$(1)$ Consider a free $H{e}^{2+}$ ion with wave function ${}_k(x)$ is accelerated along the $x$ axis

towards $x=+-$ from rest through a potential drop of $1\mathrm{.}000kV.$

$($a$)$ Calculate its final de Broglie wavelength. Would the wavelike properties be

very apparent? The diameter of an atom ranges from about $0\mathrm{.}1$ to $0\mathrm{.}5$

nanometer.

$($b$)$ Can the wave function be normalized? Justify theoretically your answer.

$($c$)$ Show that ${}_k$ is an eigenfunction of the momentum operator. What is the

eigenvalue?

$($d$)$ Prove that the kinetic energy operator for the ion is Hermitian. Why is the

quantum mechanical operator of the kinetic energy required to be Hermitian?