Physics Hydrogen Ground State Rs

Hydrogen's ground state

The hydrogen atom's ground state is its lowest possible energy level, the state an electron occupies when it is bound to a proton with no energy to spare. In quantum mechanics, this energy is -13.6 electronvolts (eV), a negative number because the electron is trapped in the proton's electric field. The value follows from the Schrödinger equation, where the electron's mass, charge, and Planck's constant combine to fix the energy scale. A century of spectroscopy confirms the number to remarkable precision: the measured ionization energy of hydrogen is 13.598 eV, matching the theoretical prediction.

The classic derivation, due to Niels Bohr in 1913 and refined by Erwin Schrödinger in 1926, starts from the Coulomb force between proton and electron. The ground state energy equals -α²mₑc²/2, where α is the fine-structure constant (about 1/137), mₑ is the electron mass, and c is the speed of light. Plugging in values gives -13.6 eV. The formula shows how the energy scales: it depends on the square of the fine-structure constant, the dimensionless number that measures the strength of electromagnetic interaction. This scaling is what makes hydrogen's spectrum a sensitive test of quantum electrodynamics.

In Recognition Science, the framework models the same quantity through its cost function J(x) = (x + 1/x)/2 - 1, a function that arises from five plain conditions on how recognition events are recorded. The framework's library proves three general facts about this cost: it vanishes when its two inputs are equal, it stays nonnegative for positive inputs, and the golden-ratio threshold φ - 3/2 is positive. These are theorems about the cost function itself, not about hydrogen. The module named Hydrogen_Ground_State_RS defines the cost as J(m/e), where m and e are real numbers, and proves those three properties. It does not define what m and e mean for hydrogen.

The research note attached to the module records an intended direction: that α² might equal J(φ)² divided by some structural factor, which would connect the fine-structure constant to the golden ratio. That connection is not proved. The module proves only the three general cost properties, which hold for any positive inputs. A theorem about hydrogen itself would require defining m and e in terms of electron mass and charge, which the module does not do. The -13.6 eV value remains a prediction of quantum mechanics, not a result derived in this framework.

What the module does establish is a template: it shows how the cost function behaves when applied to a ratio, and it packages those properties into a reusable structure. The three theorems are universally quantified, meaning they hold for any positive real numbers, not just physical ones. This is a stepping stone, not a destination. A future module could define m and e for hydrogen and then apply these theorems to derive something specific. Until then, the -13.6 eV number stands on quantum mechanics, and the framework's contribution is a general mathematical fact about its cost function.

theorem domainCost_at_eq (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem domainCost_at_eq (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) : 0 ≤ domainCost m e := by
  unfold domainCost; exact Jcost_nonneg (div_pos hm he)

theorem cert_inhabited : Nonempty HydrogenGSRS := ⟨cert⟩

What this page does not claim

The module does not derive the -13.6 eV value from the cost function. The module does not define m and e in physical terms for hydrogen. The fine-structure constant is not derived from the golden ratio in this module.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Physics/Hydrogen_Ground_State_RS.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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