What is the minimum energy that must be given to a H-atom in ground state so that it can emit an Hγ line in Balmer series? If the angular momentum of the system is conserved, what would be the angular momentum of such Hγ photon?

Hint: The energy required will be equal to the energy difference between the transition levels.
Hγ in Balmer series corresponds to transition n = 5 to n = 2. So, the electron in the ground state, i.e., from n = 1 must first be placed in state n = 5.
The energy required for the transition from n = 1 to n = 5 is given by=E5-E1=-0.54+13.6=13.06 eV
Step 2: Conserve the angular momentum.
Since, angular momentum is conserved,
angular momentum corresponding to Hγ photon = change in angular momentum of the electron
=L5-L2=5h2π-2h2π=3h2π=3×1.06×10-34=3.18×10-34 kg-m2/s