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.

For ground level, n₁ = 1

Let E₁be the energy of this level. It is known that E₁ is related with m as:

E₁ =-13.6/n²₁ eV

= -13.6 /1² = -13.6 eV

The atom is excited to a higher level, n2 = 4. Let E2 be the energy of this level.
The amount of energy absorbed by the photon is given as:
Therefore E2 =-13.6/n²2 eV 
                          =-13.6/4² = -13.6 /16 eV

The amount of energy absorbed by the photon is given as :

E = E2-E1

= -13.6 /16 -(-13.6 ) 

= 13.6 x 15/16  eV

= 13.6 x 15/16 x 1.6 x 10⁻¹⁹ = 2.04 x 10⁻¹⁸ J

For a photon of wavelength λ, the expression of energy is written as:

E = hc/λ

Where,

h = Planck’s constant = 6.6 x 10⁻³⁴ Js and c = Speed of light = 3 x 10⁸ m/s.

Therefore , λ = hc/E = (6.6 x 10⁻³⁴) x (3 x 10⁸ ) /(2.04 x 10⁻¹⁸ )

= 9.7 x 10⁻⁸ m = 97nm

And, frequency of a photon is given by the relation,

ν =c/ λ  = (3 x 10⁸)/(9.7 x 10⁻⁸)  ≈ 3.1 x 10¹⁵ Hz

Hence, the wavelength of the photon is 97 nm while the frequency is 3.1 x 10¹⁵ Hz.