The Unapologetic Mathematician

Mathematics for the interested outsider

Dirac notation I

There’s a really neat notation for inner product spaces invented by Paul Dirac that physicists use all the time. It really brings to the fore the way a both slots of the inner product enter on an equal footing.

First, we have a bracket, which brings together two vectors

\displaystyle\langle w,v\rangle

the two sides of the product are almost the same, except that the first slot is antilinear — it takes the complex conjugate of scalar multiples — while the second one is linear. Still, we’ve got one antilinear vector variable, and one linear vector variable, and when we bring them together we get a scalar. The first change we’ll make is just to tweak that comma a bit

\displaystyle\langle w\vert v\rangle

Now it doesn’t look as much like a list of variables, but it suggests we pry this bracket apart at the seam

\displaystyle\langle w\rvert\lvert v\rangle

We’ve broken up the bracket into a “bra-ket”, composed of a “ket” vector \lvert v\rangle and a “bra” dual vector \langle w\rvert (pause here to let the giggling subside) (seriously, I taught this to middle- and high-schoolers once).

In this notation, we write vectors in V as kets, with some signifier inside the ket symbol. Often this might be the name of the vector, as in \lvert v\rangle, but it can be anything that sufficiently identifies the vector. One common choice is to specify a basis that we would usually write \left\{e_i\right\}. But the index is sufficient to identify a basis vector, so we might write \lvert1\rangle, \lvert2\rangle, \lvert3\rangle, and so on to denote basis vectors. That is, \lvert i\rangle=e_i. We can even extend this idea into tensor products as follows

\displaystyle e_i\otimes e_j=\lvert i\rangle\otimes\lvert j\rangle=\lvert i,j\rangle

Just put a list of indices inside the ket, and read it as the tensor product of a list of basis vectors.

Bras work the same way — put anything inside them you want (all right, class…) as long as it specifies a vector. The difference is that the bra \langle w\rvert denotes a vector in the dual space V^*. For example, given a basis for V, we may write \langle i\rvert=\epsilon^i for a dual basis vector.

Putting a bra and a ket together means the same as evaluating the linear functional specified by the bra at the vector specified by the ket. Or we could remember that we can consider any vector in V to be a linear functional on V^*, and read the bra-ket as an evaluation that way. The nice part about Dirac notation is that it doesn’t really privilege either viewpoint — both the bra and the ket enter on an equal footing.

June 30, 2009 Posted by | Algebra, Linear Algebra | 14 Comments