[mnq]

Determine the Normalization constant for a particle in a 1-D box given that the eigenfunctions for a particle in a box can be expressed as Psi(x)=N Sin((n*pi*x)/a) where N is the normalization constant. Use the relationship Sin^2 y = 1/2(1-cos 2y)

Any help would be appreciated.

[MrTeo]

I don't know... maybe I'm wrong but I can't find it...
Here's your function:



The normalizing constant N makes the area under the graph of the function equal to 1:



So we find:



But the following limit doesn't exist:



Also, why do they give you the half-angle formula as a hint? I don't know where you can use it...

[tamim83]

Remember, you are only integrating between 0 and a (the length of the box) since this is a boundary condition. 

[MrTeo]

Remember, you are only integrating between 0 and a (the length of the box) since this is a boundary condition.  

Thanks for the advice  
So:



and:



Knowing that $$ \cos\left(n\pi\right)=-1 /$$ and that $$ \cos\left(0\right)=1 /$$ we have:



Right?

[Jorriss]

Mr. Teo, you forgot that you are taking the intergral of the wave function squared. It would be the integral from 0 to a of N^2Sin^2(stuff), etc, etc.

Man, how are you writing those symbols lol?

[MrTeo]

Mr. Teo, you forgot that you are taking the intergral of the wave function squared. It would be the integral from 0 to a of N^2Sin^2(stuff), etc, etc.

Can't convince myself that I'm not working with a gaussian... 
So (Erratum):

$$ \int_0^a\left(\psi\left(x\right)\right)^2dx=1 \\
N\int_0^a\left(\sin\frac{n\pi x}{a}\right)^2dx=1 \\
\int\left(\sin\frac{n\pi x}{a}\right)^2dx=\int\frac{1-\cos\frac{2n\pi x}{a}}{2}dx=\frac{1}{2}x-\frac{a}{4n \pi}\sin\frac{2n\pi x}{a}+k \\
\int_0^a\left(\psi\left(x\right)\right)^2dx=N\left[\frac{1}{2}x-\frac{a}{4n \pi}\sin\frac{2n\pi x}{a}\right]_0^a=1 \\
N=\frac{2}{a} /$$

Man, how are you writing those symbols lol?

http://en.wikibooks.org/wiki/LaTeX
http://www.chemicalforums.com/index.php?topic=28176.0

[Jorriss]

http://en.wikibooks.org/wiki/LaTeX
http://www.chemicalforums.com/index.php?topic=28176.0

Thanks man!

[tamim83]

Don't forget to square the "N" as well, that will change your answer a bit. 

I am actually having some problems getting LaTex to work.  I guess my browser (Safari) isn't recognizing the code?  It just shows up as written. 

[MrTeo]

Don't forget to square the "N" as well, that will change your answer a bit. 

$$ N=\sqrt{\frac{2}{a}} /$$

I am actually having some problems getting LaTex to work.  I guess my browser (Safari) isn't recognizing the code?  It just shows up as written. 

I use Safari too (5.0.1) and LaTeX works like a charm... but I also have a TeX distribution installed.

[tamim83]

I had an older version of Safari.  Lets see if this works:

 

Success!

[mnq]

Thanks you guys

[love48]

The wave function Ψ(theta), for the motion of a particle in a ring is  Ψ= Ne^(imφ).
Determine the normalization constant.