Selection rules: why some hydrogen transitions emit light and others do not

1. From an energy interval to a transition amplitude

Two states can have different energies while their direct coupling to a chosen radiation field vanishes. To calculate a transition, introduce an interaction V(t) in addition to the atomic Hamiltonian. In the electric-dipole approximation, the electric field is nearly uniform across the atom and

Here r points from the proton to the electron and d is the atom’s electric dipole operator in the internal coordinate. The charge sign affects amplitude conventions; the transition strength involves a squared magnitude. First-order time-dependent perturbation theory gives an amplitude proportional to

The time integral selects the relevant resonance condition. The matrix element determines whether that resonant field can connect the states. These are different pieces of the calculation. MIT’s time-dependent perturbation notes provide the standard framework used here. [MIT, Time Dependent Perturbation Theory]

For a temporal atom model, this distinction identifies two targets: the construction of stationary records and the rule by which those records exchange energy and angular momentum with propagating radiation.

2. The parity rule follows from the dipole operator

A hydrogen orbital has parity (−1)^ℓ: reversing r multiplies its wavefunction by that sign. The vector r is odd under the same inversion. Consider one component of the spatial dipole integral. After the change of variables r → −r, the integral acquires the factor (−1)^{ℓ_a + ℓ_b + 1}.

If this factor is −1, the integral equals its own negative and must vanish. A nonzero electric-dipole amplitude therefore requires opposite parities for the two states. In particular, an S–S transition has zero electric-dipole amplitude in the unperturbed central-potential problem, regardless of the size of its energy difference.

Parity alone does not give the entire rule. It permits any odd change in orbital parity, but the vector character of the operator supplies a further angular-momentum restriction. Combining the two facts gives Δℓ = ±1. This derivation shows why the rule belongs to the interaction and its transformation properties, rather than to a numerical coincidence in the energies.

External fields or interactions that mix opposite-parity states change the state basis and can open a weak channel. Such a result should be calculated from the mixing interaction, with its field dependence included.

3. Angular momentum, polarization, and spin

The three spherical components of a vector are labelled q = −1, 0, +1. Their angular matrix elements obey m_b = m_a + q. Thus Δm = 0, ±1, with the accessible component determined by the field’s polarization and the chosen quantization axis. These statements assume that m is a useful state label in the specified field regime.

For electron states resolved by total angular momentum j, a vector operator requires Δj = 0, ±1, except that j = 0 → j = 0 is excluded. Opposite parity is still required. The leading electric-dipole operator acts on the spatial coordinate and leaves electron spin unchanged in the uncoupled approximation. In a coupled basis, angular coefficients carry the same underlying spin independence.

When nuclear spin is resolved, corresponding total-F selection rules apply: ΔF = 0, ±1 with F = 0 → F = 0 excluded for an electric-dipole transition. The state labels and coupling regime should always accompany a quoted rule. No general electric-dipole rule requires a particular Δn; the radial integral and energy conservation impose the remaining restrictions.

A useful transition explorer therefore asks for angular labels as well as principal numbers. Selecting only n can display a gross energy interval, but it cannot identify every component or polarization of a physical line.

4. Worked example: the 1S–2P dipole matrix element

Let a = ℏ/(μcα) be the reduced-mass Coulomb length. The normalized radial functions needed for this example are R₁₀ = 2a^−3/2 exp(−r/a) and R₂₁ = [1/(2√6)]a^−3/2(r/a) exp(−r/2a). A radial dipole matrix element includes an extra factor r in addition to the volume element r²dr:

The calculation reduces to ∫₀^∞ r⁴ exp(−βr)dr = 24/β⁵ with β = 3/(2a). The result is about 1.2903a and has dimensions of length, as required.

For the z-polarized component connecting 1S to 2P with m = 0, z = r cosθ and the angular integral is 1/√3. Consequently,

The corresponding dipole amplitude is −e times this result. By contrast, the angular integral between 1S and 2S contains an odd factor cosθ and vanishes. This pair of calculations explains an allowed and a forbidden electric-dipole channel without assigning an arbitrary brightness to either arrow on an energy diagram.

5. Read common hydrogen channels correctly

“Forbidden” always identifies an approximation and an operator. It does not mean that every possible process between the two states is impossible. Cascades, magnetic or electric higher multipoles, field mixing, collisions, and multiple-photon processes have their own amplitudes.

Conversely, an allowed angular channel need not dominate an observed spectrum. A weak radial overlap, an unpopulated initial state, or an unfavourable polarization can suppress its contribution. The spectral explorer separates the energy calculation from these additional physical choices.

6. Rates, lifetimes, and observed line strength

For an isolated electric-dipole emitter in free space, the spontaneous-emission rate into a specified lower-state channel has the structure

The vector norm includes the required components; sums over final sublevels and averages over prepared initial sublevels must follow the stated convention. The angular frequency is ω = ΔE/ℏ, not ΔE/h. Using ordinary frequency in the ω³ formula without the factors of 2π produces a large numerical error.

A state’s total spontaneous decay rate is the sum over its available channels. Its radiative lifetime is the reciprocal of that total rate within the isolated-atom model. Branching fractions are individual rates divided by the total. Observed brightness additionally depends on populations, pumping, propagation, and detection.

This is why a schematic animation should not use an arbitrary tick count as a physical lifetime. A temporal model could derive a rate from its interaction dynamics, but it must first define the radiation state space, density of final states, normalization, and coupling matrix elements. Those ingredients make the prediction reproducible.

7. Two photons can share one atomic energy difference

In a two-photon transition, the sum of the photon energies equals the atomic energy difference. Individual photon frequencies need not equal the full gap divided by h. For a degenerate two-photon excitation with equal photon frequencies, each photon supplies half the energy, so the required laser frequency is ΔE/(2h).

The leading amplitude contains a sum over intermediate states, including virtual P states for an S–S electric-dipole pair. Schematically, it contains products such as ⟨b|d·ε₂|k⟩⟨k|d·ε₁|a⟩ divided by an intermediate-state energy denominator, plus the term with the two photons exchanged. A complete calculation includes the appropriate discrete and continuum intermediate spectrum.

The intermediate state need not be populated as a real stationary step. This distinguishes a coherent two-photon process from a sequential cascade through an actual populated level. The different mechanisms have different spectral and timing properties.

For the 1S–2S pair, quoting both the full atomic interval and the excitation-photon frequency prevents a factor-of-two ambiguity. The same distinction matters whenever spectroscopy data are used to check a theoretical frequency formula or an outer-time clock reading.

8. The SMM/DTT transition-operator problem

A source model can be tested at the level of more than energies. It can ask whether its completed states carry the correct rotation and parity representations, whether a source-derived radiation coupling transforms as a vector in the long-wavelength limit, and whether the resulting matrix elements have the hydrogen radial dependence.

Paper 21 explicitly separates the state dictionary from the transition dictionary. This is a useful foundation: a record associated with an atomic state does not automatically specify the operator connecting it to another state or to radiation. [Paper 21, §28]

A first constructive result could derive parity and angular selection rules from the same source symmetries that organize the states. A stronger result would calculate one nonzero matrix element, such as the 1S–2P example, together with a vanishing amplitude protected by symmetry. Recovering both outcomes is more restrictive than reproducing a list of energy gaps.

This provides a focused collaboration problem for mathematical physics: construct the operator, prove its transformation law, compute its matrix elements, and explain the clock and normalization needed to convert the amplitudes into rates.

Continue reading

• The energy ladder and spectral series
• Explore intervals and allowed E1 channels
• Spin-resolved hydrogen states
• Collaborate on source-derived interactions