PHY2022 Quantum Mechanics I

2011-2012

Code: PHY2022
Level: 2
Title: Quantum Mechanics I
Instructors: Dr A. Usher
CATS Credit Value: 15
ECTS Credit Value: 7.5
Pre-requisites: N/A
Co-requisites: N/A
Duration: T1:01-11
Availability: unrestricted
Background Assumed: -

Total Student Study Time

150 hours, to include: 22×1-hour lectures; 44 hours directed self-study; 10 hours of problems class support; 3 hours of tutorial support; 72 hours private study.

Aims

Quantum Mechanics is one of the fundamental building-blocks of Physics. It affects profoundly the way we think about the universe and is the basis for much of condensed-matter, nuclear and statistical physics. It also has a strong influence on technological developments, for instance in optical and electronic devices. The purpose of this module is to introduce students to the basic principles of quantum mechanics and to the solutions of problems, which permit straightforward mathematical analysis. These solutions demonstrate many of the general features of the subject and prepare the students for subsequent courses such as PHY3052 Nuclear and High-Energy Particle Physics.

Intended Learning Outcomes

Students will be able to:

1. Module Specific Skills:
   1. describe the definition and interpretation of the wavefunction and of operators in quantum mechanics;
   2. discuss the origin of energy quantisation and quantum tunnelling effects;
   3. describe the general properties of the stationary states of quantum particles confined to simple symmetric potentials;
   4. perform calculations on wavefunctions, and solve the Schrödinger equation for a range of problems;
   5. use time-independent perturbation theory to solve problems and interpret results;
   6. explain the origin of the un-coupled set of quantum numbers for the hydrogen atom and the form of the associated eigenfunctions;
2. Discipline Specific Skills:
   1. use the principles of quantum mechanics to solve problems;
   2. explain quantum mechanics to a lay-person in an informed manner;
3. Personal Transferable Skills:
   1. meet deadlines for completion of work for problems classes and develop appropriate time-management strategies;
   2. construct arguments that explain observations;
   3. solve problems by using mathematics;
   4. use a range of resources to develop an understanding of topics through independent study.

Learning / Teaching Methods

Lectures, e-Learning resources (ELE PHY2022), and problems classes.

Assessment and Assignments

Contribution	Assessment/Assignment	Size (duration/length)	When
10%	Problem Sets	8×2hrs	Weekly
15%	Mid-term Test	30 minutes	Week T1:06
75%	Final examination	120 minutes	Week T2:00
Formative	Guided self-study	5×6-hour packages	Fortnightly

Syllabus Plan and Content

1. Introduction
   Brief historical survey; recap of PHY1022; what is required of the theory; the wave equation; time-dependent Schrödinger equation
2. Wave Functions and their Interpretation
   The Born probability interpretation; normalization of the wave function; first postulate; wave function of a free particle; wave function of a confined particle; Gaussian wave packets (Self-study pack): the uncertainty principle; time evolution of wave packets
3. Dynamical Variables
   Observables as operators; the second postulate; the third postulate; physical significance of eigenfunctions; Schrödinger equation revisited
4. Stationary States and the Time-Independent Schrödinger Equation
   Time-independent probability distributions; the time-independent Schrödinger equation; stationary states: eigenfunctions of the Hamiltonian; example: region of constant potential; method of solution ; boundary conditions
5. Particle in a Box - the Infinite Square Well
   Internal solution; boundary conditions; energy quantization; normalized wave functions
6. The Finite Square Potential Well (Self-study pack)
   Interior and exterior solutions; boundary conditions; symmetric solutions - energies and wave functions; antisymmetric solutions - energies and wave functions
7. Flow of Particles
   Probability flux; continuity equation; persistence of normalization; derivation of probability flux
8. Barrier Problems
   Boundary conditions at a potential discontinuity; a potential step; tunnelling: reflection and transmission by a barrier; practical examples of tunnelling
9. Quantum Measurement and the Structure of Quantum Mechanics
   Properties of Hermitian operators; the superposition principle: fourth postulate; measurements of a general quantum state; commutation relations and simultaneous observables; the uncertainty principle; commutation with the Hamiltonian; summary: the postulates of quantum mechanics
10. The Quantum Harmonic Oscillator
    Hamiltonian in operator form; ladder operators; eigenvalues and eigenfunctions
11. The 3D Time-Independent Schrödinger Equation
    Momentum eigenfunctions in 3D; Schrödinger equation in 3D Cartesian coordinates (Self-study pack); example: particle in a 3D box; Schrödinger equation in spherical polar coordinates
12. 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
13. The Hydrogen Atom
    Solutions of the angular equation; solutions of the radial equation; energy eigenvalues - the hydrogen spectrum; electron density distributions
14. First-Order Time-Independent Perturbation Theory
    Perturbation theory for non-degenerate levels; perturbation theory for degenerate levels

Core Text

Rae A.I.M. (2007), Quantum Mechanics (5^th edition), Chapman and Hal, ISBN 1-584-88970-5 (UL: 530.12 RAE)

Supplementary Text(s)

McMurry S.M. (1994), Quantum Mechanics, Addison Wesley, ISBN 0-201-54439-3 (UL: 530.12 MCM)

IOP Accreditation Compliance Checklist

• QM-05: Wave function and its interpretation
• QM-06: Standard solutions and quantum numbers to the level of the hydrogen atom
• QM-07: Tunnelling
• QM-08: First order time independent perturbation theory

Formative Mechanisms

The problems that students are set on this module are marked and discussed in detail in the problems classes and in tutorials. Students monitor their own progress by attempting the problems set. Students who need additional guidance are encouraged to discuss the matter with their tutor or the lecturer.

Evaluation Mechanisms

The module will be evaluated using information gathered via the student representation mechanisms, the staff peer appraisal scheme, and measures of student attainment based on summative assessment.