An electron in the 2s state of hydrogen cannot decay directly to the 1s ground state via electric dipole radiation. Why not?
AThe energy difference is too small to produce a detectable photon
BThe 2s and 1s states have the same angular momentum quantum number, violating Δl = ±1
CThe 2s state has higher energy than 1s, so emission would violate energy conservation
DThe 2s→1s transition is in the infrared and too weak to measure
The electric dipole selection rule requires Δl = ±1. Both the 2s (l = 0) and 1s (l = 0) states have the same l value, so Δl = 0 — a forbidden transition. A photon carries one unit of angular momentum, and if the electron's angular momentum doesn't change, angular momentum cannot be conserved. This makes the 2s state metastable; it can only decay via much weaker processes. Comparing to the 2p→1s transition (Δl = −1, allowed): that produces the bright Lyman-alpha line at 121.6 nm.
Question 2 Multiple Choice
The H-β line of the Balmer series corresponds to which transition, and roughly where in the electromagnetic spectrum does it appear?
A3→1 transition; ultraviolet
B4→2 transition; visible (blue-green)
C5→3 transition; near-infrared
D4→1 transition; ultraviolet
The Balmer series collects all transitions ending at n = 2. H-α is the 3→2 transition (red, 656 nm); H-β is the 4→2 transition (blue-green, ~486 nm). Option A describes a Lyman series line (ends at n = 1, UV). Option C describes a Paschen series line (ends at n = 3, infrared). Option D is also a Lyman line. The Balmer series is the only hydrogen series with lines in the visible range, which is why it was the first to be empirically catalogued.
Question 3 True / False
The selection rule Δl = ±1 follows from angular momentum conservation: a photon carries exactly one unit of angular momentum, so the electron's orbital angular momentum must change by ±1 in any electric dipole emission.
TTrue
FFalse
Answer: True
This is exactly correct. A photon has spin-1 and carries one unit of angular momentum. When it is emitted, that angular momentum must come from somewhere — the electron's orbital angular momentum quantum number l must change by ±1 to conserve total angular momentum. This is why transitions with Δl = 0 or |Δl| > 1 are forbidden by the electric dipole selection rule, though they can occur through weaker higher-order processes.
Question 4 True / False
The Lyman series lines appear in the visible part of the electromagnetic spectrum because the hydrogen ground state is at the lowest energy.
TTrue
FFalse
Answer: False
The Lyman series appears in the ultraviolet, not the visible. Because the ground state (n = 1) is the lowest energy level, Lyman series transitions have the largest energy differences of any hydrogen series (the photon must carry away the full gap to or from the ground state). Large energy differences correspond to high-frequency, short-wavelength photons — ultraviolet radiation. The Balmer series (ending at n = 2) produces the visible lines, and the Paschen series (ending at n = 3) falls in the near-infrared.
Question 5 Short Answer
Why do excited hydrogen atoms emit only discrete spectral lines rather than a continuous spectrum of wavelengths?
Think about your answer, then reveal below.
Model answer: Because hydrogen's energy levels are quantized — the electron can only occupy specific allowed energies (En = −13.6 eV / n²). Photon emission occurs when the electron transitions between two levels, and the photon's energy must exactly equal the difference between those levels. Since only discrete level differences are possible, only discrete photon frequencies are emitted. A continuous spectrum would require the electron to occupy continuously variable energies, which is not permitted in quantum mechanics.
This is the central triumph of quantum mechanics applied to the hydrogen atom: the same mathematical framework that quantizes energy levels automatically predicts exactly the spectral line positions observed experimentally. The Rydberg formula, which was empirically known, is derived exactly from E_n = −13.6 eV / n². The discreteness of the spectrum is direct experimental evidence that atomic energy levels are quantized.