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

[Marcaias]

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.

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 $$|P\rangle = \int \psi(q') |q'\rangle dq'$$ ; defining the wave function $$\psi(q)$$ in this way.)

Certainly the Fourier transform does this (the "basis" $$\{e^{iwx} \mid w \in \mathbb{R} \}$$ 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

 

[Hurkyl]

The two most common ways (that I know of) for deal with such things are as follows:

(1) You can construct a rigged Hilbert space, by selecting a subset of test functions on which you can define linear functionals which can act as a realization of the idea of 'inner products with generalized functions'. For better or worse, it seems traditional to write such linear functionals with integral-like notation. e.g. the functional $F$ defined by $F[\varphi] = \varphi(0)$ is usually written as

$$\int_{-\infty}^{+\infty} \delta(x) \, \underline{\quad \quad} \, dx$$

where I've put an underscore where you would plug in a test function. (Actually physicists seem to use a slightly different syntax than mathematicians here, and I've never managed to work out exactly what it is) This doesn't have an (obvious) literal meaning as an integral -- instead it's just one giant symbol used to represent a distribution.


(2) You can eschew generalized functions for a more measure-theory based approach, such as in the chapter on the spectral theorem in appendix B of this course. It's very much like using cumulative probability functions to work with probability distributions. This approach doesn't give you generalized eigenfunctions for your observables, but instead directly gives you the integrals you probably wanted to use anyways.

In other words, is the true state space in quantum mechanics a Hilbert space, or this bigger thing with extra vectors?

The state space is the Hilbert space; this is not changed if we decide to use auxiliary mathematical structures for theoretical / computational purposes.