Hydrogen and temporal-model glossary

Use the terms in context

This glossary combines the vocabulary of the declared atomic model with the terms used in its proposed SMM/DTT interpretation. The definitions describe their role in this section; a complete physical correspondence is an additional research task. For the broader theory, use the main site glossary. [Hydrogen II] [Synthesis]

Definitions and related articles

Single Monad Model (SMM)

The proposed framework in which physical multiplicity is related to a common generative source. In this section, its atomic interpretation is connected to specified records, states and measurements rather than to a literal miniature object located inside the atom. Read the related explanation →

Duality of Time Theory (DTT)

The framework distinguishing inner generative activity from outer completed observation. A physical model must specify how those roles produce accessible quantities such as distance, energy and clock readings. Read the related explanation →

Inner generative level

A nested role within the proposed construction of a completed physical state. The six inner levels are not treated as six independent observable time axes. Their relation to atomic state labels requires an explicit correspondence. Read the related explanation →

Outer completion

The proposed transition to a retained, accessible observation or record. A completion rule must be distinguished from the stationary time evolution of an already specified quantum state. Read the related explanation →

Record

A retained distinction accessible under a specified class of operations or tests. A graph or collection of records acquires physical meaning only after its readouts and dynamics have been defined. Read the related explanation →

Charge neutrality

Cancellation of the total charge or the relevant charge character of two constituents. Neutrality constrains the system but does not alone establish an attractive interaction, binding energy or stable resonance. Read the related explanation →

Phase relation or phase locking

A relation between constituent or record phases under a specified dynamics. Opposite phases can be a structural condition; stability and a physical energy gap require additional arguments. Read the related explanation →

Hamiltonian

The operator specifying energy and generating time evolution in the effective quantum model. Its coefficients, domain and physical interpretation are part of the model, not consequences of its name. Read the related explanation →

Coulomb Hamiltonian

The nonrelativistic relative-coordinate operator H = −ℏ²∇²/(2μ) − αℏc/r for the point-nucleus hydrogen benchmark. Its gross spectrum omits spin, radiative and nuclear-structure corrections. Read the related explanation →

Reduced mass μ

The combination μ = mₑmₚ/(mₑ + mₚ) that appears in relative motion after separating the centre of mass. It is slightly smaller than the electron mass for ordinary hydrogen. Read the related explanation →

Relative coordinate

The vector from one constituent to the other, here r = rₑ − rₚ. Its conjugate momentum describes relative motion; the electron’s speed in the centre-of-mass frame has an additional mass factor. Read the related explanation →

Quantum state

The mathematical object used to calculate outcome probabilities. A pure spatial state in this benchmark is a square-integrable complex wavefunction; spin and other degrees of freedom enlarge the state space. Read the related explanation →

Wavefunction ψ

A complex amplitude whose squared modulus gives the position probability density for the specified normalized pure state. An amplitude, a density and a shell probability are different mathematical quantities. Read the related explanation →

Position probability density

The probability per unit volume, |ψ(r)|². For 1s it is largest at the origin, while the radial probability density includes a spherical-shell factor and peaks away from the origin. Read the related explanation →

Radial probability density

The probability per unit radial interval after angular integration. For the normalized 1s state, P(r) = 4r²exp(−2r/a)/a³. Its integral over all positive radii equals one. Read the related explanation →

Cumulative radial probability

The probability that the separation is at most r. For 1s and x = r/a, F(x) = 1 − exp(−2x)(1 + 2x + 2x²). It grows from zero to one. Read the related explanation →

Characteristic Coulomb length aμ

The scale aμ = ℏ/(μcα) appearing in the exponential ground state. It equals the most probable 1s radial separation but differs from the mean separation, which is 3aμ/2. Read the related explanation →

Bohr radius a₀

The standard constant ℏ/(mₑcα), using the electron mass. The relative-coordinate length aμ includes finite proton mass and is therefore slightly larger. Read the related explanation →

Ground state

The lowest-energy state of the specified Hamiltonian. Proving a global lower bound and finding a normalizable state that attains it establishes the ground state of the benchmark operator. Read the related explanation →

Binding energy

The energy required to reach the separated-constituent threshold from a bound state. With the threshold set to zero and a negative level energy, its magnitude is the positive binding energy. Read the related explanation →

Principal quantum number n

The positive integer labelling the gross Coulomb energy levels. Its allowed values arise from the radial boundary and normalizability conditions in the effective model. Read the related explanation →

Orbital angular momentum ℓ

The nonnegative integer specifying the angular part of a spatial hydrogen state, with 0 ≤ ℓ ≤ n−1. The familiar labels s, p, d and f correspond to ℓ = 0, 1, 2 and 3. Read the related explanation →

Magnetic quantum number m

The integer eigenvalue label for one component of orbital angular momentum, with −ℓ ≤ m ≤ ℓ. It is a state label relative to a chosen axis, not a new generative time level. Read the related explanation →

Spin and total angular momentum j

Electron spin is an additional quantum degree of freedom. Combining spin s = 1/2 with orbital angular momentum gives j = ℓ ± 1/2 when allowed. Deriving this representation from a temporal model is a separate task. Read the related explanation →

Degeneracy

The occurrence of several independent states with the same energy. The scalar Coulomb level has n² angular states; spin and additional interactions change the full state counting and level splitting. Read the related explanation →

Energy interval and transition

An energy interval is a difference between levels. A radiative transition also requires an interaction and a nonzero relevant matrix element; its intensity depends on further dynamics and preparation. Read the related explanation →

Electric-dipole transition (E1)

A leading one-photon transition generated by the electric-dipole operator. Its angular and parity selection rules restrict which state pairs can be connected in that approximation. Read the related explanation →

Fine-structure constant α

The dimensionless electromagnetic coupling used in the benchmark. Its value is an explicit reference input in the explorers. A structural counting result requires an additional matching calculation to become a prediction of this coupling. Read the related explanation →

Structural integer 137

The integer obtained in the specified reconstruction or counting construction. The count and its physical identification are separate claims; a measured coupling correction cannot be silently inserted into the structural theorem. Read the related explanation →

Root-mean-square speed

The square root of a mean squared speed. In the Coulomb eigenstate it is inferred from momentum variance. A stationary state can have nonzero RMS momentum while its probability current vanishes. Read the related explanation →

Fine structure

The relativistic and spin-dependent splitting of the gross hydrogen levels at the appropriate order in α. The displayed leading formulas have a declared approximation and do not contain the complete precision theory. Read the related explanation →

Lamb shift

Radiative and related contributions that lift degeneracies remaining in the ideal Dirac Coulomb spectrum. A complete calculation requires more than the dimensional scale of a proposed loop correction. Read the related explanation →

Hyperfine structure

Splitting associated with nuclear and electronic magnetic degrees of freedom. Its treatment requires nuclear spin and magnetic data in addition to the gross Coulomb inputs. Read the related explanation →

Recoil

Effects of the motion of the nucleus. Reduced mass supplies the leading nonrelativistic two-body kinematics; additional relativistic and radiative recoil terms belong to a more precise treatment. Read the related explanation →

Finite nuclear size

The modification caused by the spatial distribution of nuclear charge compared with a point nucleus. A leading short-range term predominantly affects S states; its coefficient and uncertainty must be distinguished from another contact-like mechanism. Read the related explanation →

Holonomy

The transformation accumulated by transport around a specified loop. Connecting a loop invariant to an atomic energy requires a defined coupling or operator, not just a numerical comparison of dimensionless quantities. Read the related explanation →

Residual

The difference between an observation and a declared comparison calculation. Omitted standard contributions, calibration choices and uncertainties must be assessed before interpreting a residual as new physics. Read the related explanation →

Identifiability

The ability to distinguish a proposed contribution from other allowed parameter changes or effects. If their response patterns coincide within the available observables, more precise values alone do not separate the mechanisms. Read the related explanation →

Calibration and independent prediction

Calibration uses data to fix an input or parameter. An independent prediction is assessed with data not used to make that choice. Both can be useful, but their evidential roles differ. Read the related explanation →

Controlled limit

A mathematical statement that a family of finite or regulated models approaches a specified limit in a defined sense, with suitable error control. A visually smooth diagram or a few stable numerical values is not by itself such a proof. Read the related explanation →