A simplified verbal description of the hydrogen atom’s first four bound‐state…
PHYSICS · Multi-part · A Level · Physics · Particles and radiation · Particles and radiation
Problem
A simplified verbal description of the hydrogen atom’s first four bound‐state energies is provided below.
• n = 1: −13.6 eV (ground state)
• n = 2: −3.40 eV
• n = 3: −1.51 eV
• n = 4: −0.85 eV
An electron in a hydrogen atom is initially found in the n = 4 state and ultimately returns to the ground state. All energetically allowed spontaneous radiative transitions are possible, and the atom may emit photons in one-step or multi-step cascades.
(a) Determine the total number of distinct photon wavelengths that could appear in the complete emission spectrum originating from the n = 4 state and terminating in the ground state.
(b) Describe, without performing numerical calculations, why exactly that number of distinct wavelengths occurs. In your explanation, refer to both the number of possible pairs of initial and final states and the constraint that the photon energy equals the difference in atomic energy levels.
(c) Without using the numerical energy values, rank the photon energies associated with the n = 4 → 3, 4 → 2, 4 → 1, 3 → 2, 3 → 1, and 2 → 1 transitions from greatest to least. Justify your ordering qualitatively in terms of level spacing.