Bohr Model Hydrogen Energy Levels Calculator mechanism

Model scope

The core Bohr-model level and radius formulas are standard under the stated hydrogen-like assumptions. Before implementation, define n1/n2 direction, use photon_energy=abs(E_final−E_initial), specify wavelength units and constants, validate n and Z, and restrict the claim to one-electron ions. The honest-label warning is appropriate and should be prominent.

• The transition relation ΔE=E_final−E_initial is signed, but photon energy must be E_photon=|ΔE|; for emission ΔE<0 and for absorption ΔE>0. The specification does not state this sign handling.
• The inputs n1 and n2 do not define which is initial and which is final, so emission versus absorption is ambiguous.
• The Bohr radius and energy formulas require positive integer n and positive integer Z, and the ion must be hydrogen-like (one electron); the specification does not state these constraints.
• The spectral-series name is determined by the lower principal quantum number of the transition (n=1 Lyman, n=2 Balmer, n=3 Paschen), not universally by n_final unless direction and naming convention are explicitly defined.
• The stated 13.6 eV value is rounded. If CODATA-consistent precision is intended, the Rydberg energy constant should be specified explicitly rather than mixed with a CODATA Bohr radius.

Units and conventions

• Use 13.6 eV exactly for the supplied energy formula, with energies reported in eV.
• Use a0=5.29177×10^-11 m for the Bohr radius.
• Use hc=1239.841984 eV·nm when calculating wavelengths in nm; convert wavelength to meters by multiplying nm by 10^-9.
• Interpret n1 as the initial principal quantum number and n2 as the final principal quantum number in the test vectors.
• Use Z as a positive integer atomic number for a one-electron hydrogen-like ion.

Inspectable vector

One of 4 source-bound review vectors:

{
  "inputs": {
    "Z": 1,
    "n1": 2,
    "n2": 1,
    "transition": "emission",
    "wavelength_units": "nm"
  },
  "expected_outputs": {
    "E_initial_eV": -3.4,
    "E_final_eV": -13.6,
    "radius_initial_m": 2.116708e-10,
    "radius_final_m": 5.29177e-11,
    "delta_E_eV": -10.2,
    "photon_energy_eV": 10.2,
    "wavelength_nm": 121.553135686,
    "wavelength_m": 1.21553135686e-7,
    "spectral_series": "Lyman"
  },
  "working": "E_initial=−13.6×1²/2²=−13.6/4=−3.4 eV. E_final=−13.6×1²/1²=−13.6 eV. r_initial=2²a0/1=4(5.29177×10^-11)=2.116708×10^-10 m. r_final=1²a0=5.29177×10^-11 m. ΔE=E_final−E_initial=−13.6−(−3.4)=−10.2 eV. Emitted photon energy=|ΔE|=10.2 eV. λ=hc/E=1239.841984/10.2=121.553135686 nm=1.21553135686×10^-7 m. The lower level is n=1, so this is Lyman."
}

Failure boundary

• Reject n=0, negative, noninteger, or otherwise invalid principal quantum numbers because n² appears in a denominator for energy.
• Reject Z≤0; require a positive integer Z.
• The formulas are physically applicable only to one-electron hydrogen-like ions, not neutral multi-electron atoms with the stated atomic number.
• A transition with n1=n2 gives ΔE=0 and causes division by zero when calculating wavelength; it must be rejected or reported as no photon.
• Wavelength calculations require an explicit energy unit and a matching hc constant; mixing joules with eV·nm produces incorrect results.
• Very large n or Z values may expose floating-point rounding; displayed precision should not imply precision beyond the constants used.
• The energy-level diagram should label the Bohr-model approximation and should not imply full quantum-mechanical accuracy, especially for multi-electron atoms, fine structure, Lamb shifts, reduced-mass corrections, or external-field effects.

Other records

Sources and evidence

These pointers bind the page to evidence. They do not grant authority or collapse verification into deployment.

• VERIFIED_LEDGER.json
• evidence-tranche6/tranche6-results.json