Position representation of a wavefunction - technical issue

[spaghetti3451]

Hey, I have been churning the idea of Dirac notation around in my head and I am thinking about the position and momentum basis representation of a wavefunction in a Hilbert space.

Wikipedia mentions the following in the article 'Bra-ket notation' under the heading 'Position-space wave function':

<r|ψ> = ψ(r) by definition.

Now, I can appreciate how there's can be many infinitely many wavefunctions for a quantum system and for each possible wavefunction, there is mapping from the position space onto the field of complex numbers. That's the idea encoded by the representation ψ(r) of the wavefunction, right?

What I am having trouble trouble to understand is (r, ψ). ψ is, surely, an element of a Hilbert space that corresponds to the quantum system, but r is not an element from that Hilbert space. So, how can you have an inner product between the two?

Any ideas?

 

[cgk]

What I am having trouble trouble to understand is (r, ψ). ψ is, surely, an element of a Hilbert space that corresponds to the quantum system, but r is not an element from that Hilbert space. So, how can you have an inner product between the two?

Actually, it is[1]. The Hilbert space in this particular case would have to be tought of as "the vector space of all functions mapping one coordinate to a scalar". In that case phi(r), i.e., |phi> would be a vector in that space, and |r> would be described as Dirac delta, which one could also see as a vector in this space[2].

This still makes some sense if you think of the formalism in a different defining basis. E.g., if your actual phi is given in momentum representation, you could form
$$\langle r|\phi\rangle = \int_{p\in R^3}\langle r|p\rangle\langle p|\phi\rangle\,\mathrm{d}^3p$$
where |p><p| is the resolution of the identity in the momentum basis, and <p|phi> = phi(p) now denotes the original wave function's components in this momentum basis. The main point about the Dirac notation is that it makes it easy to jump around in different representations and to express projectors onto various subspaces.

[1] Modulo technicalities with rigged vs non-rigged Hilbert spaces etc. The physicist's Hilbert space is not actually a mathematical Hilbert space, so quantities like <r| (corresponding to distributions) would have to be interpreted as some limit, e.g., a Gauss wave packet with increasinly smaller sigma. For a start, just consider all those Hilbert spaces to be finite or countably infinite, and act like you would in a finite vector space.
[2] In a more mathematical way, <r| would be the linear functional mapping a function to its value at place r; so it would technically be in the dual vector space of the |phi>, but that is not the point.