Fourier Transform and Wave Function

You just need to finish it off.In summary, the conversation discusses finding the normalization constant N for a Gaussian wave packet and verifying the normalized Fourier Transform. The solution involves manipulating the integrand and combining exponents, but the final evaluation of the integral is still needed.
  • #1
atomicpedals
209
7

Homework Statement



a) Find the normalization constant N for the Gaussian wave packet [itex]\psi (x) = N e^{\frac{-(x-x_{0})^{2}}{2K^{2}}}[/itex]. b) Find the Fourier Transform and verify it is normalized.

2. The attempt at a solution

a) I think I've got
[tex]\psi (x) = N e^{\frac{-(x-x_{0})^{2}}{2K^{2}}}[/tex]
[tex]\int |N e^{\frac{-(x-x_{0})^{2}}{2K^{2}}}|^{2}dx = 1[/tex]
[tex]N^{2}=\frac{1}{\sqrt{\pi}K}[/tex]
[tex]N = \frac{1}{\pi^{1/4}\sqrt K}[/tex]
b) This is where the trouble starts...
[tex]\psi (x) = \frac{1}{\pi^{1/4}\sqrt K} e^{\frac{-(x-x_{0})^{2}}{2K^{2}}}[/tex]
[tex]F(\omega)=\frac{1}{\sqrt{2\pi}} \int \frac{1}{\pi^{1/4}\sqrt K} e^{\frac{-(x-x_{0})^{2}}{2K^{2}}} e^{i \omega x}dx[/tex]
I think I can pull the normalization constant out of the integrand and get
[tex]F(\omega)=\frac{1}{\sqrt{2\pi}}\frac{1}{\pi^{1/4}\sqrt K} \int e^{\frac{-(x-x_{0})^{2}}{2K^{2}}} e^{i \omega x}dx[/tex]
And the exponents should combine (here I'm not so sure)
[tex]F(\omega)=\frac{1}{\sqrt{2\pi}}\frac{1}{\pi^{1/4}\sqrt K} \int e^{\frac{-(x-x_{0})^{2}}{2K^{2}}+i \omega x}dx[/tex]
Assuming I've not gone horribly wrong earlier, evaluation of this integral stumps me. Any help and suggestions are much appreciated.
 
Last edited:
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  • #2
If I crank through it, I think the result is

[tex]F(\omega)=\frac{1}{\pi^{1/4}\sqrt{1/K^{}2}\sqrt K}e^{\frac{-K^{2}k{2}}{2}+i x_{0} \omega}[/tex]
 
Last edited:
  • #3
There should be an ##\omega^2## somewhere in the exponent. Your answer seems to have other typos as well. It would help to see what you actually did, but I think you have it under control.
 

1. What is a Fourier transform?

A Fourier transform is a mathematical operation that decomposes a function into its individual frequency components. It converts a function from its original domain (such as time or space) to a representation in the frequency domain.

2. How is a Fourier transform related to wave functions?

A Fourier transform is often used to analyze and understand wave functions. The Fourier transform of a wave function represents the amplitude and phase of the different frequency components that make up the wave.

3. What is the difference between a Fourier transform and an inverse Fourier transform?

A Fourier transform converts a function from its original domain to the frequency domain, while an inverse Fourier transform converts it back to the original domain. The two operations are essentially inverse of each other.

4. What is the significance of the Fourier transform in science and engineering?

The Fourier transform has many applications in science and engineering, including signal processing, image reconstruction, and solving differential equations. It allows us to analyze and manipulate complex functions and signals in a more intuitive way.

5. Are there any limitations to using the Fourier transform?

While the Fourier transform is a powerful tool, it does have some limitations. It assumes that the function being transformed is periodic and infinite, which may not always be the case. It also cannot accurately represent functions with sharp discontinuities or singularities.

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