Year:12/13
Department:Physics
Level:Part II (yr 2)
Learning Hours:150
Credit Points:15
Weight:0.5
Course Convenor:Dr J Gratus
Status:Live
Syllabus Rules
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Prior to PHYS211, the student must have successfully completed:
Assessment Rules
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CMod description
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Linear algebra: Systems of coupled linear
equations. Linear transformations, diagonalisation of matrices. Pauli
matrices and practicing in operations with them. Eigenvalues and
eigenvectors. Symmetric and Hermitian matrices and their diagonalisation using
orthogonal and unitary matrices. Cyclotron motion in a magnetic field. Normal
modes of coupled oscillators. Commutation relations involving matrices,
invariants of linear transformations.
Hilbert
Space: Wave equation in 1D with boundary conditions, separation of variables
using standing waves. Wave equation in 3D: separation of variables and
resonances in a drum. Bases of functions. Fourier series as an example of a
basis in the Hilbert space. Orthogonality of harmonic functions, Kronekker
delta-symbol, and completeness of a basis.
Angular harmonics. Operators and their
eigenfunctions. Angular harmonics in 2 dimensions, relation between plane
waves and cylindrical waves, Bessel functions. Laplace operator in Cartesian,
cylindrical and spherical coordinates. Spherical harmonic functions in 3
dimensions. Representation of operators as matrices acting in the Hilbert
space, commutation relations between operators. Symmetry of space in relation
to the method of separation of variables.
Curriculum Design: Outline Syllabus
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Linear algebra: Systems of coupled linear equations. Linear transformations. Determinant of a matrix. Diagonalisation of matrices.
Pauli matrices and practicing in operations with them. Eigenvalues and eigenvectors. Symmetric and Hermitian matrices and their diagonalisation using orthogonal and unitary matrices. Normal modes of coupled oscillators. Commutation relations involving matrices, invariants of linear transformations.
Hilbert Space
: Wave equation in 1D with boundary conditions, separation of variables using standing waves. Wave equation in 3D: separation of variables and resonances in a drum. Bases of functions. Fourier series as an example of a basis in the Hilbert space.
Orthogonality of harmonic functions, Kronekker delta-symbol, and completeness of a basis.
Angular harmonics.
Operators and their eigenfunctions.Angular harmonics in 2 dimensions, relation between plane waves and cylindrical waves, Bessel functions. Laplace operator in Cartesian, cylindrical and spherical coordinates. Spherical harmonic functions in 3 dimensions. Representation of operators as matrices acting in the Hilbert space, commutation relations between operators.
Symmetry of space in relation to the method of separation of variables.
Educational Aims: Subject Specific: Knowledge, Understanding and Skills
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The LTA strategy is fourfold. Each week the core physics material is developed in the lectures. Students are expected to reinforce and extend the lecture material by private study of the course textbook and other sources. Student's understanding is consolidated and assessed via the fortnightly work sheet, which is completed by students independently, then marked and discussed by the lecturer at the seminar.
Learning Outcomes: Subject Specific: Knowledge, Understanding and Skills
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On completion of the module, students should be able:
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to solve problems involving systems of coupled linear equations
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to solve the wave equation in 3D using Cartesian, cylindrical and spherical polar coordinates.
Curriculum Design: Select Bibliography
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(E) Mathematical Techniques, D W Jordan, P Smith, OUP
