A rectangular corral of widths ${\mathbf{L}}_{\mathbf{x}}\mathbf{=}\mathbf{L}$and ${\mathbf{L}}_y\mathbf{=}\mathbf{L}$contains seven electrons. What multiple of${\mathit{h}}^{\mathbf{2}}\mathbf{/}\mathbf{8}{\mathbf{mL}}^{\mathbf{2}}$ gives the energy of the ground state of the system? Assume that the electrons do not interact with one another, and do not neglect spin.

A rectangular coral of widths,${\mathrm{L}}_x=L$ androle="math" localid="1661496986140" $L_y=L$

Pauli's exclusive principle states that no two electrons in the same atom can have the same values in all four of their quantum numbers.

Using Eq. 39-20, define that the lowest four levels of the rectangular coral are non-degenerate. Using the concept of Pauli's exclusion principle, define the total energy of the non-degenerate levels in the multiple of our required value ${\mathit{h}}^{\mathbf{2}}\mathbf{/}\mathbf{8}{\mathbf{mL}}^{\mathbf{2}}$

Using the concept, the energies of the lowest four levels can be given as:

$$\begin{gathered} {\mathrm{E}}_{1,1}=1.25{\mathrm{h}}^2/8{\mathrm{mL}}^2 \\ {\mathrm{E}}_{1,2}=2.00{\mathrm{h}}^2/8{\mathrm{mL}}^2 \\ {\mathrm{E}}_{1,3}=3.25{\mathrm{h}}^2/8{\mathrm{mL}}^2 \\ {\mathrm{E}}_{1,4}=4.25{\mathrm{h}}^2/8{\mathrm{mL}}^2 \end{gathered}$$

Now, using Pauli’s exclusion Principle the total energy in the ground state can be given as follows:

$$\begin{gathered} {\mathrm{E}}_{ground}=2\left(1.25{\mathrm{h}}^2/8{\mathrm{mL}}^2\right)+2\left(2.00{\mathrm{h}}^2/8{\mathrm{mL}}^2\right)+2\left(3.25{\mathrm{h}}^2/8{\mathrm{mL}}^2\right)+4.25{\mathrm{h}}^2/8{\mathrm{mL}}^2 \\ =17.25{\mathrm{h}}^2/8{\mathrm{mL}}^2 \end{gathered}$$

Hence, the required value of the multiple is 17.25.