12.6 A hydrogen atom initially in the ground level absorbs a photon, which excites it to the n = 4 level. Determine the wavelength and frequency of photon.

Hint: Energy recieved by the hydrogen atom equals to the energy difference between the levels.
Step 1: Find the energy of the atom in the ground level.
For ground level, n1=1
Let E1 be the energy of this level.
It is known that E1 is related with n1 as:
E1=13.6n12eV=13.612=13.6eV
Step 2: Find the energy of the atom in the excited level.
The atom is excited to a higher level, n2 = 4.
Let E2 be the energy of this level.
=13.616(13.61)=13.6×1516eV=13.6×1516×1.6×1019=2.04×1018  J
Step 3: Find the wavelength and frequency of the photon.
For a photon of wavelength λ, the expression of energy is written as:
E=hcλ
where,
h=planck's constant
=6.62×10-34   Js and c=Speed of light=3×108m/s
λ=hcE=6.62×10-34×3×1082.04×10-18=9.74×10-8 m=97.4  nm
And, the frequency of a photon is given by the relation,
\(\nu=\frac{c}{\lambda}=\frac{3 \times 10^{8}}{9.74 \times 10^{-8}} \approx 3.1 \times 10^{15} \mathrm{~Hz}\)
Hence, the wavelength of the photone is 97.4 nm while the frequency is  3.1×1015Hz.