Using the Schr$$dinger equation for a particle of mass, $m,$ moving in a one$-$dimensional region:

$-\frac{h^2}{2m}\frac{d^2}{d{x}^2}+V=E$

Show that the permitted energies for the "particle in a box" are:

$E=\frac{n^2{h}^2}{8m{L}^2},n=1,2,3$dots

Use $=sin\frac{nx}{L}$ where $L$ is the length of the box.

$2.$ A consistent set of units that may be used in de Broglie's Equation is: $$ in $cm,$ mass in $g,$ velocity in $cm{s}^{-1}$ and $h($Planck$\text{'}$s constant$)$ in $gc{m}^2{s}^{-1}.$ What is the wavelength in $cm$ and in $$ of a$)$ an electron travelling at ${10}^{6}cm{s}^{-1},$ a velocity typical in an electron microscope? b$)$ a baseball or cricket ball thrown at ${10}^{3}cm{s}^{-1}?$ Assume that the mass equals $2\mathrm{.}00{10}^{2}g.$ c$)$ Prof. Aquino moving at $1m{s}^{-1}$ with a mass $95kg?($I know, I know, I need to lose a few pounds!!$)$

$3.$ Chlorophyll absorbs blue light, $=460nm,$ and emits red light, $=660nm.$ Calculate the net energy change in the chlorophyll system $($in $kJ)$ when a mole of $460nm$ photons is absorbed$,$ and a mole of $660nm$ photons is emitted.

$4.$ State the principle and orbital angular momentum quantum numbers for each of the following orbitals: $3s,5p,7g,3f,6d.$ Which set, if any, are $($is$)$ not permissible?

$5.$ How many orbitals are there in a shell of principal quantum $n?$ Derive an equation for this relationship.

$6.$ Consider $s$ and $p$ orbital wavefunctions on two separate atoms $($i$.$e$.s,p_x,p_y,$ and $p_z).$ Which overlaps $,$ and $s+p$ will lead to the overlap integral being $0($i$.$e$.S=0).$ Sketch these overlaps out with the proper phasing $(+$ and$/$or $-).$