Answer
The energy of the electron’s ground state is $1.25$ times of $\frac{h^2}{8mL^2}$.
Work Step by Step
The quantized energies for an electron trapped in a two-dimensional infinite potential well that forms a rectangular corral are
$E_{nx,ny}=\frac{h^2}{8m}\Big(\frac{n_x^2}{L_x^2}+\frac{n_y^2}{L_y^2}\Big),$
where $m$ is the electron mass and $n_x$ is a quantum number for well width $L_x$ and $n_y$ is a quantum number for well width $L_y$.
The ground state is the $(1,1)$ state, with an energy of
$E_{1,1}=\frac{h^2}{8m}\Big(\frac{1}{L_x^2}+\frac{1}{L_y^2}\Big)$
Substituting, $L_x=L$, and $L_y=2L$, we get the ground state energy of the electron in the given rectangular corral
$E_{1,1}=\frac{h^2}{8m}\Big\{\frac{1}{L^2}+\frac{1}{(2L)^2}\Big\}$
or, $E_{1,1}=1.25\times\frac{h^2}{8mL^2}$
$\therefore$ The energy of the electron’s ground state is $1.25$ times of $\frac{h^2}{8mL^2}$.
