Study-Unit Description

Study-Unit Description


TITLE Quantum Mechanics

LEVEL 02 - Years 2, 3 in Modular Undergraduate Course



DESCRIPTION This study-unit introduces physics students to quantum mechanics. It introduces the basic concepts of quantum mechanics, and focuses on Dirac notation, operator methods, and mixed quantum states. The study-unit will begin with an exploration of the Stern--Gerlach experiment, which will be used to introduce the concept of Dirac notation. The time evolution of quantum systems will be investigated, leading to a discussion of the Schrödinger and Heisenberg pictures.

Some properties and implications of the wavefunction will be discussed, including reflection and transmission through a barrier, quantum tunnelling, the notion of the probability current, and the continuity equation. The rate of change of expectation values with time and the Ehrenfest theorem will be derived. The quantum harmonic oscillator will be used as a concrete example of several of these concepts. The theory of angular momentum, and the corresponding operators, will be discussed and used to introduce the abstract concept of spin. Density matrices will be introduced with an emphasis on their connection with, and differences from, pure quantum states. The concepts of measurement and quantum correlations will be introduced through a discussion of Bell's inequality and its relation with the EPR paradox. An exploration of time-independent perturbation theory will see the study-unit going beyond exactly-solvable problems. Finally, the theory of open quantum systems will be briefly introduced towards the end of the course, providing a connection to real-world situations.

Study-unit Aims:

This study-unit will provide students with a comprehensive introduction to the concepts and main mathematical tools used in modern quantum mechanics. It will put them in a better position to appreciate recent advances in the field as well as some of the many outstanding issues in quantum mechanics.

Learning Outcomes:

1. Knowledge & Understanding
By the end of the study-unit the student will be able to:

- describe and analyse the Stern--Gerlach experiment;
- explain Dirac notation and its use in solving quantum mechanics problems;
- calculate the time evolution of states of a simple quantum system starting from its Hamiltonian;
- describe how the Hamilton--Jacobi equation relates to the classical limit of quantum mechanics;
- describe the assumptions underlying Bell's inequality, and explain why quantum mechanics violates it;
- compare situations that require pure states versus those calling for mixed states;
- describe how time-independent perturbation theory can be used to find approximate results in complex potentials;
- explain the major differences between closed and open quantum systems.

2. Skills
By the end of the study-unit the student will be able to:

- use Dirac notation to solve quantum mechanics problems involving the evolution of systems governed by simple Hamiltonians and ones undergoing measurements;
- use operator methods to transform to the interaction picture and to solve quantum mechanics problems;
- analyse the quantum harmonic oscillator, and calculate statistical properties (e.g., means and variances) related the states of such a system;
- derive the Hamilton--Jacobi equation and use it to obtain an approximate (WKB) solution to particles in particular potentials;
- use time-independent perturbation theory to calculate modifications of wave functions and eigenenergies in simple situations;
- calculate the evolution of mixed quantum states for simple open quantum systems.

Main Text/s and any supplementary readings:


- J J Sakurai, "Modern Quantum Mechanics" (Addison-Wesley); chs. 1 to 5


- N Zettili, "Quantum Mechanics: Concepts and Applications" (John Wiley & Sons)
- J Townsend, "A Modern Approach to Quantum Mechanics" (University Science Books)


Assessment Component/s Assessment Due Resit Availability Weighting
Worksheets SEM2 No 20%
Oral Examination SEM2 Yes 40%
Assignment SEM2 Yes 40%

LECTURER/S Andre Xuereb

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The availability of optional units may be subject to timetabling constraints.
Units not attracting a sufficient number of registrations may be withdrawn without notice.
It should be noted that all the information in the description above applies to study-units available during the academic year 2021/2. It may be subject to change in subsequent years.