Study-Unit Description

Study-Unit Description


CODE PHY2180

 
TITLE Quantum Mechanics 2

 
UM LEVEL 02 - Years 2, 3 in Modular Undergraduate Course

 
MQF LEVEL Not Applicable

 
ECTS CREDITS 4

 
DEPARTMENT Physics

 
DESCRIPTION This study-unit constitutes an advanced course in quantum mechanics.

Study-unit Aims

This study-unit will provide:

- the derivation of the continuity equation for quantum mechanics and the definition of probability current;
- the derivation of the time evolution of expectation values;
- the derivation of Ehrenfest theorem or equations;
- a discussion on the correspondence between quantum and classical mechanics;
- the definition of the reflection and transmission coefficients and the practical application for one dimension problems;
- harder examples of how to solve Schrödinger equation in one dimension when the potential is a constant;
- the mathematical treatment of the linear quantum harmonic oscillator in one dimension, including the solution of the Schrödinger equation for the system using ladder operators;
- the mathematical treatment of free particles in one dimension including its evolution with time;
- the extension of Schrödinger equation to three dimension including its form in spherical coordinates;
- the definition of the angular momentum operators including their corresponding commutation relations;
- the derivation of the eigenfunction of the total angular momentum;
- an analysis of Schrödinger equation for a central potential;
- the derivation of the solution of Schrödinger equation for the hydrogen atom;
- a detailed review of linear algebra;
- the extension of angular momentum to spin;
- the derivation and definition of the Pauli spin matrices;
- a heuristic description of the Stern-Gerlach experiment that verifies the existence of spin.

Learning Outcomes

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

- explain the meaning of the continuity equation as well as the meaning of each term it contains;
- explain how quantum mechanics links with classical mechanics;
- explain what the reflection and transmission coefficients are and why their sum is always equal to one;
- explain what is meant by quantum tunnelling;
- explain what is meant by ladder operators;
- explain what is meant by free particles;
- define the eigenfunction of the total angular momentum;
- explain how the formulation of angular momentum is extended to spin;
- list at least one particle that has spin a half;
- explain how the Stern-Gerlach experiment is carried out.

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

- work out the expectation value of an operator;
- calculate the probability current for a given wavefunction;
- show that a given wavefunction satisfies the continuity equation;
- calculate the time evolution of expectation variables;
- calculate the reflection and transmission coefficients for a given wave function;
- solve Schrödinger equation in one dimension when the potential is piecewise constant and hence derive the general solution;
- use the ladder operators for a linear harmonic oscillator to obtain the possible states of the system;
- obtain the wavefunction in k or p -space from its spatial space equivalent and vice versa;
- use the transformation between k or p -space and spatial space to determine how a given wave-packet will evolve in time;
- work out the commutation relations of the angular momentum operators;
- use the results for the solution of Schrödinger equation for the hydrogen atom to calculate the possible states of the system.

Main Text/s and any supplementary readings

Recommended textbooks:
Zettili, N., Quantum Mechanics, concepts and applications, John Wiley and Sons
or
Griffiths, D. J., Introduction to Quantum Mechanics, Prentice Hall

Bibliography:
Liboff, R. L., Introductory Quantum Mechanics, Addison-Wesley
Cassels, J. M., Basic Quantum Mechanics, Macmillan
Pebbles, P. J. E., Quantum Mechanics, Princeton University Press
Goswani, A., Quantum Mechanics, Wm C Brown Publishers
Eisberg R. and Resnick, R., Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles, John Wiley and Sons
Park, D., Introduction to the Quantum Theory, McGraw-Hill

Additional reading:
Al Khalili, J., Quantum: A Guide for the Perplexed, Weidenfeld Nicolson

The use of other books with similar content would be equivalently good.

 
ADDITIONAL NOTES Pre-Requisites: Basic knowledge of Mathematics including calculus. Waves and quantum mechanics or a course with similar material.

 
STUDY-UNIT TYPE Lecture

 
METHOD OF ASSESSMENT
Assessment Component/s Sept. Asst Session Weighting
Examination (2 Hours) Yes 100%

 
LECTURER/S

 

 
The University makes every effort to ensure that the published Courses Plans, Programmes of Study and Study-Unit information are complete and up-to-date at the time of publication. The University reserves the right to make changes in case errors are detected after publication.
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 2023/4. It may be subject to change in subsequent years.

https://www.um.edu.mt/course/studyunit