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 
denoting a vector 
and the bra 
denoting a linear functional 
. The evaluation 
is then denoted by the bra-ket pairing 
.
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 
is some sort of reflection of the vector space 
, 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 
to the linear functional 
. In Dirac notation, we send the ket to the bra 
. Then the value of this linear functional on a vector 
(the ket 
) is the pairing 
, just as it should be.
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July 1, 2009 -
Posted by John Armstrong |
Algebra, Linear Algebra
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!)
Yes, I’ve mentioned it explicitly before, but I should still make it explicit here.
Have you looked at rigged Hilbert spaces at all? They made taking QM from physicists much more tolerable. =)
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.
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