Particle trapped in an infinite well (1d) - find probability

[Brianrofl]

Homework Statement

http://puu.sh/bTtVx/ba89b717b8.png

Homework Equations

I've tried using the integral method of Schrödinger's eq, getting:

(X/L - (1/4pi)sin(4xpi/L) from x1 to x2.

The Attempt at a Solution

I've tried plugging in the values of x given in the problem to the above equation and got reasonable answers, but they weren't right. I've tried multiple things, and I'd love to just keep trying to figure this problem out but unfortunately if I try any more I'll just miss out on the points and not bother learning it anyway (I really hate online HW systems for this reason. Puts cheating above learning the material).

Also, if you have a width L, and you're finding the probability of it being in either the first third of the width, the second third of the width, or the last third of the width, why is the probability of finding it in each region not just 100/3 or 33.33%?

Overall, I don't understand these probability questions much. Why can I use the generic (2/L)sin^2(pix/L) for small regions but have to do the integral of P(x) for others? Explanation would really be appreciated.

 

[Brianrofl]

Also, I'm not even sure where to begin on this question: http://puu.sh/bTxr5/7109166e04.png .

I'm really not understanding this stuff, his lectures were super confusing and the textbook isn't helping either.

 

[unscientific]

1. First derive the wavefunction $\psi(x) = \sqrt{\frac{2}{L}} sin(\frac{n \pi x}{L})$. Make sure you know how this is derived, trust me - it's important for you to understand this.
2. To find probability of the particle being in region $x_0$ to $x_1$, take the overlap of wavefunction: $\int_{x_0}^{x_1} (\psi)(\psi^*) dx$

Also, if you have a width L, and you're finding the probability of it being in either the first third of the width, the second third of the width, or the last third of the width, why is the probability of finding it in each region not just 100/3 or 33.33%?

Hint: It is not equally likely to find the electron anywhere (why?).

 

[BvU]

Also, I'm not even sure where to begin on this question

Well, you could make a table for nx = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 .. ny = 0,1,2,3,4,5,6,7,8,9 ..

Oh, and : try to avoid intertwining two threads. Start a new one when you have a new exercise.