Fourier Transform of Gaussian Function

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Homework Help Overview

The discussion revolves around finding the Fourier transform of the function exp((a x^2)/2), a topic relevant to Advanced Quantum Mechanics. Participants share their attempts and challenges in solving the problem without relying on computational tools.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • The original poster attempts various methods, including splitting the Fourier factor into sine and cosine and expressing the exponential term as an infinite series. They express frustration over reaching results that lack detail or lead to the error function.
  • Some participants suggest completing the square in the exponent, while others question how this approach might help, noting concerns about the introduction of new variables.
  • There are expressions of impatience regarding the pace of responses and the perceived difficulty of the question.

Discussion Status

The discussion includes attempts to clarify the completion of the square method, with one participant successfully resolving their issue by correctly applying this technique. However, there is no explicit consensus on the best approach, and multiple interpretations of the problem remain evident.

Contextual Notes

Participants mention the challenge of dealing with infinite series and improper integrals, as well as the desire to solve the problem independently for class purposes. There is an emphasis on the need for detailed steps and clarity in the discussion.

TheDestroyer
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Hello guys,

I have got a homework for Advanced Quantum Mechanics, actually I've tried to solve it in many ways my own, but I'm always forced to use computer at the end (For infinite series or improper integrals), I want to solve it my self, so I can do it also in the class!

The problem is:

Find the Fourier transform of exp((a x^2)/2)

Actually the first way I tried is by splitting the Fourier factor exp(ikx) to sine and cosine, by Euler form, but no use! because I reached "Abramowitz and Stegun integral", which have a result without details.

The second way is in the PDF, where I tried to release the exp((a x^2)/2) term to an infinite series, But also no use, I reached the Erf error function due to an infinite series!

Thanks for your help :)
 
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TheDestroyer said:
Hello guys,

I have got a homework for Advanced Quantum Mechanics, actually I've tried to solve it in many ways my own, but I'm always forced to use computer at the end (For infinite series or improper integrals), I want to solve it my self, so I can do it also in the class!

The problem is:

Find the Fourier transform of exp((a x^2)/2)

Actually the first way I tried is by splitting the Fourier factor exp(ikx) to sine and cosine, by Euler form, but no use! because I reached "Abramowitz and Stegun integral", which have a result without details.

The second way is in the PDF, where I tried to release the exp((a x^2)/2) term to an infinite series, But also no use, I reached the Erf error function due to an infinite series!

Thanks for your help :)

Try completeing the square in the exponent...

;)
 
You mean in the PDF? or from the begining?
 
Sorry, I didn't understand how can this benefit me, If I completed the square this will cause the appearance of a new exponential multiplicand!

I also failed because in the first term inside the exponential I have a, and in the second no a! so how can I do it?
 
Why you guys aren't helping? Is the question too difficult? or I don't deserve the help??
 
TheDestroyer said:
Why you guys aren't helping? Is the question too difficult? or I don't deserve the help??

calm down. I'm not online 24 hours a day. You completed the square in the exponent of the integrand of the Fourier transform and what did you end up with. Show up your work, if possible. Also, if possible can you write your work down using LaTeX.
 
Thank you :), It's solved, I succeeded, my mistake was in the completion of the square using variables, I used the constants and it worked ;)

Thanks :)
 

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