The equations describing many important physical phenomena are linear differential equations. For example, in quantum mechanics the Schrödinger equation contains no power of the wavefunction or its derivatives greater than the first, and so is a linear differential equation. Similarly Maxwell's equations contain no power of the electric and magnetic fields or their derivatives greater than the first, and so Maxwell's theory of electromagnetism is also linear. One  consequence of a linear theory is that the sum of any two independent solutions is also a solution; this leads to the phenomenon of interference in both optics and quantum mechanics. What is more surprising is that the solutions may be regarded as ‘vectors”, where the “angle” between them is important. In this module we shall study the mathematics which underlies and is common to all these different linear systems. The mathematical techniques and ideas are illustrated by drawing on familiar examples from both classical and quantum physics.

By the end of the module the student should be able to:

• Understand the axiomatic development of mathematical structures such as groups, fields and vector spaces;
• Understand the ideas of vector space, function space and inner product and test a given system against the relevant axioms;
• Understand and test for linear independence and linear dependence of a set of vectors or functions;
• Understand the ideas of basis, orthogonal basis and orthonormal basis;
• Construct an orthonormal basis by the Gram-Schmidt process;
• Understand the idea of a linear operator and construct its matrix representation in a given basis;
• Understand change of basis and construct and use a change of basis matrix;
• Understand the idea of an operator eigenvalue problem;
• Convert second-order differential equations into Sturm-Liouville form;
• Use power series solutions to solve differential equations;
• Use power series solutions and boundary conditions to find eigenvalues of Sturm-Liouville problems;
• Be able to use generating functions for orthogonal polynomials;
• Be able to use Rodrigues formulae for orthogonal polynomials;
• Understand and use the Rayleigh-Ritz variational method to estimate the lowest eigenvalue 10 of a given problem;
• Apply the above mathematical techniques to problems of classical and quantum mechanics