Finding the Right Normalization Constant for Gaussian Integrals

[core1985]

Hello I have tried gaussian integrals does gaussian integrals have this general form formula? if not then weather i do integration by parts or what just needed a hint to solve it correctly

 

[BvU]

Hello Core,

The template is there for a reason, don't erase it but use it; it will be to your benefit.

What is the question ? and what is the relationship between your first line and the second ?

 

[core1985]

I want to say in the pic I have tried many things here should I show the steps I tried?? just want a hint that how to correctly start this nothing more weather I substitute or use gaussian integral formula for expomentional that is sqrt(pie/a) then do integration by parts??

 

[BvU]

My point is it seems you are trying to normalize the wave function $$\Psi(x,t) = A\, e^{-x^2/a^2} e^{-i\omega t} \sin kx $$ on the first line.
But the second line does not reflect that ( it says ##\ \sin (2kx) \ ## instead of ##\ \sin^2 (kx) \ ## ).

So :

What is the question ? and what is the relationship between your first line and the second ?

 

[core1985]

yes yes it is sin^2(kx) we can use 1-cos2(x)/2 formula here

 

[core1985]

but that nasty exponential how to handle that

 

[core1985]

it is liboff problem 3.15 I have found <p> that is zero but I stuck at A?

 

[core1985]

these steps I have tried now where is mistake??

 

[core1985]

Now see here in this formula list there no formula for x^2 that's why I am stuck at this step needed a hint

 

[core1985]

how to integrate E^x^2/a^2 sinkx

 

[Mark44]

yes yes it is sin^2(kx) we can use 1-cos2(x)/2 formula here

For the latter expression, if you mean ##\frac{1 - \cos^2(x)}2##, use parentheses around the terms in the numerator. What you wrote means ##1 - \frac{\cos(2x)}2##. In any case, ##\sin^2(kx) \ne \frac{1 - \cos(2x)}{2}##. You have to consider that k mulitplier.

 

[core1985]

thanks I am new to this website

 

[BvU]

Maybe you want to check out number 6 here ?

Otherwise there is CRC handbook of chemistry and physics, or Abramowitz (7.4.6)

 

[core1985]

so if I use number 6 then can I change limits to 0 to infinity multiplied by 2 then It can be applied ?

 

[core1985]

one thing more can I change sin(kx) into exponentionals and then try to solve will it work or not??

 

[BvU]

so if I use number 6 then can I change limits to 0 to infinity multiplied by 2 then It can be applied ?

IF the function is even (##\ f(x) = f(-x)\ ##) then yes.

one thing more can I change sin(kx) into exponentionals and then try to solve will it work or not??

You can give it a try...

 

[core1985]

what do you suggest now changing sin to exponential using euler formula or use this

 

[core1985]

but cos(kx) is even ?? so I can use this to solve this nasty integral

 

[core1985]

ok I am solving it by both methods and will tell you what I got

 

[BvU]

but cos(kx) is even ?? so I can use this to solve this nasty integral

Yes you can

what do you suggest now changing sin to exponential using euler formula or use this

That would be the idea. But it doesn't look clean and quick to me, such a complex exponential...

After all, integrating ##\ e^{-x^2}\ ## alone already requires ingenious mathematical manipulating...

 

[core1985]

ok then I use cos formula but can normalization have ? e term? according to the formula number 6 means I can write exponential in normalization

 

[BvU]

The normalization process is to determine your ##A## such that ##\displaystyle \int \Psi(x,t)^* \Psi(x,t) \ = 1 ##. The result of the indefinite integral is basically just a number.