91Ó°ÊÓ

The expression of number of states is given by,

$\mathbf{N}\mathbf{=}\frac{\mathbf{n}}{{\mathbf{E}}_{\mathbf{2}}\mathbf{−}{\mathbf{E}}_{\mathbf{1}}}$

a)

Given information is

The ${\mathrm{E}}_{\mathrm{n}}$ is proportional to single quantum number $n$.

To complete the table up to ${10}^{\text{th}}$ state, $\mathrm{n}$ should varies such that it gives the lowest amount of energy. The complete table is

(b)

The expression of number of states is given by,

$\mathrm{N}=\frac{\mathrm{n}}{{\mathrm{E}}_2−{\mathrm{E}}_1}$

The number of states for first five in 1D is calculated as,

$\begin{array}{c}\mathrm{N}=\frac{\mathrm{n}}{{\mathrm{E}}_2−{\mathrm{E}}_1}\\ \mathrm{N}\left(\text{firstfive}\right)=\frac{5}{25−1}\\ =\frac{5}{24}\end{array}$

The number of states for last five in 1Dis calculated as,

$\begin{array}{c}\mathrm{N}=\frac{\mathrm{n}}{{\mathrm{E}}_2−{\mathrm{E}}_1}\\ \mathrm{N}\left(\text{lastfive}\right)=\frac{5}{100−36}\\ =\frac{5}{64}\end{array}$

The number of states for first five in 2Dis calculated as,

$\mathrm{N}=\frac{\mathrm{n}}{{\mathrm{E}}_2−{\mathrm{E}}_1}$

$\begin{array}{c}\mathrm{N}\left(\text{firstfive}\right)=\frac{5}{10−2}\\ =\frac{5}{8}\end{array}$

The number of states for last five in 2Dis calculated as,

$\begin{array}{c}\mathrm{N}=\frac{\mathrm{n}}{{\mathrm{E}}_2−{\mathrm{E}}_1}\\ \mathrm{N}\left(\text{lastfive}\right)=\frac{5}{17−10}\\ =\frac{5}{7}\end{array}$

The number of states for first five in 3Dis calculated as,

$\begin{array}{c}\mathrm{N}=\frac{\mathrm{n}}{{\mathrm{E}}_2−{\mathrm{E}}_1}\\ \mathrm{N}\left(\text{firstfive}\right)=\frac{5}{9−3}\\ =\frac{5}{6}\end{array}$

The number of states for last five in 3Dis calculated as,

$\begin{array}{c}\mathrm{N}=\frac{\mathrm{n}}{{\mathrm{E}}_2−{\mathrm{E}}_1}\\ \mathrm{N}\left(\text{lastfive}\right)=\frac{5}{11−9}\\ =\frac{5}{2}\end{array}$

Therefore, the number of states for first five and last five in $1D$ are $\frac{5}{24}$ and $\frac{5}{64}$ respectively, in 2D are $\frac{5}{8}$ and $\frac{5}{7}$ respectively and in 3D are $\frac{5}{6}$ and $\frac{5}{2}$ respectively.

c)

From part (b), the density of the states tends to increase with the dimensionality but decrease with the energy.