Complete the Square: How to Evaluate an Integral

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Discussion Overview

The discussion revolves around the process of completing the square in the context of evaluating an integral related to a wave packet, as presented in a modern physics book. Participants are trying to understand the transformation of the expression involving the exponent in the integral.

Discussion Character

  • Exploratory
  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • One participant expresses confusion about how to arrive at the completed square form of the expression involving the integral: ikx - a^2k^2 = - (ak - \frac{ix}{2a} )^2 - \frac{x^2}{4a^2}.
  • Another participant provides a link to a general formula for completing the square, but it is not deemed helpful by the original poster.
  • There are requests for clarification on specific steps, particularly regarding the manipulation of terms like a^2k^2 and the introduction of \frac{ix}{2a}.
  • One participant attempts to break down the equation by suggesting that certain terms must be added and subtracted to reveal a linear expression in a square.
  • Another participant references a general example of completing the square, relating it to the original equation and suggesting a method to confirm the transformation.

Areas of Agreement / Disagreement

The discussion remains unresolved, with participants expressing confusion and seeking clarification on specific steps in the mathematical process. There is no consensus on the best approach to complete the square in this context.

Contextual Notes

Participants have not fully articulated their assumptions or the specific definitions they are using, which may affect their understanding of the problem. The discussion includes various interpretations of the steps involved in completing the square.

solidus_E
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this isn't really a homework problem

it's just me trying to understand a part of an example problem from a modern physics book

it's an integral of a wave packet, blah blah blah

but they go on to say "...to evaluate the integral, we first complete the square in the exponent as..."

[tex]ikx - a^2k^2 = - (ak - \frac{ix}{2a} )^2 - \frac{x^2}{4a^2}[/tex]

how in the world does one arrive at that?
 
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i appreciate the link

but it doesn't help

edit: nm, i see what's going on...but can someone explain wtf
 
Last edited:
solidus_E said:
edit: nm, i see what's going on...but can someone explain wtf

If you can elaborate on what exactly it is you're struggling with then perhaps someone can help you.
 
getting started lol

how would you work with the [tex]a^2k^2[/tex]
 
solidus_E said:
this isn't really a homework problem

it's just me trying to understand a part of an example problem from a modern physics book

it's an integral of a wave packet, blah blah blah

but they go on to say "...to evaluate the integral, we first complete the square in the exponent as..."

[tex]ikx - a^2k^2 = - (ak - \frac{ix}{2a} )^2 - \frac{x^2}{4a^2}[/tex]

how in the world does one arrive at that?
We can start with:
[tex]ikx-a^{2}k^{2}=-((ak)^{2}-ikx))=-((ak)^{2}-2(ak)\frac{ix}{2a})[/tex]
What must you add&subtract in order to generate an expression in which k is hidden away linearly within a square?
 
where did you get the [tex]\frac{ix}{2a}[/tex]
 
solidus_E said:
where did you get the [tex]\frac{ix}{2a}[/tex]
[tex]ikx=1*ikx=\frac{2a}{2a}*ikx=2(ak)\frac{ix}{2a}[/tex]
 
If you take a look at a general example it may help clarify. We use the fact that:

[tex](x+b)^2 = x^2+2bx +b^2[/tex]

which can be written as,

[tex](x+b)^2 -b^2 = x^2 +2bx[/tex]

Now if you look at what arildno has done in post number 6, he's written your equation in the form [itex]x^2+2bx[/itex] where [itex]x=-(ak)[/itex] and [itex]b=\frac{ix}{2a}[/itex]. Now I'm sure you can confirm for yourself from here where they obtained the equation in your original post.
 

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