Troubleshooting Integration for Particle Probability in an Infinite Well

[dangsy]

Homework Statement

Determine the probability of finding a particle of mass m between x=0 and x=L/10, if it is in n=3 state of an infinite well.

Homework Equations

$$P = \int_a^b\left |\psi\left(x\right)\right|^2 dx$$
$$\left|\psi\right|^2 = \frac{2}{L}Sin^2\left(\frac{nx\pi}{L}\right)$$

The Attempt at a Solution

I'm trying to integrate...

$$\int_0^{\frac{L}{10}}\frac 2 L \sin^{2}\left(\frac{3x\pi}{L}\right)dx$$
step (1)
$$\frac{1}{L}\left[ \int_0^\frac{L}{10}1-\int_0^\frac{L}{10}Cos\left(\frac{6x\pi}{L}\right)\right]dx$$step (2)

$$\frac{1}{L}\left[\frac{L}{10} - Sin\left(\frac{3\pi}{5}\right)\right]$$

When I finish solving, I end up with an L in the answer...
which I know I'm not suppose to have, did I mess up my integration somewhere?
sorry about my pooooor pooor latex

 

[rocomath]

$$\int_0^{\frac{L}{10}}\frac 2 L \sin^{2}\left(\frac{3x\pi}{L}\right)dx$$

Yes?

 

[dangsy]

yep that's right sorry =(

 

[rocomath]

Sorry, PF was like broke last night. Use this trig identity:

$$\cos{2x}=\cos^{2}x-\sin^{2}x$$

$$\sin^{2}x=\frac 1 2(1+\cos{2x})$$

But I think that's what you did, so it's good? Sorry don't have time to work it out myself.

 

[dangsy]

Yes thank you!