The possible values for the principal quantum numbers (n₁, n₂) for a hydrogen atom transition with a frequency ratio of 1/27 between initial and final states are (3, 4) or (6, 9).
In the context of the hydrogen atom, electronic transitions occur when an electron changes its orbit, and these transitions are characterized by the principal quantum numbers (n₁, n₂) corresponding to the initial and final states of the electron's orbit. According to the Bohr model, which assumes quantized orbits, the frequency of the orbital motion is inversely proportional to the square of the principal quantum number.
In this scenario, if the frequency of the orbital motion in the initial state is 1/27 of that in the final state, the possible values for (n₁, n₂) that satisfy this condition are (3, 4) or (6, 9). These values adhere to the relationship between quantum numbers and orbital frequencies as described by the Bohr model of the hydrogen atom.