4. The normalized wavefunctions for a particle-in-box (length, a = 40 nm) in the x-
direction are given by:
$\Psi(x) = \left(\frac{2}{a}\right)^{\frac{1}{2}} \sin\left(\frac{n\pi}{a}x\right)$ for $0 \le x \le a$ and $n = 1, 2, 3, 4$, etc.
(a) Using Excel and on the same diagram, plot $\Psi$ (n = 2), $\Psi$ (n = 3) and the product
$\Psi$ (n = 2) $\cdot \Psi$ (n = 3) and comment on the result.
(b) Using Excel, plot diagrams of the variation of the $\Psi(n = 5)$ and the corresponding
probability density function, $\Psi^2(n = 5)$ [6]
(c) At what values of x are the functions $\Psi$ (n = 2), $\Psi$ (n = 3), $\Psi$ (n = 2) $\cdot \Psi$ (n = 3)
and $\Psi^2(n = 5)$ above equal to zero in the range $0 < x < a$. [5]
