The wave function for a particle in a box of length L is given by . If the box extends from x = 0 to x = L, at which of the following positions is the highest probability for finding the particle?
A) x = 0.33 L
B) x = 0.04 L
C) x = 0.60 L
D) Both (a) and (b).
E) Both (b) and (c).

Question

The wave function for a particle in a box of length L is given by  . If the box extends from x = 0 to x = L, at which of the following positions is the highest probability for finding the particle?
A) x = 0.33 L 
B) x = 0.04 L 
C) x = 0.60 L 
D) Both (a) and (b). 
E) Both (b) and (c).

Quizplus · Accepted Answer

The position of highest probability is given by the square of the wave function. Taking the square of the given wave function, we get:
|ψ(x)|^2 = (2/L)sin^2((πx)/L)
To find the position of highest probability, we need to find the maximum value of sin^2((πx)/L). Since sin^2θ has a maximum value of 1, we can see that the maximum value of sin^2((πx)/L) occurs when sin((πx)/L) = 1. This occurs when (πx)/L = π/2, or x = (L/2). Therefore, the position of highest probability is at the center of the box, which is x = 0.5L. Of the given choices, the closest to the center is C, which corresponds to x = 0.6L. Therefore, the answer is C.

The Wave Function for a Particle in a Box of Length