Finite one dimensional quantum well

Figure a) shows the energy-level diagram for a finite, one-dimensional energy well that contains an electron. The non quantized region begins at $E4 =480$ eV. Figure b) gives the absorption spectrum of the electron when it is in the ground state – it can absorb at the indicated wavelengths $\lambda_a=14.588 nm$ and $\lambda_b=4.8437 nm$ and for any wavelength less than $\lambda_c=2.9108 nm$. What is the energy of the first excited state?

The lowest wavelength absorbed correspond to the transition from the lowest energy state to the non-quantized region. Therefore the energy of the first level with respect to the non quantized region is:

$E1 = h*c/\lambda(c) =6.626*10^{-34}*3*10^8/2.9108*10^{-9} =6.83*10^{-17} J =426.8 eV$

Measured from the bottom of the well the first energy level is at

$E1′ =480-426.8 =53.18 eV$

The energy of the second level (relative to the first level) is

$E2 =h*c/\lambda(b) = 6.626*10^{-34}*3*10^8/14.588*10^{-9} =1.36*10^{-17} J =85.16 eV$

The energy of the first excited state from the well bottom is just

$E2’=85.16+53.18 =138.34 eV =2.21*10^{-17} J$