Quantum Physics: Fourier transform of a function

In summary, the conversation discusses the determination and physical interpretation of the Fourier transform of e^{iax} \psi (x). The relevant equations (1) and (2) are provided and used to derive the Fourier transform of e^{iax} \psi (x), denoted as \tilde{\phi_2} (k). However, there is an algebraic error in the attempt at a solution and further discussion is needed to fully understand the significance of the result.
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Homework Statement


Let [tex]\phi (k)[/tex] be the Fourier transform of the function [tex]\psi (x)[/tex]. Determine the Fourier transform of [tex]e^{iax} \psi (x)[/tex] and discuss the physical interpretation of this result.

Homework Equations


(1) [tex]\tilde{f} (k) = \frac{1}{\sqrt{2 \pi}} \int{f (x) e^{-ikx} dx}[/tex]
(2) [tex]\psi (x,0)=\int{\phi(k)e^{ikx}dk}[/tex] (might be needed)

The Attempt at a Solution


We know that: [tex]\phi (k) = \tilde{\psi} (x) = \frac{1}{\sqrt{2 \pi}} \int{\psi (x) e^{-ikx} dx}[/tex] (eq. 1)

I'll let [tex]\tilde{\phi_2} (k)[/tex] be the Fourier transform of [tex]e^{iax} \psi (x)[/tex]

[tex]\tilde{\phi_2} (k) = \frac{1}{\sqrt{2 \pi}} \int{\psi (x) e^{iax} e^{-ikx} dx} = \frac{e^{a/k}}{\sqrt{2 \pi}} \int{\psi (x) dx}[/tex]

Can I do anything more? How's the result interesting?
 
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  • #2
You made an algebraic error: eae-b ≠ ea/b. Fix that and try again.
 

Related to Quantum Physics: Fourier transform of a function

1. What is a Fourier transform?

A Fourier transform is a mathematical operation that decomposes a function or signal into its individual frequency components. It allows us to analyze a function or signal in terms of its frequency content.

2. Why is a Fourier transform important in quantum physics?

In quantum physics, the Fourier transform is used to describe the wave-like behavior of particles. It allows us to determine the probability distribution of a particle's position and momentum, which are important concepts in quantum mechanics.

3. How is a Fourier transform of a function calculated?

The Fourier transform of a function is calculated by taking the integral of the function multiplied by a complex exponential function. The result is a new function that represents the frequency components of the original function.

4. What is the relationship between a function and its Fourier transform?

The Fourier transform is a mathematical operation that transforms a function from the time or spatial domain to the frequency domain. This means that the Fourier transform of a function is a different representation of the same information, just in a different domain.

5. Can a Fourier transform be applied to any function?

In theory, a Fourier transform can be applied to any function. However, in practice, some functions may not have a well-defined Fourier transform due to technical limitations or mathematical constraints.

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