PHY2018 Mathematics with Physical Applications

2007-2008

Code: PHY2018
Title: Mathematics with Physical Applications
Instructors: Dr F.Y. Ogrin and Prof. R. Jones
CATS credits: 20
ECTS credits: 10
Availability: unrestricted
Level: 2
Pre-requisites: Mathematics for Physicists (PHY1116)
Co-requisites: N/A
Background Assumed: N/A
Duration: Semesters I and II
Directed Study Time: 44 lectures/problems classes of 1 hour
Private Study Time: 156 hours
Assessment Tasks Time: -
Observation report: 2004/05 MEP

Aims

To give the student a deeper understanding of and greater competence in some central mathematical ideas and techniques used throughout physics with the emphasis on practical skills rather than formal proof. Students will acquire skills in some key techniques that relate directly to the advanced modules they will meet in the third and fourth years but also have wide applicability across the mathematical sciences.

Intended Learning Outcomes

Students will be able to:

• solve partial differential equations by separation of variables;
• calculate eignvalues and eigenvectors and apply the the techniques to physical problems;
• use basis vectors to transform differential operator equations to matrix form and hence apply eigen equation techniques;
• obtain approximate solutions to differential equations through the use of perturbation theory.

Transferable Skills

• Analytical and numerical skills in mathematics;
• Logical formulation of problems;
• Presentation and justification of techniques and methods;
• Group work - students are encouraged to work co-operatively together and with the demonstrators to solve guided problems.

Learning and Teaching Methods

The module is taught in two-hour blocks most of which entail a lecture, a related guided problems section with demonstrator assistance, and additional assessed coursework. On line learning resources.

Assignments

Assessed homework.

Assessment

Coursework (30%), tests (2×15%), one 180-minute examination (40%).

Syllabus Plan and Content

1. Second-order Partial Differential Equations
   1. Revision: Simple second order differential equations and common varieties: Harmonic oscillator, Schrödinger equation, Poisson's equation, wave equation and diffusion equation.
   2. Separation of variables: The Laplacian family of equations in physics, separation of variables, mechanics of the technique, form of solutions, general solutions in series form, relation to Fourier series, spatial boundary conditions, time dependence, initial conditions.
   3. Examples: rectangular drum, classical and quantum harmonic oscillator, waves at a boundary, temperature distributions, wavepacket/quantum particle in a box
   4. Role of symmetry: Cylindrical and spherical polar co-ordinates, appearance of special functions. Use of special functions by analogy to sin, cos, sinh, cosh etc.
   5. Examples: circular drum, hydrogen wave function
2. Linear Algebra
   1. Revision: Row and column vectors, matrices, matrix algebra, the solutions of systems of linear equations.
   2. Eigenvalue equations I: The matrix equation Ax=ax, solving the matrix equation, the secular determinant, eigenvalues and eigenvectors, canonical form, normal modes/harmonics, simple coupled oscillators.
   3. Eigenvalue equations II: Properties of eigenvectors: orthogonality, degeneracy, as basis vectors.
   4. Eigenvalue equations III: Differential equations as eigenvalue equations and the matrix representation Ax=ax; choosing the basis, solving the equation, the secular determinant, eigenvalues and eigenvectors.
   5. Examples: classical coupled modes, Schrödinger wave equation
   6. Approximate solutions to differential equations (perturbation theory): use of eigenvectors, first- and second-order through repeated substitution, problem of degeneracies.
   7. Examples: quantum particle in a well, a mass on drum, coupled particles

Core Text

Spiegel M.R. (1971), Advanced Mathematics for Engineers and Scientists, Schaum Outline Series, McGraw-Hill, ISBN 0-070-60216-6 (UL: 510 SPI)

Supplementary Text(s)

Constantinescu F. and Magyari E. (1971), Problems in Quantum Mechanics, Pergamon, ISBN 0-080-19008-1 (UL: 530.12 CON)
Greiner W. (1994), Quantum Mechanics: An Introduction (3^rd edition), Springer-Verlag, ISBN 3-540-58079-4 (UL: 530.12 GRE)
James G. (1993), Advanced modern engineering mathematics, Addison-Wesley, ISBN 0-201-56519-6 (UL: 510.2462 JAM)
Kreyszig E. (1993), Advanced Engineering Mathematics (5^th edition), Wiley, ISBN 0-471-55380-8 (UL: 510.2462 KRE)
Stroud K.A. (2001), Engineering Mathematics (5^th edition), Paulgrave, ISBN 0-333-91939-4 (UL: 510.2462 STR)

Formative Mechanisms

Discussion of lecture and guided problems, via demonstrators and lectures during class section, working through selected assessed coursework.

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.