I'm trying to teach myself quantum mechanics from Dirac, and I'm having trouble justifying some of the maths, in particular how we can just jump out of the confines of a Hilbert space when it's convenient.(adsbygoogle = window.adsbygoogle || []).push({});

Dirac rather liberally talks about observables that have a continuous range of "eigenvalues", whose "eigenvectors" CANNOT be normalized (as the inner product of any two of them is equal to a delta function) and hence are not in the Hilbert space, but nonetheless how these "eigenvectors" can be thought of forming a basis of the space via integration rather than summation. (i.e., if any vector in the space can be expressed as [tex]|P\rangle = \int \psi(q') |q'\rangle dq'[/tex] ; defining the wave function [tex]\psi(q)[/tex] in this way.)

Certainly the fourier transform does this (the "basis" [tex]\{e^{iwx} \mid w \in \mathbb{R} \}[/tex] spans a vector space of functions in this way) but it makes me feel uneasy to do all the work in a Hilbert space, and then occasionally jump out of it and introduce vectors not in it as if they were, when it's convenient to, without justifying it.

I realize this isn't a very coherent question (or a question at all), but that's because I'm a bit confused. So here are a few concrete questions whose answers that might help me:

1. Is there a generalized notion of a Hilbert space we can talk about, where the inner product can take on "infinite" values?

2. If so, is that the proper context in which to perform quantum mechanics (in an infinite-dimensional space) rather than a Hilbert space? In other words, is the true state space in quantum mechanics a Hilbert space, or this bigger thing with extra vectors?

3. What is the bigger thing called?

Thanks,

Mark

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# Treating operators with continuous spectra as if they had actual eigenvectors?

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