Probability of finding a particle in a box

[ahhppull]

Homework Statement

Consider ψ (x) for a particle in a box:

ψ_n(x) = (2/L)^1/2sin(n∏x/L)

Calculate the probability of finding the particle in the middle half of the box (i.e., L/4 ≤ x ≤ 3L/4). Also, using this solution show that as ''n'' goes to infinity you get the classical solution of 0.5.

Homework Equations

The Attempt at a Solution

I integrated and figured out the probability for n=1,2,3. For n=1 I got 1/2 + 1/∏ which is about 0.818. For n = 1 I got 1/2 and for n=3, I got 0.430.

I don't understand where the problem asks "Also, using this solution show that as ''n'' goes to infinity you get the classical solution of 0.5." From my calculations, as n goes to infinity, it does not approach a value of 0.5.

 

[TSny]

It will help if you can evaluate the integral for a general (unspecified) values of n and then look at the result as n goes to infinity.

From the 3 values you have obtained, you can't tell whether or not the probability is approaching any specific value as n gets large. (By the way, I agree with your answers for n = 1 and 2, but not for n = 3.)

 

[clamtrox]

Drawing a picture of some of the solutions might give you some insight into what kind of answer you are looking for.