1s 2s TransitionEdit

The 1s 2s transition is a cornerstone phenomenon in the quantum description of the hydrogen atom. It describes the connection between the ground state, denoted 1s, and the first excited singlet state, denoted 2s. Because these two states have the same orbital angular momentum quantum number (l = 0 for both), a direct single-photon electric-dipole transition between them is forbidden by standard selection rules. As a result, the transition proceeds primarily through a higher-order process that reveals both the elegance and the limits of simple atomic models: a two-photon transition where the atom simultaneously absorbs or emits two photons to conserve energy and angular momentum. This peculiar pathway has made the 1s 2s transition one of the most precisely studied processes in experimental and theoretical atomic physics, with implications for fundamental constants and tests of quantum electrodynamics (QED).

The energy difference between the 1s and 2s levels in hydrogen is 10.2 eV. If a single photon carried that energy, its wavelength would be about 121.6 nm, placing it in the ultraviolet Lyman region. However, because a one-photon E1 (electric dipole) transition is forbidden for 1s ↔ 2s, the observable signature is not a sharp single-photon line but a two-photon process in which two photons, each with approximately half the energy, are absorbed or emitted. In practice, two-photon excitation commonly uses photons near 243 nm (each photon around 5.1 eV) to promote an atom from 1s to 2s. The net effect is an energy change of 10.2 eV, but distributed over two photons, with angular momentum and parity conservation satisfied by the two-photon mechanism.

Overview

The 1s 2s transition sits at the intersection of simple hydrogen theory and high-precision tests of QED. The two-photon pathway is allowed by higher-order electromagnetic interactions, in contrast to the forbidden single-photon dipole channel.
The 2s state is metastable in free hydrogen, with a lifetime long enough to permit detailed spectroscopic interrogation. The dominant decay from 2s to 1s is a two-photon emission process, with a rate that gives a lifetime of about 0.12 seconds in vacuum. This long lifetime enables narrow-linewidth measurements and critical tests of fundamental physics.
Measurements of the 1s 2s transition frequency have driven advances in laser technology, frequency combs, and metrology, and they provide stringent checks on QED corrections, the Rydberg constant, and the proton charge radius when compared with other spectroscopic data.

Quantum mechanical framework

The hydrogen atom is a textbook realization of the Schrödinger equation in a Coulomb field. Its energy levels, to first approximation, follow E_n = -13.6 eV / n^2, with n the principal quantum number. The ground state is 1s (n = 1, l = 0) and the first excited state is 2s (n = 2, l = 0).
Selection rules for electric-dipole transitions ordinarily require Δl = ±1 and a change in parity. The 1s → 2s transition violates these rules for a single photon, rendering the E1 channel forbidden. Higher-order processes, such as two-photon absorption or emission, are allowed and dominate the observable transition between these two levels.
In practice, the 1s 2s transition probes not only the nonrelativistic hydrogen model but also relativistic corrections, radiative QED corrections (including the Lamb shift), and finite-size effects. The precise measurement of the 1s 2s energy separation thus acts as a sensitive test bed for the accuracy of QED in bound systems.

Spectroscopy and transition rates

Two-photon processes connect 1s and 2s with a rate that competes with, but is distinct from, any forbidden single-photon channel. The 2s state decays to 1s primarily via two-photon emission, a process with a total decay rate A_2s1s ≈ 8 s^-1, giving the characteristic lifetime of roughly 0.12 seconds.
The two-photon excitation route has been exploited with high-resolution laser systems, including frequency combs, to measure the 1s 2s transition frequency with extraordinary precision. The result is a precise determination of the Rydberg constant and stringent tests of bound-state QED.
Spectroscopic lines associated with the 1s 2s transition are broadened by natural linewidth, Doppler effects, and experimental conditions, but the intrinsic two-photon linewidth can be extremely narrow, reflecting the metastability of the 2s state and the weak coupling to the electromagnetic field in the two-photon channel.

Experimental realization

Experimental approaches routinely employ pulsed or continuous-wave ultraviolet light near 243 nm to induce two-photon transitions from 1s to 2s in atomic hydrogen or deuterium, often within a gas cell, beam apparatus, or trap where environmental perturbations are minimized.
Laser stabilization, frequency comb metrology, and precise control of systematic effects (such as ac Stark shifts, Zeeman shifts, and background gas collisions) enable measurements of the transition frequency with relative uncertainties reaching parts in 10^15 or better in modern experiments.
The results from these experiments feed into the determination of fundamental constants, including the R∞ (Rydberg constant) and tests of QED corrections across bound states. They also inform comparisons with measurements in other simple systems and with predictions from advanced atomic theory.

History and context

The study of the 1s 2s transition has a long history in atomic physics as a testbed for quantum mechanics and QED. Initial insights into the forbidden nature of the 1s ↔ 2s single-photon transition motivated the development of two-photon spectroscopy techniques, which in turn enabled precision measurements that constrained theoretical models.
The refined agreement between experiment and QED predictions for the 1s 2s transition stands as a milestone in the validation of quantum electrodynamics in bound systems, paralleling other precision tests in atomic, molecular, and optical physics.

Implications and applications

Precision spectroscopy of the 1s 2s transition informs the determination of fundamental constants, most notably the Rydberg constant, and provides stringent tests of bound-state QED, including relativistic and radiative corrections.
The methods developed to observe and measure the 1s 2s transition — especially two-photon excitation and frequency comb metrology — have wide applicability in optical clock development, high-resolution spectroscopy of other atoms and ions, and advances in quantum metrology.
Beyond hydrogen itself, the principles illustrated by the 1s 2s transition illuminate how selection rules shape the pathways available for quantum transitions, how metastable states can be leveraged for precision measurements, and how higher-order processes reveal the richness of light–matter interactions.