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

In summary, Dirac discusses observables with continuous eigenvalues and eigenvectors that cannot be normalized, but can be thought of as forming a basis through integration. There are two common ways to handle this concept: constructing a rigged Hilbert space or using a measure-theory based approach. However, the true state space in quantum mechanics is still a Hilbert space.
  • #1
Marcaias
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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 [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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  • #2
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 [itex]F[/itex] defined by [itex]F[\varphi] = \varphi(0)[/itex] is usually written as

[tex]\int_{-\infty}^{+\infty} \delta(x) \, \underline{\quad \quad} \, dx[/tex]

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.
 
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  • #3


I can understand your confusion and concern about treating operators with continuous spectra as if they had actual eigenvectors. This is a concept that can be difficult to grasp, but it is an important aspect of quantum mechanics.

Firstly, it is important to understand that the Hilbert space is the mathematical space in which quantum mechanics is formulated. It is a mathematical construct that allows us to describe the state of a quantum system and perform calculations. However, in reality, physical systems do not always behave in a way that can be fully described by a Hilbert space. This is where the concept of operators with continuous spectra comes in.

In quantum mechanics, observables are represented by operators. These operators can have discrete or continuous spectra, depending on the physical system being studied. When an operator has a continuous spectrum, it means that the possible values it can take on are not countable, and therefore, the corresponding eigenvectors cannot be normalized as in the case of discrete spectra.

Now, to address your questions:

1. There is a generalization of the Hilbert space called a rigged Hilbert space, which allows for the inner product to take on "infinite" values. This is often used in quantum mechanics to deal with operators with continuous spectra.

2. The true state space in quantum mechanics is still considered to be a Hilbert space. The rigged Hilbert space is used as a mathematical tool to deal with certain situations, but it is not the fundamental state space.

3. The bigger thing is called a rigged Hilbert space or a Gelfand triple.

It is important to note that the use of rigged Hilbert spaces is a mathematical tool and does not change the fundamental principles of quantum mechanics. It is simply a way to deal with certain mathematical difficulties that arise when working with operators with continuous spectra.

In conclusion, while it may feel uneasy to jump out of the confines of a Hilbert space, it is a necessary aspect of quantum mechanics when dealing with operators with continuous spectra. The rigged Hilbert space provides a way to mathematically handle these situations without changing the fundamental principles of quantum mechanics. I hope this helps to clarify some of your confusion.
 

1. What is the significance of treating operators with continuous spectra as if they had actual eigenvectors?

The treatment of operators with continuous spectra as if they had actual eigenvectors is important in the field of quantum mechanics, where it allows for the simplification of mathematical calculations and the prediction of physical outcomes of experiments.

2. How can operators with continuous spectra be treated as if they had actual eigenvectors?

This can be achieved through the use of spectral theory, which provides a framework for analyzing operators with continuous spectra and allows for the use of generalized eigenvectors.

3. What are the limitations of treating operators with continuous spectra as if they had actual eigenvectors?

One limitation is that not all operators with continuous spectra have actual eigenvectors, so this approach may not always be applicable. Additionally, the accuracy of predictions made using this method may be limited in certain cases.

4. Can the treatment of operators with continuous spectra as if they had actual eigenvectors lead to incorrect results?

In some cases, yes. This approach is based on mathematical approximations and assumptions, so there is a possibility of obtaining incorrect results. However, it has been shown to be highly accurate in many practical applications.

5. How does treating operators with continuous spectra as if they had actual eigenvectors relate to the concept of quantum superposition?

The use of generalized eigenvectors allows for the representation of a state as a linear combination of eigenstates, which is a key concept in quantum mechanics. Therefore, treating operators with continuous spectra as if they had actual eigenvectors is closely related to the concept of quantum superposition.

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