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 Schrodinger'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.

 

[HalfLostAndSearching]

I understand your frustration with online homework systems and the pressure to just get the right answer rather than truly understanding the material. However, it's important to remember that the purpose of homework is to help you learn and practice the material, not just to get points. It's okay to struggle and make mistakes as long as you are actively learning and trying to understand the concepts.

Now, let's talk about the problem at hand. It seems like you are trying to find the probability of finding the particle in a specific region of the infinite well. This can be done by using the wavefunction, which is given by the Schrodinger's equation. The wavefunction represents the probability amplitude, which is the square root of the probability. In other words, the probability of finding the particle in a specific region is equal to the square of the wavefunction in that region.

In the case of an infinite well, the wavefunction can be written as a sine function, as you have shown in your attempt at a solution. However, the probability of finding the particle in a specific region is not simply given by the value of the wavefunction in that region. Instead, it is given by the integral of the wavefunction squared over that region. This is because the wavefunction is a complex number and its absolute value (which represents the probability) can vary over the region.

To understand why the probability is not just 33.33% for each third of the width, think about how the wavefunction changes over the region. In the first third, the wavefunction starts at zero and increases to a maximum value, then decreases back to zero. In the second third, the wavefunction starts at zero again, but this time it decreases to a negative value, then increases back to zero. In the last third, the wavefunction starts at zero and decreases to a negative value, then increases back to zero. Because of these variations in the wavefunction, the probability of finding the particle in each region is not the same.

I hope this explanation helps you understand the concept of probability in quantum mechanics a little better. Remember, it's okay to make mistakes and struggle with the material as long as you are actively trying to learn and understand it. Good luck with your studies!