In Eq. $38-25$ keep both terms, putting $A=B=\psi_{0} .$ The equation then describes the superposition of two matter waves of
equal amplitude, traveling in opposite directions. (Recall that this
is the condition for a standing wave.) (a) Show that $|\Psi(x, t)|^{2}$ is
then given by
$$
|\Psi(x, t)|^{2}=2 \psi_{0}^{2}[1+\cos 2 k x]
$$
(b) Plot this function, and demonstrate that it describes the square
of the amplitude of a standing matter wave. (c) Show that the
nodes of this standing wave are located at
$$
x=(2 n+1)\left(\frac{1}{4} \lambda\right), \quad \text { where } n=0,1,2,3, \ldots
$$
and $\lambda$ is the de Broglie wavelength of the particle. (d) Write a similar expression for the most probable locations of the particle.