PHY2025 Mathematics with Physical Applications

2011-2012

Code: PHY2025
Level: 2
Title: Mathematics with Physical Applications
Instructors: Dr F.Y. Ogrin
CATS Credit Value: 15
ECTS Credit Value: 7.5
Pre-requisites: N/A
Co-requisites: N/A
Duration: T1:01-11, T2:01-11
Availability: unrestricted
Background Assumed: -

Total Student Study Time

150 hours, to include: 22×1-hour lectures; 46 hours directed self-study; 11 hours of problems class support; 3 hours of tutorial support; 68 hours private study.

Aims

This module will enable the student to build on the knowledge and skills developed in PHY1026 in order to achieve a deeper understanding of and greater competence in some central mathematical ideas and techniques used throughout physics. The emphasis is on practical skills rather than formal proofs. Students will acquire skills in some key techniques that relate directly to the advanced modules they will meet in the later stages of their degree programme, but also have wide applicability across the mathematical sciences.

Intended Learning Outcomes

Students will be able to:

1. Module Specific Skills:
   1. use probability theory to solve problems;
   2. calculate Fourier transforms and use them to solve problems
   3. solve partial differential equations by separation of variables;
   4. calculate eignvalues and eigenvectors and apply the the techniques to physical problems;
   5. use basis vectors to transform differential operator equations to matrix form and hence apply eigen equation techniques;
   6. obtain approximate solutions to differential equations through the use of perturbation theory.
2. Discipline Specific Skills:
   1. analytical and numerical skills in mathematics;
3. Personal Transferable Skills:
   1. logical formulation of problems;
   2. presentation and justification of techniques and methods;
   3. group work - students are encouraged to work co-operatively together and with the demonstrators to solve guided problems.

Learning / Teaching Methods

Lectures, e-Learning resources (ELE PHY2025), and problems classes.

Assessment and Assignments

Contribution	Assessment/Assignment	Size (duration/length)	When
20%	Problem Sets	10×3-hour sets	Fortnightly
15%	Mid-term Test 1	30 minutes	Week T1:06
15%	Mid-term Test 2	30 minutes	Week T2:06
50%	Final examination	120 minutes	Term 3
Formative	Guided self-study	8×2-hour packages	Fortnightly

Syllabus Plan and Content

1. Probability theory
   1. Random variables
   2. Conditional probability
   3. Probability distributions
      1. Discrete
      2. Continuous
2. The Dirac delta-function
3. Fourier transforms including the convolution theorem
4. Solution of linear partial differential equations
   1. 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
5. 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. (2007), Engineering Mathematics (6^th edition), Paulgrave, ISBN 1-4039-4246-3 (UL: 510.2462 STR)

IOP Accreditation Compliance Checklist

• MT-07: Solution of linear partial differential equations
• MT-12: Fourier series and transforms including the convolution theorem.
• MT-13: Probability distributions.

Formative Mechanisms

Students monitor their own progress by attempting the problem sets which will be discussed in classes. Students who need additional guidance are encouraged to discuss the matter with the lecturer or their tutor.

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.