The Unapologetic Mathematician

Mathematics for the interested outsider

Dirac Notation II

We continue discussing Dirac notation by bringing up the inner product. To this point, our notation applies to any vector space and its dual, with the ket \lvert v\rangle denoting a vector v\in V and the bra \langle\lambda\rvert denoting a linear functional \lambda\in V^*. The evaluation \lambda(v) is then denoted by the bra-ket pairing \langle\lambda\vert v\rangle.

But the neat thing about this notation is that it makes bras look like some sort of reflection of kets. And they are, in a sense. The dual space V^* is some sort of reflection of the vector space V, but there’s no clear mapping from vectors in one space to vectors in the other; unless, that is, we pick a specific isomorphism; or, equivalently, an inner product.

When we’ve got an inner product in the picture, we get a (conjugate) linear isomorphism that sends the vector v to the linear functional \langle v,\underline{\hphantom{X}}\rangle. In Dirac notation, we send the ket \lvert v\rangle to the bra \langle v\rvert. Then the value of this linear functional on a vector w (the ket \lvert w\rangle) is the pairing \langle v\vert w\rangle=\langle v,w\rangle, just as it should be.


July 1, 2009 Posted by | Algebra, Linear Algebra | 6 Comments