2s/2p energy split in the hydrogen emission spectrum, not predicted by the Dirac equation, but explained by quantum electrodynamics, which is one of the first great triumphs of that theory.
Note that for atoms with multiple electrons, 2s/2p shifts are expected: Why does 2s have less energy than 1s if they have the same principal quantum number?. The surprise was observing that on hydrogen which only has one electron.
Initial experiment: Lamb-Retherford experiment.
On the return from the train from the Shelter Island Conference in New York, Hans Bethe managed to do a non-relativistic calculation of the Lamb shift. He then published as The Electromagnetic Shift of Energy Levels by Hans Bethe (1947) which is still paywalled as of 2021, fuck me: journals.aps.org/pr/abstract/10.1103/PhysRev.72.339 by Physical Review.
The Electromagnetic Shift of Energy Levels Freeman Dyson (1948) published on Physical Review is apparently a relativistic analysis of the same: journals.aps.org/pr/abstract/10.1103/PhysRev.73.617 also paywalled as of 2021.
TODO how do the infinities show up, and how did people solve them?
Lamb shift by Dr. Nissar Ahmad (2020)
Source. Whiteboard Lecture about the phenomena, includes description of the experiment. Seems quite good.Murray Gell-Mann - The race to calculate the relativistic Lamb shift by Web of Stories (1997)
Source. Quick historical overview. Mentions that Richard Feynman and Julian Schwinger were using mass renormalization and cancellation if infinities. He says that French and Weisskopf actually managed to do the correct calculations first with a less elegant method.www.mdpi.com/2624-8174/2/2/8/pdf History and Some Aspects of the Lamb Shift by G. Jordan Maclay (2019)
Freeman Dyson - The Lamb shift by Web of Stories (1998)
Source. Mentions that he moved to the USA from the United Kingdom specifically because great experiments were being carried at Columbia University, which is where the Lamb-Retherford experiment was done, and that Isidor Isaac Rabi was the head at the time.
He then explains mass renormalization briefly: instead of calculating from scratch, you just compare the raw electron to the bound electron and take the difference. Both of those have infinities in them, but the difference between them cancels out those infinities.
Hans Bethe - The Lamb shift (1996)
Source. A way to write the wavefunction such that the position operator is:i.e., a function that takes the wavefunction as input, and outputs another function:
If you believe that mathematicians took care of continuous spectrum for us and that everything just works, the most concrete and direct thing that this representation tells us is that:equals:
the probability of finding a particle between and at time
TODO are there experiments, or just theoretical?
How many logical qubits are needed to run Shor's algorithm? Updated 2025-07-14 +Created 1970-01-01
Integrations React integration:
- github.com/shakacode/react_on_rails: webpack and server-side rendering
- github.com/reactjs/react-rails Official on the React side only. Demo app linked from package: github.com/BookOfGreg/react-rails-example-app and how it fails: github.com/BookOfGreg/react-rails-example-app/issues/30... The related projects section has some good links:
- shakacode/react_on_rails
- github.com/hyperstack-org/hyperstack transpiles Ruby to JavaScript + React. What could possibly go wrong? :-)
Selected Papers on Quantum Electrodynamics by Julian Schwinger (1958) Updated 2025-07-14 +Created 1970-01-01
Time-independent Schrödinger equation for a free one dimensional particle Updated 2025-07-14 +Created 1970-01-01
so the solution is:We notice that the solution has continuous spectrum, since any value of can provide a solution.
There are unlisted articles, also show them or only show them.