4 The hydrogen atom, general dimension

Hydrogen/, on top of Systems/Orthogonal/, mirroring Chapter 3’s shape exactly. The primary story: hidden symmetry \(\mathfrak {so}(n{+}1)\), any spatial dimension \(n\).

4.1 The concrete operators

Definition 41 Hydrogen \(\mathfrak {so}(n{+}1)\) data

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Ordinary angular momentum \(L\) together with a generalized Runge–Lenz vector \(A\), satisfying the \(\mathfrak {so}(n)\)/vector-transform/self-closure relations that promote \(\mathrm{Fin}(n{+}1)\to \mathrm{Fin}(n{+}1)\to \mathrm{End}_K(V)\) into a genuine SoSystem.

Theorem 42 The induced \(\mathfrak {so}(n{+}1)\) system

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HydrogenSoData packages exactly the data SoSystem (Definition 21) needs.

4.2 Schwartz space

Definition 43 Concrete angular momentum

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Physlib’s angular-momentum operators on Schwartz space \(\mathcal S(\mathbb R^d,\mathbb C)\).

Theorem 44 \(\mathfrak {so}(d)\) relations, proved

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physlibGen genuinely satisfies SoSystem’s exact structure constants.

Theorem 45 The Runge–Lenz vector transforms as a vector under rotations

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Closes a gap Physlib itself leaves @[sorryful]: proved independently here, reusing only Physlib’s own already-proved companion commutators.

Theorem 46 The bridge to HydrogenSoData

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Builds a HydrogenSoData from any operator family satisfying the angular-momentum relations and a scalar Runge–Lenz self-commutator hypothesis — the general constructor Theorem 48 specializes.

4.3 Spectrum: the one analytic admission

Theorem 47 Runge–Lenz self-commutator scalar — admitted

\([A(\varepsilon )_i,A(\varepsilon )_j]\)’s operator-valued coefficient (from Physlib’s lrl_commutation_lrl) collapses to the plain scalar \(-i/\hbar \) — true exactly on a genuine bound-state energy eigenspace, where the Hamiltonian and \(\mathbf r(\varepsilon )^{-3}\) each act by their eigenvalue/expectation value. The entire missing hydrogen spectral-theory front, converted into one precise, directly-admitted statement: Physlib itself lists the hydrogen point spectrum as an unstarted TODO.

Theorem 48 A genuine, concrete HydrogenSoData for the real Physlib atom

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Unconditional, given Theorem 47: zero further sorry.

4.4 Classification

Theorem 49 Hydrogen’s irreducibility, from concrete highest-weight data

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The literal shared-with-the-oscillator invocation: a direct specialization of Theorem 7 to HydrogenSoData’s \(\mathfrak {so}(n{+}1)\)-module — no new representation theory, only type-matching. Not yet fed the concrete root-datum/highest-weight-vector data (needs genuine spectral theory, per Theorem 47), so nothing downstream invokes it at a concrete instance yet.

4.5 Quantum numbers

Theorem 50 Irreducibility under the full hidden symmetry — admitted

For hydrogen there is no bigger dynamical group to descend from (unlike the oscillator’s \(\mathfrak {gl}(d)\), strictly bigger than the physical rotation group), so the entire level-\(N\) bound-state eigenspace being one \(\mathfrak {so}(n{+}1)\)-irrep is this fact. Blocked on the same missing type-\(B_r/D_r\) root-datum gap as Theorem 49.

Theorem 51 The Cartan family acts semisimply — admitted

Standard once \(\mathfrak {so}(n{+}1)\)’s Weyl/root-datum machinery exists, not otherwise attacked here. Mirrors Theorem 36 exactly.

Theorem 52 Weight decomposition

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\(V\) decomposes into simultaneous cartanGen-eigenspaces, via Theorem 4 applied directly to \(\mathfrak {so}(n{+}1)\)’s Cartan family — the same Foundations theorem Theorem 37 calls for \(\mathfrak {gl}(d)\).

Theorem 53 The full quantum-number decomposition

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At a fixed (implicit) principal quantum number \(N\): \(V\) is irreducible under the full hidden symmetry (Theorem 50) and splits into simultaneous eigenspaces of the concrete Cartan subalgebra, the magnetic numbers (Theorem 52).

4.6 Summary leaves

Theorem 54 Summary: a genuine HydrogenSoData for real Physlib hydrogen

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Hydrogen/Summary.lean item 4.

Theorem 55 Summary: irreducibility from cyclic highest-weight data

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Hydrogen/Summary.lean item 5 — the same Foundations theorem the oscillator’s WeylClassification.lean targets, \(\mathfrak {so}(n{+}1)\) in place of \(\mathfrak {gl}(d)\).

Theorem 56 Summary: the full quantum-number decomposition

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Hydrogen/Summary.lean item 6.

Theorem 57 Summary: the oscillator’s own instance — cross-system evidence

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Hydrogen/Summary.lean item 7: this leaf reaches directly into HarmonicOscillator/QuantumNumbers.lean, citing Theorem 37 from the other physical system’s file — concrete, checkable evidence the sharing claimed throughout this blueprint is real, not aspirational.