skills enabling them to apply quantum mechanical principles to the study and analysis of phenomena in Physics.

LEARNING OUTCOMES

After successfully completing this course, students should be able to explain and apply the following to solve problems in Quantum Mechanics:

• Seminal experiments and effects theoretically which led to the birth of quantum mechanics. The behaviour of particles as waves and waves as particles; De Broglie’s formulation of wave-particle duality; Bohr’s planetary model of the hydrogen atom and its energy spectrum.
• Schrödinger equation for a plane wave from energy considerations and the De Broglie relation. Separation of variables applied to the equation leading to the time-independent Schrödinger equation. Analyse the probability interpretation of the wave function. Define momentum and energy operators. Calculate uncertainties in position and momentum.
• Plane wave solutions to the Schrödinger equation for a particle moving freely in an unbounded region in one dimension. The energy spectrum and wave functions. Solutions for a particle trapped in a region of finite length-the infinite square well. Discrete energy spectrum, wave functions and position and momentum uncertainties. Reflection and transmission probabilities for a particle with energy less that the height of a rectangular potential barrier. The tunnel effect. Applications of tunnelling. Schrödinger’s equation for the linear harmonic oscillator. Solution of Schrödinger’ sequation using ladder operators. The spectrum of the harmonic oscillator. Applications.
• The three-dimensional Schrödinger equation. Schrödinger’s equation in Cartesian coordinates. The free particle in 3D. The energy levels and wave functions of a particle trapped in a cubical box. Schrödinger’s equation in spherical coordinates. Separation of variables in spherical coordinates. Applications to the hydrogen atom. The spectrum of the hydrogen atom derived from the Schrödinger equation.
• Schrödinger equation as an eigenvalue equation. The Hamiltonian operator, its eigenfunctions and eigenvalues. Normalization and orthogonality of the eigenfunctions. Application of these concepts to the particle in an infinite square well. The expansion postulate (Superposition principle) and its physical interpretation. The momentum operator, its eigenvalue equation and momentum eigenfunctions.
• Definition of orbital angular momentum,L, in terms of linear momentum. Use of the linear momentum operator to construct the 3D orbital angular momentum operator. Representation of L2 and Lz in Cartesian and spherical coordinates. Eigenfunctions and eigenvalues of L2 and Lz. Orbital magnetic moment.The Stern-Gerlack experiment and the spin hypothesis of Ulenbeck and Goudsmit. Theory of spin 1/2 and matrix representations of spin. The spin magnetic moment of the electron. Spin 1/2 particles in magnetic fields.

Assessment