The energy levels of the hydrogen atom are quantized and their energy is given by the approximated formula
![E=- \frac{13.6}{n^2} [eV]](https://tex.z-dn.net/?f=E%3D-%20%20%5Cfrac%7B13.6%7D%7Bn%5E2%7D%20%5BeV%5D%20)
where n is the number of the level.
In the transition from n=2 to n=6, the variation of energy is
![\Delta E=E(n=6)-E(n=2)=-13.6 ( \frac{1}{6^2}- \frac{1}{2^2} )[eV]=3.02 eV](https://tex.z-dn.net/?f=%5CDelta%20E%3DE%28n%3D6%29-E%28n%3D2%29%3D-13.6%20%28%20%5Cfrac%7B1%7D%7B6%5E2%7D-%20%5Cfrac%7B1%7D%7B2%5E2%7D%20%20%29%5BeV%5D%3D3.02%20eV)
Since this variation is positive, it means that the system has gained energy, so it must have absorbed a photon.
The energy of photon absorbed is equal to this
![\Delta E](https://tex.z-dn.net/?f=%5CDelta%20E)
. Converting it into Joule,
![\Delta E=3.02 eV=4.84 \cdot 10^{-19}J](https://tex.z-dn.net/?f=%5CDelta%20E%3D3.02%20eV%3D4.84%20%5Ccdot%2010%5E%7B-19%7DJ)
The energy of the photon is
![E=hf](https://tex.z-dn.net/?f=E%3Dhf)
where h is the Planck constant while f is its frequency. Writing
![\Delta E=hf](https://tex.z-dn.net/?f=%5CDelta%20E%3Dhf)
, we can write the frequency f of the photon:
![f= \frac{\Delta E}{h}= \frac{4.84 \cdot 10^{-19}J}{6.63 \cdot 10^{-34}m^2 kg/s}=7.29 \cdot 10^{14}Hz](https://tex.z-dn.net/?f=f%3D%20%5Cfrac%7B%5CDelta%20E%7D%7Bh%7D%3D%20%5Cfrac%7B4.84%20%5Ccdot%2010%5E%7B-19%7DJ%7D%7B6.63%20%5Ccdot%2010%5E%7B-34%7Dm%5E2%20kg%2Fs%7D%3D7.29%20%5Ccdot%2010%5E%7B14%7DHz%20%20)
The photon travels at the speed of light,
![c=3 \cdot 10^8 m/s](https://tex.z-dn.net/?f=c%3D3%20%5Ccdot%2010%5E8%20m%2Fs)
, so its wavelength is
![\lambda = \frac{c}{f}= \frac{3 \cdot 10^8 m/s}{7.29 \cdot 10^{14}Hz}=4.11 \cdot 10^{-7}m=411 nm](https://tex.z-dn.net/?f=%5Clambda%20%3D%20%20%5Cfrac%7Bc%7D%7Bf%7D%3D%20%5Cfrac%7B3%20%5Ccdot%2010%5E8%20m%2Fs%7D%7B7.29%20%5Ccdot%2010%5E%7B14%7DHz%7D%3D4.11%20%5Ccdot%2010%5E%7B-7%7Dm%3D411%20nm%20%20)
So, the initial sentence can be completed as:
The n = 2 to n = 6 transition in the bohr hydrogen atom corresponds to the "absorption" of a photon with a wavelength of "411" nm.