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. |