Fourier analysis and continuous spectra

In summary, the conversation discusses the topic of eigenfunctions and operators in quantum mechanics, specifically focusing on continuous spectrum eigenvalues and non-normalizable eigenfunctions. The participants also discuss the concept of expressing a function as a linear combination of its eigenvectors and the use of orthogonality to find expansion coefficients. The conversation ends with a suggestion to read about the "Rigged Hilbert Space" formulation of modern quantum mechanics for further understanding.
  • #1
ThereIam
65
0
So I've been self-studying from Griffiths Intro to QM to get back in shape for graduate school this fall, and I guess I'd just like some confirmation that I'm on the right track...

So while I am sure there are many other applications, the one I am dealing with is eigenfunctions of an operator having a continuous spectrum of eigenvalues, making the eigenfunctions non-normalizable.

So from the top, qualitatively, as I understand it, essentially how this works if one has solved the eigenvalue equation for a particular operator, and this yields an eigenfunction which is at first glance non-normalizable (not in Hilbert space). However, one notes that some linear combination of the eigenvectors CAN be normalized, and the "linear combination" manifests as an integral over some function (called the "expansion coefficient") * the eigenvectors, all of which we want to be equal to some ψ(x). I guess we're trying to figure out a way to express ψ(x) as a linear combination of it's eigenvectors (which in the case of QM would tell you a lot about the state of the system ψ(x) represents)

Let's say the eigenvectors are of the form e^(ikx) so
ψ(x) = ∫Θ(k)*e^(ikx)dk

Then in terms of Fourier stuff, we're seeking to exploit the fact that e^(-imx) is orthogonal to e^(ikx) unless m = k. So we say
Θ(k) = ∫ψ(x)*e^(-ikx)dx

And that Θ(k) can be plugged back into the first equation for ψ(x)...

More generally, one the point is just that we're exploiting the orthogonality of the e^[itex]\pm[/itex]kx or the orthogonality of whatever function is present in the integral in order to find the expansion coefficient... right?

Am I hopeless?
 
Physics news on Phys.org
  • #2
ThereIam said:
More generally, one the point is just that we're exploiting the orthogonality of the e^[itex]\pm[/itex]kx or the orthogonality of whatever function is present in the integral in order to find the expansion coefficient... right?
That's the idea.

I think you would enjoy reading about the "Rigged Hilbert Space" formulation of modern quantum mechanics. Ballentine has a gentle introduction in ch1 of his textbook. A more extensive introduction can be found in this paper by Rafael de la Madrid. If you want even more, search for his PhD Thesis on Google Scholar.

Am I hopeless?
I see no reason whatsoever for that conclusion. :biggrin:
 
  • Like
Likes 1 person

What is Fourier analysis?

Fourier analysis is a mathematical technique used to break down a complex signal or function into its individual component frequencies. It is based on the idea that any periodic function can be represented as a sum of simple sinusoidal functions. This allows for the analysis of signals and systems in terms of their frequency components.

What is a continuous spectrum?

A continuous spectrum is a type of spectrum that contains an infinite number of frequencies. It is characterized by a continuous distribution of energy across all frequencies, rather than distinct lines or peaks. This type of spectrum is commonly seen in natural phenomena such as sunlight, where a continuous range of frequencies is present.

What is the difference between Fourier series and Fourier transform?

Fourier series and Fourier transform are two different ways of representing a signal or function in terms of its frequency components. Fourier series is used for periodic functions, while Fourier transform is used for non-periodic functions. Fourier series decomposes a function into a sum of sinusoidal functions, while Fourier transform decomposes a function into a continuous spectrum.

What is the relationship between Fourier analysis and signal processing?

Fourier analysis and signal processing are closely related. Fourier analysis is used to analyze signals and systems in terms of their frequency components, while signal processing uses this information to manipulate and modify signals. Techniques such as filtering and compression rely on Fourier analysis to identify and manipulate specific frequency components within a signal.

Why is Fourier analysis important in science and engineering?

Fourier analysis is a fundamental tool in many areas of science and engineering. It is used in fields such as physics, mathematics, engineering, and signal processing to analyze and understand complex systems and signals. It allows for the efficient representation and manipulation of signals, making it an essential tool in many modern technologies such as telecommunications, audio and image processing, and data compression.

Similar threads

  • Quantum Physics
Replies
2
Views
923
  • Quantum Physics
Replies
1
Views
2K
Replies
5
Views
1K
  • Quantum Physics
Replies
3
Views
2K
Replies
8
Views
1K
Replies
4
Views
307
  • Advanced Physics Homework Help
Replies
6
Views
1K
  • Quantum Physics
Replies
11
Views
3K
  • Quantum Physics
Replies
2
Views
888
  • Quantum Physics
Replies
2
Views
995
Back
Top