Answer
$8$ different frequencies of light
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 $L_y$.
Substituting, $L_x=L$, and $L_y=2L$, we get
$E_{nx,ny}=\frac{h^2}{8mL^2}\Big(n_x^2+\frac{n_y^2}{2^2}\Big),$
The sates that corresponds to the lowest five energy levels are
$(1,1),\;(1,2),\;(1,3),\;(2,1)$ and, $(2,2)\;\text{and}\;(1,4)$ degenerate states respectively. Thus the lowest five energy levels are
$E_{1,1}=1.25\Big(\frac{h^2}{8mL^2}\Big)$
$E_{1,2}=2.00\Big(\frac{h^2}{8mL^2}\Big)$
$E_{1,3}=3.25\Big(\frac{h^2}{8mL^2}\Big)$
$E_{2,1}=4.25\Big(\frac{h^2}{8mL^2}\Big)$
$E_{2,2}=E_{1,4}=5.00\Big(\frac{h^2}{8mL^2}\Big)$
Now the possible transition with the corresponding energy difference are given below:
$\text{Transition}\;\;\;\;\;\;\;\;\text{Energy difference}(\Delta E)$
$E_{2,2}\leftrightarrow E_{1,1}\;\;\;\ \;\;\;\;\;\;\;\; 3.75\Big(\frac{h^2}{8mL^2}\Big)$
$E_{2,2}\leftrightarrow E_{1,2}\;\;\;\ \;\;\;\;\;\;\;\; 3.00\Big(\frac{h^2}{8mL^2}\Big)$
$E_{2,2}\leftrightarrow E_{1,3}\;\;\;\ \;\;\;\;\;\;\;\; 1.75\Big(\frac{h^2}{8mL^2}\Big)$
$E_{2,2}\leftrightarrow E_{2,1}\;\;\;\ \;\;\;\;\;\;\;\; 0.75\Big(\frac{h^2}{8mL^2}\Big)$
$E_{2,1}\leftrightarrow E_{1,1}\;\;\;\ \;\;\;\;\;\;\;\; 3.00\Big(\frac{h^2}{8mL^2}\Big)$
$E_{2,1}\leftrightarrow E_{1,2}\;\;\;\ \;\;\;\;\;\;\;\; 2.25\Big(\frac{h^2}{8mL^2}\Big)$
$E_{2,1}\leftrightarrow E_{1,3}\;\;\;\ \;\;\;\;\;\;\;\; 1.00\Big(\frac{h^2}{8mL^2}\Big)$
$E_{1,3}\leftrightarrow E_{1,1}\;\;\;\ \;\;\;\;\;\;\;\; 2.00\Big(\frac{h^2}{8mL^2}\Big)$
$E_{1,3}\leftrightarrow E_{1,2}\;\;\;\ \;\;\;\;\;\;\;\; 1.25\Big(\frac{h^2}{8mL^2}\Big)$
$E_{1,2}\leftrightarrow E_{1,1}\;\;\;\ \;\;\;\;\;\;\;\; 0.75\Big(\frac{h^2}{8mL^2}\Big)$
Form the energy difference value, we can say that there are $8$ different frequencies of light.
