Answer
See explanation.
Work Step by Step
Calculate the probability density from the given wavefunction.
$$|\psi(x)|^2=\frac{2}{L}\sin^2 \frac{2 \pi x}{L}$$
a. The probability is zero where the probability density is zero.
$ \sin \frac{2 \pi x}{L}=0$
$\frac{2 \pi x}{L}=n \pi$, where n = 0, 1, 2,…
$ x=0, \frac{L}{2}, L$ are the only solutions for x inside the box.
b. The probability is highest where the probability density is maximum. A sine function’s maximum magnitude is 1.
$ \sin \frac{2 \pi x}{L}=\pm 1$
$\frac{2 \pi x}{L}=\frac{n \pi}{2}$, where n = 1, 3, 5,…
$ x=\frac{L}{4}, \frac{3L}{4}$ are the only solutions for x inside the box.
c. Yes, this matches the picture.
