Bethe’s concise paper, “The Electromagnetic Shift of Energy Levels” delivered the first successful theoretical calculation of the Lamb shift, the energy difference between the hydrogen atom’s 2s_{1/2} and 2p_{1/2} states.
The paper tackles the problem of the electron’s self-energy arising from its interaction with the vacuum fluctuations (zero-point energy) of the quantized electromagnetic field. Standard perturbation theory yielded a linearly divergent result, proportional to an integral over virtual photon momentum k, ∫ k⁻¹ dk.
Bethe’s conceptual stroke was the realization of mass renormalization. The measured electron mass m already includes the divergent self-energy of the free electron. Therefore, the physical energy shift ΔE for a bound electron is the difference between the self-energy of the bound electron and the already-accounted-for self-energy of the free electron.
By calculating this difference using a non-relativistic approximation and imposing a high-frequency cutoff at k ≈ mc² / ħ, the divergent terms cancel. The remaining finite term depends logarithmically on the characteristic energy difference (ΔE_{avg}) between the states. This difference calculation yielded ΔE ≈ 1040 MHz, a value in close agreement with Willis Lamb’s experimental data.
This paper’s core achievement was demonstrating that divergences in Quantum Field Theory are manageable through a physical redefinition of parameters (renormalization), directly confirming the theory’s structure and paving the way for the full relativistic development of Quantum Electrodynamics by Feynman, Schwinger, and Tomonaga.
If you’d like a PDF of this paper for your own research or review, send me a direct message. [Bethe, H. A. (1947). The electromagnetic shift of energy levels. Physical Review, 72(4), 339–341.]