Notation can help us substantially in thinking about and manipulating symbolic representations meant to describe complex physical phenomena. The brain's working memory can only manipulate a small number of ideas at once ("7 ± 2"). We handle complex ideas by "chunking" -- binding together many things and manipulating them as a single object. Another way we extend our range is by storing information outside of our brains temporarily and manipulating external objects or symbols, like an abacus or equations written on a piece of paper.
Notation -- the way we choose to organize our symbology to represent something -- can play a powerful role in helping us think about a complex situation. In Maxwell's day, the equations for electric and magnetic fields were written out component by component, so his equations took up a full page of text. Looking at those equations, it's clear that there is a regularity to the equations that should allow for some compression. When Gibbs introduced his vector notation, Maxwell's equations could be collapsed into 4 lines. Furthermore, they had the advantage that they did not depend on the choice of coordinate system. You could use the same equations, manipulate them as you wished, and then introduce a particular choice of coordinate after you were done (e.g., a particular orientation of rectangular coordinates or a convenient set of curvilinear coordinates) .
A similar situation pertains for dealing with linear spaces. In some cases, we might want to describe a system of coupled oscillators with the coordinates of the masses. In other cases, we might want to describe them in terms of how much of each normal mode is excited. This change corresponds to a change of coordinates in the linear space describing the state of the system. We would like to have a representation that describes the state without specifying the particular coordinates used to describe them.
Other cases where linear spaces are useful include cases where complex numbers are helpful in describing the physical system. Some examples of this include: polarization of electromagnetic waves (linear vs. circular), wave motion of mechanical systems (Fourier analysis), and quantum physics.
The Dirac notation for states in a linear space is a way of representing a state in a linear space in a way that is free of the choice of coordinate but allows us to insert a particular choice of coordinates easily and to convert from one choice of coordinates to another conveniently. Furthermore, it is oriented in a way (bra vs. ket) that allows us to keep track of whether we need to take complex conjugates or not. This is particularly useful if we are in an inner-product space. To take the length of a complex vector, we have to multiply the vector by its complex conjugate -- otherwise we wont get a positive number. The orientation of the Dirac representation allows us to nicely represent the inner product in a way that keeps careful track of complex conjugation.
Last revision 25. October, 2005.