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.

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July 1, 2009 - Posted by | Algebra, Linear Algebra

6 Comments »

  1. Assuming we’re working over the complex numbers, the map sending v to the linear functional is surely conjugate-linear, not linear? (Apologies for lack of LaTeX!)

    Comment by Matt Daws | July 1, 2009 | Reply

  2. Yes, I’ve mentioned it explicitly before, but I should still make it explicit here.

    Comment by John Armstrong | July 1, 2009 | Reply

  3. Have you looked at rigged Hilbert spaces at all? They made taking QM from physicists much more tolerable. =)

    Comment by Douglas | July 2, 2009 | Reply

  4. Yes, I’ve looked at them, Douglas, but I’m not sure why you bring them up in the context of a purely notational discussion.

    Comment by John Armstrong | July 2, 2009 | Reply

  5. […] Notation III So we’ve got Dirac notation and it’s nice for inner product spaces, but remember we’re not just interested in vectors and vector spaces. We’re even more […]

    Pingback by Dirac Notation III « The Unapologetic Mathematician | July 6, 2009 | Reply

  6. […] and Bilinear Forms on Inner Product Spaces in Dirac Notation Now, armed with Dirac notation, we can come back and reconsider matrices and forms. For our background, we’ve […]

    Pingback by Matrices and Bilinear Forms on Inner Product Spaces in Dirac Notation « The Unapologetic Mathematician | July 8, 2009 | Reply


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