B$-$Carotene $($a linear polyene below$)$ conjugated system can be treated as a one$-$dimensional box. If the length $L$ of the molecular box is $L=0\mathrm{.}294nm,$ calculate the wavelength $($in nanometer$)$ of the radiation that an electron should absorb to transfer from $n=11($ground state$)$ to $n=12.$

$(4$ marks$)$

a$)300nm$

A point mass rotates in a circle with $l=1,$ calculate the magnitude of its angular momentum $.$

a$)1\mathrm{.}0{10}^{-34}$

b$)1\mathrm{.}49{10}^{-34}$

c$)3\mathrm{.}49{10}^{-4}$

$($d$)2\mathrm{.}49{10}^{-34}$

e$)4\mathrm{.}49{10}^{-4}$

Q$2.$ A particle is moving in one$-$dimension between $x=$a and $x=b.$ the potential energy is such that the particle cannot be outside these limits and that the wave function in between is

$=\frac{A}{x}$

a$.$ Determine the normalization constant $A.$

b$.$ Calculate the average value of $x.($Note: hat$(x)=x$

Q$3.$Assuming that the vibrations of $N_2$ molecule are equivalent to those of a harmonic oscillator with force constant $k=2293\mathrm{.}8\frac{N}{m},$ what is the zero point energy of vibration of $N_2$ molecule $.$