An atom recoils when it emits a photon. The energy of the photon and the recoil kinetic energy are derived from the energy difference between the states involved in the transition. Consider a hydrogen atom that changes its state from n = 3 to n = 2 . Determine the fractional change in the wavelength of the emitted light due to this recoil.

Step-by-step solution

The energy difference between the states for a transition from n = 3 to n = 2 in a hydrogen atom is given by: E = 13.6 ( 1 2^2 - 1 3^2 ) eV = 13.6 5 36 eV 1.89 eV Let E be the energy of the emitted photon and K be the recoil kinetic energy of the atom. From the conservation of energy: E = E + K From the conservation of momentum, the momentum of the recoiling atom equals the momentum of the photon: p = E c The recoil kinetic energy is: K = p^2 2m = E^2 2mc^2 ( E)^2 2mc^2 The energy of the emitted photon is E = E - K