Ahydrogen atom is in the ground state. it absorbs energy and makes a transition to the n = 6 excited state. the atom returns to the ground state by emitting two photons, one in dropping to n = 5 state, and one in further dropping to the ground state. what are the photon wavelengths of (a) the first and (b) the second transitions?

The energy levels of the hydrogen atom are given by

where n is the level number. Let's use this formula to calculate the energy of the levels n=1 (ground state), n=5 and n=6:

- ground state:

- level n=5:

- level n=6:

Now that we have the energy for all the levels we are interested in, we can calculate the energy of the emitted photons.

a) In the first transition, the atom goes from n=6 to n=5. The energy of the emitted photon is equal to the energy difference between these two levels:

The energy of a photon is also equal to

where 

h is the Planck constant

c is the speed of light

f is the photon frequency

is its wavelength

Re-arranging this relationship and using the photon's energy, we find its wavelength:

b) in the second transition, from n=5 to n=1, the energy of the emitted photon is equal to the difference in energy between the two levels:

Similarly to part a), the wavelength of the photon is given by: