Quantum numbers and the hydrogen state space

What the labels describe

In the leading hydrogen model, n = 1, 2, 3, … labels the negative energy levels. The orbital label ℓ takes the values 0 through n−1, and m takes the integer values −ℓ through +ℓ. The corresponding orbital letters are s, p, d, f, and so on. These labels classify states and measurement outcomes; they are not three coordinates locating a tiny orbiting object.

The label m refers to one chosen quantization axis. Rotating that axis mixes the basis states within a fixed angular-momentum representation without changing the physics. An atomic state can also be a superposition of several such basis states.

Spin introduces additional labels and a larger Hilbert space. Relativistic and hyperfine descriptions use combinations such as total electronic angular momentum j and total atomic angular momentum F. Those extensions should be added through their operators, rather than folded into the scalar notation without explanation.

Paper 12 supplies the effective Coulomb spectral problem; Paper 11 identifies the emergence of its angular labels from temporal records as a further target. [Paper 12] [Paper 11, §9]

Separate the angular and radial problems

For H = −D∇² − C/r, the potential depends only on distance. The Hamiltonian therefore commutes with the usual angular momentum operators. Write ψ(r,θ,φ) = R(r)Y_ℓm(θ,φ), where the spherical harmonics satisfy:

Set u(r) = rR(r). Removing the spherical radial measure gives a one-dimensional equation on r > 0:

The first term in brackets is the centrifugal contribution associated with the angular dependence. It does not represent an additional force introduced independently of the kinetic operator. The radial boundary condition u(0) = 0 selects the regular Coulomb realization, and normalization requires ∫₀^∞|u|²dr to be finite.

These steps depend on the ordinary spatial measure and rotational representation. A temporal reconstruction must recover those structures or explicitly state a different comparison model.

Derive the principal integer from normalizability

For a bound energy E < 0, write E = −Dκ² with κ > 0. Near the origin the regular radial solution behaves as r^ℓ+1; at large distance the acceptable solution decays as e^−κr. Use these behaviors to set u = r^ℓ+1e^−κrv(ρ), with ρ = 2κr and a = 2D/C.

The regular solution must terminate as a polynomial to avoid an exponentially growing contribution after the prefactor is restored. Its degree is a nonnegative integer n_r, giving:

The principal integer is thus a consequence of the radial equation and its boundary conditions. The spectrum has not been inserted as a list of eigenvalues. Paper 12 uses this effective operator route to establish its conditional leading spectrum. [Paper 12, §6]

The continuum begins at zero, and the bound levels approach that threshold as n increases. Their accumulation is part of the Coulomb spectral structure.

Read the radial and angular shapes

The number n_r = n−ℓ−1 counts radial nodes at positive r for these bound radial states. The 1s state has none. The 2s state has one radial node, while the 2p states have no radial node and carry angular dependence.

Up to their normalization constants, representative spatial shapes are:

The 2s radial amplitude changes sign at r = 2a. The m = 0 2p amplitude changes sign across the plane cos θ = 0. The signs matter for interference and transition integrals, although the position probability is the squared magnitude.

A plotted surface of constant density does not show an electron boundary. It shows a chosen level of a probability function. The ground-state explorer starts with the simpler distinction between volume density, radial density, and cumulative probability. The same care should be used when adding higher-state visualizations.

Count the states and identify the extra symmetry

For a fixed ℓ there are 2ℓ+1 possible m values. Summing over the allowed angular sectors at fixed n gives:

Thus the scalar Coulomb level has n² orbital states. Adding an uncoupled spin-one-half factor doubles this count. Spin-dependent interactions later change how the states are grouped and how their energies split.

Rotational symmetry accounts for degeneracy among the m values at fixed ℓ. The equality of energies for different ℓ at the same n is more restrictive: it is special to the Coulomb problem and is associated with its additional conserved structure. A generic central potential preserves rotational symmetry while breaking this larger degeneracy.

This offers a demanding temporal reconstruction target. Producing three-dimensional rotations alone is not enough to explain the full n² pattern. Paper 12 explicitly includes the origin of the Coulomb hidden symmetry among its correspondence questions. [Paper 12, §19]

Distinguish a level interval from an allowed line

Two energies determine an interval hν = |E_a−E_b|. Whether a particular emission or absorption process can connect the states also depends on its interaction operator. In the ordinary electric-dipole approximation, that operator contains the position vector, and the transition amplitude involves a matrix element such as ⟨ψ_b|r|ψ_a⟩.

For the supplied orbital model, the angular integral gives Δℓ = ±1 and Δm = 0, ±1, subject to polarization. The electric dipole does not change spin in this approximation. The 2p to 1s transition can therefore be electric-dipole allowed, while the 2s to 1s electric-dipole matrix element vanishes. Other processes require their own interactions.

The leading interval between n = 2 and n = 1 is 3B/4. Between n = 3 and n = 2 it is 5B/36. Their frequency ratio is 5/27 when both are evaluated in this gross Coulomb approximation.

The spectral explorer identifies its approximation and distinguishes interval calculations from a full prediction of line strengths or lifetimes. Energy eigenvalues alone do not supply those quantities.

Worked example: the n = 3 family

For n = 3, the allowed angular sectors are ℓ = 0, 1, and 2. Their dimensions are 1, 3, and 5, respectively, giving nine orbital states with the same leading energy −B/9. An uncoupled electron-spin factor would give eighteen states.

The transitions to n = 2 share one gross energy interval, even though their participating wavefunctions and dipole matrix elements differ. Resolving the line structure requires the additional interactions discussed in fine structure and radiative corrections.

This example shows why an energy ladder is only one view of the atom. A state description also needs angular content, phases, and transition operators. A temporal model that assigns only one record to each energy would need to explain where the remaining state information resides.

Connect temporal construction to observable labels

The six inner generative levels and the seventh outer completion describe the architecture of the SMM/DTT proposal. They are not automatically the quantum numbers derived above. A correspondence should show how completed records support the same rotational action and spectral projectors as the effective hydrogen states.

For example, an embedding J could be required to preserve rotations: J U_source(R) = U_atom(R)J. A dynamical comparison would additionally connect the source update to the effective Hamiltonian through the same physical clock. The transition operator would need its own compatible map if selection rules are to be derived.

These requirements make an angular-label theorem specific. It should explain the multiplicities, allowed labels, and matrix elements, not merely assign familiar names to preselected records. Paper 11’s angular-emergence target is a natural place to develop that argument. [Paper 11, §9]

A successful construction would connect the temporal interpretation to the full observable state space, including information that the leading energy values alone cannot reveal.

Continue reading

• Energy intervals and the spectral series
• Explore the leading hydrogen intervals
• Add spin and relativistic structure
• Return to nested temporal generation