Answer
See explanation.
Work Step by Step
Calculate the probability density from the given wavefunction.
$$|\Psi(x)|^2=A^2\sin^2 kx$$
a. The probability is highest where the probability density is maximum. A sine function’s maximum magnitude is 1.
$\sin kx = \pm 1$
$kx=\frac{n \pi}{2}$, where n = 1, 3, 5,…
$ \frac{2 \pi x}{\lambda}=\frac{n \pi}{2}$, where n = 1, 3, 5,…
$ x=\frac{n \lambda}{4}$, where n = 1, 3, 5,…
$ x=\frac{ \lambda}{4}, \frac{3 \lambda}{4}, \frac{5 \lambda}{4}…$
b. The probability is zero where the probability density is zero.
$\sin kx = 0$
$kx=n \pi$ where n = 0, 1, 2,…
$ \frac{2 \pi x}{\lambda}=n \pi$, where n = 0, 1, 2,…
$ x=\frac{n \lambda}{2}$, where n = 0, 1, 2,…
$ x=0, \frac{ \lambda}{2}, \lambda, \frac{3 \lambda}{2},…$
