Help with Eigenvalue Equation and Fourier Transform

In summary, the conversation discussed the Fourier transform of the eigenvalue equation and the uniqueness of the energy value in this representation. The speaker suggests working on enforcing <φE(p)|φE'(p)> = δ(E–E') and mentions how E needs to be positive and unique.
  • #1
Helloaksdoq
1
0

Homework Statement


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Homework Equations


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The Attempt at a Solution



I did Fourier transform directly to the eigenvalue equation and got

Psi(p)=a*Psi(0)/(p^2/2m-E)

But the rest, I don't even know where to start.
Any opinion guys?
 
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  • #2
Hi. What you got so far seems right.
Now i can't think of any easy "trick" to show the uniqueness of E in this representation, so i would suggest you work on how to enforce:
E(p)|φE'(p)> = δ(E–E')
I suppose you can look up the integral; given the result you should see:
1 - how E needs to be positive
2 - how E must be = E' (i.e.: E is unique)
That's a start...
 
Last edited:

Related to Help with Eigenvalue Equation and Fourier Transform

1. What is an eigenvalue equation?

An eigenvalue equation is a mathematical formula that describes the relationship between an operator and its corresponding eigenvalues and eigenvectors. It is commonly used in linear algebra to solve for unknown variables.

2. How is the eigenvalue equation used in Fourier transforms?

The eigenvalue equation is used in Fourier transforms to decompose a function into a series of sinusoidal functions. The eigenvalues correspond to the frequencies of the sinusoidal functions, and the eigenvectors represent the amplitudes of each frequency component.

3. What is the significance of eigenvalues in the context of Fourier transforms?

Eigenvalues in the context of Fourier transforms are important because they allow us to analyze and manipulate complex functions in terms of simpler sinusoidal functions. This simplifies the mathematical calculations involved in solving problems in fields such as signal processing, image processing, and quantum mechanics.

4. Can you explain how eigenvalues and eigenvectors are related in the Fourier transform?

In the Fourier transform, eigenvalues and eigenvectors are related through the eigenvalue equation. The eigenvalues represent the frequencies of the sinusoidal functions, and the eigenvectors represent the amplitudes of each frequency component. Together, they allow us to decompose a function into a series of simpler components.

5. How can I use the eigenvalue equation and Fourier transform to solve real-world problems?

The eigenvalue equation and Fourier transform can be used to solve a variety of real-world problems, such as analyzing signals in communication systems, processing images in medical imaging, and understanding the behavior of quantum systems. By decomposing complex functions into simpler components, we can gain insight and make predictions about these systems and their behavior.

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