Final answer:
The quantum numbers of the hydrogen atom's initial state for the violet light emission of 410 nm correspond to n = 6, while the final state quantum number is n = 2 according to the Balmer series.
Step-by-step explanation:
The question asks to determine the quantum numbers of the atom's initial and final states for a violet light emission of wavelength 410 nm from a hydrogen atom. According to the Balmer series in the emission spectrum of hydrogen, every line corresponds to an electron transition from a higher energy orbit to the second orbit (n = 2). Specifically, the transition n = 6 to n = 2 gives rise to the violet light at 410 nm.
Therefore, the quantum numbers of the atom's initial state are n = 6 (since this is the transition that results in violet light). The final state quantum number is n = 2, which is given by the Balmer series for any transitions resulting in visible light emission. The violet light observed at 410 nm signifies a transition in this series.