Solve ∫xp e-x2 eix dx | Get Hints for Integration

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Discussion Overview

The discussion revolves around evaluating the integral ∫xp e-x2 eix dx from -∞ to ∞. Participants seek hints and strategies for solving this integral, which is related to concepts in quantum mechanics.

Discussion Character

  • Homework-related, Exploratory, Mathematical reasoning

Main Points Raised

  • One participant expresses confusion and requests hints for evaluating the integral.
  • Another participant questions whether the inquiry is related to homework.
  • A participant clarifies that their interest in the integral stems from learning quantum mechanics.
  • There is a suggestion that posting in a homework section might yield better responses.
  • One participant proposes that if p is even, only cos(x) is needed, while if p is odd, isinx is required, referencing the complex exponential form.
  • A suggestion is made to integrate by parts to reduce the exponent from p to p-1 until reaching p = 0, ultimately leading to a simpler integral involving cos(x).
  • It is mentioned that starting from the integral of e^{-\frac{x^2}{2}}cos(x) might be a more straightforward approach.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the best approach to solve the integral, and multiple strategies are proposed without resolving which is preferable.

Contextual Notes

There are assumptions regarding the nature of p (even or odd) that influence the proposed methods, and the discussion does not clarify the implications of these assumptions on the integral's evaluation.

Ananthan9470
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I need to evaluate ∫xp e-x2 eix dx from -∞ to ∞ Can someone please give me some pointers on how to do this? I am completely lost. I just need some hints or something.
 
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Ananthan9470 said:
I need to evaluate ∫xp e-x2 eix dx from -∞ to ∞ Can someone please give me some pointers on how to do this? I am completely lost. I just need some hints or something.

Is this homework?
 
PeroK said:
Is this homework?
No. I'm trying to learn quantum mechanics and this thing keeps popping up.
 
You'll get a better response if you post it in homework. Even if you're learning on your own, it still counts.

Can you integrate it without the complex exponential?
 
To get you started, if p is even you need only cosx. If p is odd you need only isinx, where e^{ix}=cosx+isinx.
Next integrate by parts to reduce exponent from p to p-1, and continue until you get p = 0.
At the end you should have \int_{-\infty}^{\infty}e^{-\frac{x^2}{2}}cosxdx.

As an afterthought, it might be easier to start from
\int_{-\infty}^{\infty}e^{-\frac{x^2}{2}}cosxdx. Then integrate by parts to increase the exponent of x.
 
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