An atomic radius, a spectral frequency and a binding energy are different observables. A physical theory becomes especially informative when it relates them using a common set of constants. Those relations also show why reproducing one famous number is only one part of explaining the scales of matter.
Numbers and units
A length written as 1 metre is the same length written as 100 centimetres. A theory of nature should not select one of those numerical values as more physically fundamental merely because a chosen unit makes it convenient. Dimensionless ratios remove that particular freedom: the ratio of two lengths is unchanged when both units change together.
The fine-structure constant α is dimensionless. Its reciprocal is near 137, but is not exactly the integer 137. The 2022 CODATA adjustment gives approximately 137.035999177. Specifying the adjustment matters because recommended measured values come with uncertainties, correlations and a defined evidence base.
Within the SI, some dimensional constants have exact assigned numerical values because they define units. This does not mean that every experiment involving them is exact. Realizing a metre or a second still requires an apparatus and an uncertainty evaluation.
Three related atomic scales
For a nonrelativistic electron in a Coulomb field with an infinitely heavy nucleus of charge +e, useful characteristic scales are the Bohr radius a₀, the Rydberg energy ER and a time scale tA. Writing ℏ for h/(2π), their relations are
The approximate scales are 5.29 × 10⁻¹¹ metres, 13.6 electronvolts and 2.42 × 10⁻¹⁷ seconds. The time scale is ℏ divided by the Hartree energy; it is not a claim that the electron circles the nucleus on a literal classical orbit with that period.
These formulas have a defined domain. A real hydrogen atom requires finite nuclear mass, and precise spectroscopy requires further physical corrections. Recovering the leading scales is a necessary benchmark for many models, but it does not complete the spectroscopic test.
Change one input and follow the consequences
For a sensitivity exercise, increase α by 1% while keeping mₑ, c and ℏ fixed as parameters of this model. The radius is divided by 1.01, the binding-energy scale is multiplied by 1.01², and the time scale is divided by 1.01².
| Quantity | New / old | Approximate change |
|---|---|---|
| Radius a₀ | 1/1.01 | −0.9901% |
| Energy ER | 1.0201 | +2.01% |
| Time tA | 1/1.0201 | −1.9704% |
This is a conditional comparison of mathematical models, not evidence that α changes in nature. Its value is that one altered input affects several linked outputs. A proposed explanation should face those linked consequences rather than independently retuning each observable.
Dimensionless checks expose the structure
Define the reduced Compton length λ̄C = ℏ/(mₑc). Then a₀/λ̄C = 1/α and ER/(mₑc²) = α²/2 in the stated model. These ratios reveal the role of α without relying on whether lengths are reported in metres or nanometres.
A purely dimensional calculation can identify possible combinations of constants. It cannot generally select every numerical coefficient or justify a particular physical Hamiltonian. Conversely, a discrete mathematical count can be exact while its identification with a measured coupling remains an additional hypothesis.
Reading a constants proposal
The DTT Constants Explorer lets readers follow proposed dependencies and inspect open matching steps. Use it alongside the question: which quantities are independently supplied, which follow from equations, and which are identified with measured observables?
For the Single Monad programme, a persuasive advance would connect the inner generative mechanism to a physical scale or dimensionless relation without using the target value to choose the input. The resulting account should also determine what happens to related observables. The atomic-test essay develops that next step through transition ratios.
Sources and further reading
- P. Mohr, D. Newell, B. N. Taylor and E. Tiesinga, CODATA recommended values: 2022; published tables and discussion.
- Recognizing circular derivations; the constants research section.
- The duality-of-time postulate and book reader.