Question

Determine the probability that a particle in 1-dimension box of length a, with no forces acting on it, in state corresponding to n=10,will be found in the region between a/3 and 2a/3.

Answer 1

Answer:

$P=0.36$

Explanation:

The probability to find a particle in a 1 dimension box of length is:

$P=\int^{2a/3}_{a/3}\Psi^{2}dx$

Let's recall the wave function of a particle in this candiction will be:

$\Psi=\sqrt{\frac{2}{a}}sin\left(\frac{n\pi x}{a}\right)$

So the propability:

$P=\int^{2a/3}_{a/3}\Psi^{2}dx=\frac{2}{a}\int^{2a/3}_{a/3}sin^{2}\left(\frac{n\pi x}{a}\right)dx$  

Solving the integral, it is:

$P=\frac{2}{a}(\frac{x}{2}-\frac{asin(2\pi nx/a)}{4\pi n})|^{2a/3}_{a/3}$    

$P=\frac{2}{a}\frac{a(2\pi n+3sin(2\pi n/3)-3sin(4\pi n/3))}{12\pi n}$    

We can cancel out a and using n = 10, the probabilty finally will be:

$P=0.36$

I hope it helps you!