# Imperial college of science, technology and medicine

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IMPERIAL COLLEGE OF SCIENCE, TECHNOLOGY AND MEDICINE

DEPARTMENTS OF MATHEMATICS & PHYSICS

ANCILLARY - MATHS & PHYSICS PART 2 (MPC2)

LECTURE COURSE INFORMATION

## Chemistry Co-ordinator:Professor John Seddon

Lecture Course: MPC2 - Mathematics & Physics for Chemists Part 2

Year : Second (c. 40 students)

Duration : Maths: 20 lectures+10 problem classes. Physics: 20 lectures +10 problem classes

Terms : Maths: Autumn (20+10). Physics: Spring (20+10)

Lecturers : Maths: Prof. N. Bingham. Physics: Dr. J. Pasternak (QM) and Dr. D. Eakins (EM)

AIMS : To continue Mathematics and Physics as logical and structured disciplines; to introduce students to further mathematical and physical techniques that will be of value to them as professional chemists.

OBJECTIVES: By the end of the MPC2 course, students will be able to:

1. Solve simple linear inhomogeneous equations by the method of variation of one parameter and understand the basic principle of Green function methods;

2. Manipulate complex numbers in standard and polar form and apply de Moivre’s theorem;

3. Solve some simple linear homogeneous partial differential equations by the method of separation of variables in both Cartesian and polar forms, understand the origin and use of spherical harmonics;

4. Solve sets of homogeneous and inhomogeneous linear algebraic equations by the method of Gaussian elimination, calculate eigenvalues and eigenvectors of square matrices and use them for diagonalisation;

5. Find Fourier series for periodic functions and Fourier transforms for non-periodic functions;

6. Understand the basic concepts of vector field theory and use the basic differential operators, understand the basic integral theorems;

7. Understand the motivation of the Schrödinger equation and its relationship to the classical energy equation through the replacement of energy and momentum with differential operators;

8. Be able to separate the Schrödinger equation into time and space components;

9. Understand what is meant by an operator and the relationship between an operator and a measurable quantity;

10. Know the postulates that make up quantum mechanics and be able to expand on the ideas contained within these postulates to give a more detailed interpretation of the theory;

11. Be competent in using the mathematical formulation of the theory and be able to apply it to a variety of simple physical systems, understand the importance of boundary conditions;

12. Know the importance of superposition and interference in quantum systems and be able to handle the mathematical expressions associated with this;

13. Be able to determine the wavefunctions and energy levels of particles in simple potentials by solution of the Schrödinger equation, understand the origin of discrete energies;

14. Understand the fundamentals of the quantum theory of angular momentum;

15. Be able to separate the three-dimensional Schrödinger equation for a particle in a central potential into radial and angular parts and solve the angular part in terms of spherical harmonics;

16. Understand the role of exchange symmetry in simple two electron systems;

17. Be familiar with the concepts of charge, electric force and electric field;

18. Be able to calculate electric fields using Coulomb’s Law and Gauss’s Law;

19. Know what an electric dipole is and know their behavior in uniform electric fields;

20. Know what capacitance is and be able to find the energy stored in the electric field of a capacitor;

21. Know the effect of dielectric media on electric fields and to know about dielectric relaxation, and relaxation frequencies;

22. Know what magnetism is; know that it can also be represented by a field, and what things experience a force due to a magnetic field;

23. Be able to calculate the magnetic field in symmetrical configurations;

24. Be able to calculate the motion of a charged particle moving in a combination of electric and magnetic fields (e.g. a mass spectrometer);

25. Know Faraday’s Law and calculate electric fields generated due to magnetic induction;

26. Be able to write Maxwell’s equations in differential form including magnetic induction and displacement current;

27. Be familiar with the electromagnetic wave equation and their properties over the different frequency ranges.

COURSE TOPICS in chronological order are as follows:

TOPICS NUMBER OF LECTURES

MATHEMATICS

1 Inhomogeneous ODEs 3 AUTUMN TERM

2 Complex Numbers 2 AUTUMN TERM

3 Partial differential equations 4 AUTUMN TERM

4 Linear Algebra 4 AUTUMN TERM

5 Fourier methods 4 AUTUMN TERM

6 Field theory 3 AUTUMN TERM

PHYSICS

7 Quantum mechanics 10 SPRING TERM

8 Electricity and Magnetism 10 SPRING TERM

TUTORIAL ARRANGEMENTS

(i) A weekly problem class/tutorial.

(ii) 6 Maths and 6 Physics problem sheet will be issued regularly.

SYLLABUS: Mathematics: Mathematical Methods

1. Inhomogeneous Ordinary Differential equations:

Variation of one parameter and Green function methods.

2. Complex numbers:

Algebraic properties and geometrical interpretation, polar representation, de Moivre’s theorem, functions.

3. Partial differential equations:

Separation of variables, application to Wave, heat-conduction, Laplace’s and Schroedinger’s equations, use of polars and spherical harmonics.

4. Linear Algebra:

Solution of sets of simultaneous linear equations by Gaussian elimination, Eigen problems and diagonalisation of matrices. Cayley-Hamilton theorem.

5. Fourier series and transforms:

Standard formulae for Fourier series of periodic functions, role of symmetry, convergence and the Gibbs’ phenomenon, Parseval’s theorem and applications. Standard formulae for Fourier transforms of non-periodic functions, basic introduction and application to signal processing.

6. Vector Field theory:

Calculus of scalar and vector fields, introduction and physical interpretation of the basic operators - div, grad and curl, Integral theorems (if time permits).

SYLLABUS: Physics: Quantum Mechanics

1. The Schrodinger Equation:

Meaning of H,  probabilities, the quantum description of a particle.

2. One dimensional problems:

The wave equation, wavepackets, superposition, potential steps and wells. Particle in a box, simple harmonic oscillator.

3. Postulates of quantum mechanics:

Uncertainty relations, Hermitian operators, mathematical properties of the wavefunction.

4. Three dimensional problems:

Spherical coordinates and separation of variables, angular momentum.

5. The hydrogen atom:

The radial equation, quantum numbers, hydrogen and the periodic table.

6. Two electron systems:

Pauli exclusion and spin, helium and the H2 molecule.

SYLLABUS: Physics: Electricity and Magnetism

1. Electric fields (4 lectures)

Charges and currents, Coulomb’s Law, electric fields, electric flux, Gauss’s Law, electric potential, circuital law, differential forms of Gauss’s law and circuital law, electric dipoles

2. Electric properties of materials (2 lectures)

Capacitance, energy stored in electric field, dielectrics, dipoles, classical hydrogen atom, types of electrostatic bonding

3. Magnetic fields (2 lectures)

Magnetic force, motion of particles in magnetic fields, motors, MRI, Ampere’s Law, magnetic materials, mass spectrometer

4. Maxwell’s equations and electromagnetic waves (2 lectures)

Faraday’s law, Induction, Displacement current, Maxwell’s equations, the wave equation, electromagnetic waves

"Foundations of Physics for Chemists", Ritchie & Sivia, 6th Ed. (Oxford University Press, 2000)

ASSESSMENT:

1. Examination: One 3-hour written examination in May/June in two sections: section A consists of 6 Mathematics questions and section B consists of 5 Physics questions. Section A questions are marked out of 15 and section B questions are marked out of 20.

Rubric: “Answer 4 questions from section A and 3 questions from section B. Use separate answer books for section A and section B. The same maximum mark is available for section A and section B.” The exam contributes 90% to the course assessment.

1. Coursework: Either a 1-hour progress test at the end of each term or an assessed question in each problem sheet or some combination of these, contributing 10% to the course assessment.

## RECOMMENDED TEXTS:

M.C.R.Cockett & G.Doggett: Maths for Chemists Vol. I and II, Tutorial Chemistry Texts, RSC (2003)

## K.A.Stroud: Engineering Mathematics (MacMillan, 1995)

H.D.Young & R.A.Freedman: University Physics - International Edition, 11th Ed. (Addison Wesley, 2004)

H.J.Pain: The Physics of Vibrations and Waves, 6th Ed. (Wiley, 2005)

A.P.French & M.G.Ebison: Introduction to Classical Mechanics, (Van Nostrand Reinhold, 1986)

David J. Griffiths: Introduction to Quantum Mechanics, 2nd ed. (Pearson Prentice Hall, 2005)

2012/2013 TIMETABLE:

Autumn Term: 1 October - 14 December 2012 (Autumn Reading Week: 5-9 Nov 2012)

Spring Term: 7 January - 22 March 2013 (Spring Reading Week: 11-15 Feb 2013)

Summer Term: 29 April - 28 June 2013

Commemoration Day (no academic work): Wednesday 17th October 2012

AUTUMN TERM

Mathematics: Professor Nick Bingham

SPRING TERM

Physics: (Quantum Mechanics): Dr. J. Pasternak

Physics: (Electricity & Magnetism): Dr. D. Eakins

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