The energy levels of the hydrogen atom are given by
[tex]E_n=-13.6 \frac{1}{n^2} [eV] [/tex]
where n is the level number.
In order to make transition from n=2 state to n=5 state, the electron should acquire an energy equal to the difference between the two energy levels:
[tex]\Delta E= E_5 - E_2 = -13.6 \frac{1}{5^2}-(-13.6 \frac{1}{2^2})= -0.54+3.4=2.86 eV[/tex]
Keeping in mind that [tex]1 eV = 1.6 \cdot 10^{-19}eV[/tex], we can convert this energy in Joules
[tex]\Delta E = 2.86 eV \cdot 1.6 \cdot 10^{-19} J/eV=4.58 \cdot 10^{-19} J[/tex]
This is the energy the photons of the laser should have in order to excite electrons from n=2 state to n=5 state. Their frequency can be found by using
[tex]\Delta E=hf[/tex]
where h is the Planck constant and f is the photon frequency. Re-arranging it, we find
[tex]f= \frac{\Delta E}{h}= \frac{4.58 \cdot 10^{-19} J}{6.6 \cdot 10^{-34} Js}=6.94 \cdot 10^{14} Hz [/tex]
and finally we can use the relationship between frequency, wavelength and speed of light which holds for photons, in order to find their wavelength:
[tex]\lambda= \frac{c}{f}= \frac{3 \cdot 10^8 m/s}{6.94 \cdot 10^{14} Hz}=4.32 \cdot 10^{-7} m =432 nm [/tex]
and this is the laser wavelenghth.
