Bohr Model

Bohr's postulates

Niels Bohr (1913) proposed three radical ideas:

1. Stationary states. Electrons orbit in specific allowed circular orbits without radiating energy. Classical physics says they should — Bohr declared they don't.
2. Quantization of angular momentum. Allowed orbits have L = mvr = nℏ, where n = 1, 2, 3, ...
3. Photon emission. When electron jumps from orbit n_initial to n_final (lower), photon emitted with energy E = hf = E_initial − E_final.

Derived results for hydrogen

Combining quantization with classical mechanics for circular orbits:

Quantity	Formula	n=1 value
Orbital radius	r_n = n²·a_0	0.529 × 10⁻¹⁰ m
Bohr radius (constant)	a_0 = ε₀h²/(πm·e²)	0.529 × 10⁻¹⁰ m
Energy	E_n = -13.6/n² eV	-13.6 eV
Velocity	v_n = (e²/2ε₀h)/n	2.19 × 10⁶ m/s (n=1)
Period	T_n = 2π·a_0·n³ / v_1	1.5 × 10⁻¹⁶ s

Hydrogen emission spectrum

Photon emitted from transition n_i → n_f (i > f):

E_photon = E_i - E_f = -13.6 · (1/n_i² - 1/n_f²) eV (for emission)
        = 13.6 · (1/n_f² - 1/n_i²) eV (positive, emitted)

This is the Rydberg formula rewritten.

Series	n_f	Range
Lyman	1	UV
Balmer	2	Visible (656, 486, 434, 410 nm)
Paschen	3	Near IR
Brackett	4	IR
Pfund	5	IR

JavaScript — Bohr model calculations

// Bohr radius and energy levels
const a_0 = 5.29e-11;  // m
const E_1 = -13.6;     // eV (hydrogen ground state)

function bohrRadius(n, Z = 1) {
  return n * n * a_0 / Z;  // m
}

function bohrEnergy(n, Z = 1) {
  return E_1 * Z * Z / (n * n);  // eV
}

console.log(`H n=1: r = ${(bohrRadius(1) * 1e10).toFixed(2)} Å, E = ${bohrEnergy(1).toFixed(2)} eV`);
console.log(`H n=2: r = ${(bohrRadius(2) * 1e10).toFixed(2)} Å, E = ${bohrEnergy(2).toFixed(2)} eV`);
console.log(`H n=3: r = ${(bohrRadius(3) * 1e10).toFixed(2)} Å, E = ${bohrEnergy(3).toFixed(2)} eV`);

// Photon energy from transition
function transitionEnergy(n_i, n_f, Z = 1) {
  // Returns positive value (emitted photon)
  return Math.abs(bohrEnergy(n_i, Z) - bohrEnergy(n_f, Z));
}

// Photon wavelength from energy (E in eV → λ in nm)
function eVtoNm(E_eV) {
  return 1240 / E_eV;
}

// Balmer series (n_f = 2)
console.log('Balmer series:');
[3, 4, 5, 6].forEach(n_i => {
  const E = transitionEnergy(n_i, 2);
  console.log(`  ${n_i}→2: ${E.toFixed(2)} eV = ${eVtoNm(E).toFixed(0)} nm`);
});

// Hα (3→2) should be 656 nm — visible red
// H ionization energy
const E_ion_H = 13.6;
console.log(`H ionization: ${E_ion_H} eV`);

// Helium+ (Z=2)
console.log(`He+ n=1: ${bohrEnergy(1, 2)} eV`);  // -54.4

// Velocity in nth orbit
function bohrVelocity(n, Z = 1) {
  return 2.19e6 * Z / n;  // m/s
}

console.log(`H ground state v: ${bohrVelocity(1).toExponential(2)} m/s`);
// 0.7% of c — non-relativistic but close

Where Bohr model applies

• Hydrogen and hydrogen-like atoms. H, He⁺, Li²⁺ — single electron systems. Bohr formula works exactly (small relativistic and QED corrections).
• Education. Intuitive intro to atomic physics; pictures quantization, transitions, level diagrams.
• Astrophysics. Hydrogen lines (e.g., 21 cm H I, Lyman-α at 121 nm) used to map gas in galaxies.
• Spectroscopy basics. Most atomic spectra teaching starts with Bohr.
• Plasma physics. Hydrogen plasma analysis, Stark effect, Zeeman effect — Bohr framework as starting point.
• Quantum mechanics history. Bohr atom was bridge between classical and quantum — key historical step.
• Atomic clocks. Cesium clock uses transitions; Bohr-style energy level analysis (with full QM corrections).

Common mistakes

• Treating Bohr orbits as classical. Quantum mechanics replaces orbits with probability distributions (orbitals). Bohr model is a simplification.
• Applying to multi-electron atoms. Bohr fails for atoms with electron-electron interactions. Use Hartree-Fock or DFT for multi-electron systems.
• Forgetting Z dependence. Bohr formulas have Z² scaling. He⁺ has 4× the binding energy of H; Li²⁺ has 9×.
• Using only ground state. Excited states (n > 1) are equally important for transitions and excited-atom spectroscopy.
• Confusing Bohr radius with nuclear radius. Bohr radius (5.29e-11 m) = electron orbit. Nuclear radius (~10⁻¹⁵ m) is 100,000× smaller.
• Believing electrons orbit like planets. Bohr model has classical picture but quantum reality is fuzzy probability cloud. Modern picture: orbitals, not orbits.