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### Topic: Determine the normalization constant?  (Read 21936 times)

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#### mnq

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##### Determine the normalization constant?
« on: September 07, 2010, 10:14:23 PM »

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

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##### Re: Determine the normalization constant?
« Reply #1 on: September 08, 2010, 03:21:38 AM »
I don't know... maybe I'm wrong but I can't find it...

${ \psi\left(x\right)=N\sin \frac{n\pi x}{a} }$

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

${
\int_{-\infty}^{+\infty} \!\psi\left(x\right)dx=1 \\
N\int_{-\infty}^{+\infty} \! \sin \frac{n\pi x}{a}dx=1
}$

So we find:

${ \int\! \sin \frac{n\pi x}{a}dx=-\frac{a}{n\pi}\cos\frac{n\pi x}{a}+k }$

But the following limit doesn't exist:

${ -\lim_{x \to \pm \infty}\frac{a}{n\pi}\cos\frac{n\pi x}{a}=}$

Also, why do they give you the half-angle formula as a hint? I don't know where you can use it...
The way of the superior man may be compared to what takes place in traveling, when to go to a distance we must first traverse the space that is near, and in ascending a height, when we must begin from the lower ground. (Confucius)

#### tamim83

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##### Re: Determine the normalization constant?
« Reply #2 on: September 08, 2010, 08:24:15 AM »
Remember, you are only integrating between 0 and a (the length of the box) since this is a boundary condition.

#### MrTeo

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##### Re: Determine the normalization constant?
« Reply #3 on: September 08, 2010, 12:57:03 PM »
Remember, you are only integrating between 0 and a (the length of the box) since this is a boundary condition.

So:

${ \int_0^a \!\psi\left(x\right)dx=1 \\
N\int_0^a \! \sin \frac{n\pi x}{a}dx=1 \\ }$

and:

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

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

${ N=\frac{n\pi}{2a} }$

Right?

The way of the superior man may be compared to what takes place in traveling, when to go to a distance we must first traverse the space that is near, and in ascending a height, when we must begin from the lower ground. (Confucius)

#### Jorriss

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##### Re: Determine the normalization constant?
« Reply #4 on: September 10, 2010, 09:58:49 PM »
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

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##### Re: Determine the normalization constant?
« Reply #5 on: September 11, 2010, 02:35:08 AM »
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
The way of the superior man may be compared to what takes place in traveling, when to go to a distance we must first traverse the space that is near, and in ascending a height, when we must begin from the lower ground. (Confucius)

#### Jorriss

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##### Re: Determine the normalization constant?
« Reply #6 on: September 11, 2010, 08:32:24 PM »

#### tamim83

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##### Re: Determine the normalization constant?
« Reply #7 on: September 11, 2010, 11:15:07 PM »
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

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##### Re: Determine the normalization constant?
« Reply #8 on: September 12, 2010, 02:23:14 AM »
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.
The way of the superior man may be compared to what takes place in traveling, when to go to a distance we must first traverse the space that is near, and in ascending a height, when we must begin from the lower ground. (Confucius)

#### tamim83

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##### Re: Determine the normalization constant?
« Reply #9 on: September 12, 2010, 10:44:36 AM »
I had an older version of Safari.  Lets see if this works:

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

Success!

#### mnq

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##### Re: Determine the normalization constant?
« Reply #10 on: September 14, 2010, 07:20:11 PM »
Thanks you guys

#### love48

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##### Re: Determine the normalization constant?
« Reply #11 on: January 30, 2011, 12:18:40 PM »
The wave function Ψ(theta), for the motion of a particle in a ring is  Ψ= Ne^(imφ).
Determine the normalization constant.