Bohr’s Postulates
By the early twentieth century, physicists faced a profound puzzle: classical electromagnetism predicted that an electron orbiting a nucleus should continuously radiate energy, spiralling inward and collapsing into the nucleus in less than a nanosecond. Yet atoms are stable, and they emit light only at discrete, sharply defined wavelengths. Something was fundamentally wrong with the classical picture. In 1913, Danish physicist Niels Bohr proposed a bold set of rules — now called Bohr's postulates — that broke from classical physics and explained the hydrogen spectrum with stunning precision. Bohr's key insight was that electrons do not occupy just any orbit; they are restricted to certain allowed orbits in which they do not radiate energy. These stable orbits correspond to specific, quantised values of angular momentum. Think of it like a staircase: an electron can stand on any step, but it cannot hover between steps. When an electron jumps from a higher step (higher energy orbit) to a lower one, it releases a photon whose energy equals exactly the difference between the two levels — which is why atoms emit light at specific colours rather than a continuous rainbow. Although Bohr's model was eventually superseded by full quantum mechanics, it remains one of the most important conceptual breakthroughs in physics. It introduced the idea of energy quantisation into atomic structure, correctly predicted the wavelengths of hydrogen's spectral lines, and established the framework of quantum numbers. Understanding Bohr's postulates is the essential gateway into modern atomic theory and quantum mechanics.
- Postulate 1 — Allowed Orbits (Stationary States)
- Postulate 2 — No Radiation in Stationary States
- Postulate 3 — Quantum Jumps and Photon Emission/Absorption
- Deriving the Bohr Radius
Postulate 1 — Allowed Orbits (Stationary States)
An electron in a hydrogen atom moves in a circular orbit around the nucleus under the Coulomb electrostatic force. However, only certain orbits are permitted: those in which the electron's angular momentum L is an integer multiple of the reduced Planck constant ħ = h/2π. Mathematically, L = mvr = nħ, where n = 1, 2, 3, … is called the principal quantum number. In these special orbits, called stationary states, the electron does not radiate energy — defying classical electromagnetism. The smallest allowed orbit (n = 1) is called the ground state; higher n values are excited states.
Postulate 2 — No Radiation in Stationary States
While an electron remains in a stationary orbit, it does not emit electromagnetic radiation, even though it is accelerating (centripetal acceleration). This directly contradicts classical electrodynamics, which demands that any accelerating charge radiates. Bohr simply asserted this as a rule — a necessary departure from classical physics. The total mechanical energy of the electron in orbit n can be calculated by combining the quantisation condition mvr = nħ with the condition that Coulomb attraction provides centripetal force, giving E_n = −13.6 eV / n². The negative sign means the electron is bound to the nucleus.
Postulate 3 — Quantum Jumps and Photon Emission/Absorption
An electron can transition between stationary states by absorbing or emitting a single photon. When it falls from a higher orbit (n_i) to a lower orbit (n_f < n_i), it emits a photon whose energy equals the energy difference: E_photon = E_i − E_f = 13.6 eV (1/n_f² − 1/n_i²). Since E = hf, this fixes the frequency — and therefore the wavelength — of the emitted light precisely. This explains why hydrogen emits light only at specific wavelengths: the Lyman series (transitions to n=1), Balmer series (to n=2, producing visible light), Paschen series (to n=3), and so on. The formula 1/λ = R_H(1/n_f² − 1/n_i²), where R_H = 1.097 × 10⁷ m⁻¹ is the Rydberg constant, matches experiment to extraordinary accuracy.
Deriving the Bohr Radius
Combining quantised angular momentum (mvr = nħ) with Newton's second law applied to circular motion (the Coulomb force provides centripetal force: ke²/r² = mv²/r), one can solve for the orbit radius. For the ground state (n = 1), this gives r₁ = a₀ = ħ²/(mₑke²) ≈ 0.529 × 10⁻¹⁰ m, known as the Bohr radius. It sets the natural length scale of the hydrogen atom. For orbit n, the radius scales as r_n = n²a₀, so the orbit spacing increases rapidly with n — the atom expands dramatically in excited states.
The Key Equation
- E_n — Energy of level n (eV (electron-volts) or J): Total mechanical energy of the electron in the nth orbit; negative because the electron is bound to the nucleus.
- 13.6 \text{ eV} — Ground state ionisation energy (eV): The energy required to remove an electron from the ground state (n = 1) of hydrogen to infinity. Equal to 2.18 × 10⁻¹⁸ J.
- n — Principal quantum number (dimensionless): A positive integer (1, 2, 3, …) labelling each allowed orbit. n = 1 is the ground state; larger n means higher energy and larger orbit.
Worked Example
Problem. An electron in a hydrogen atom drops from the n = 3 orbit to the n = 2 orbit. Calculate the wavelength of the photon emitted and identify which spectral series this transition belongs to.
Answer. The emitted photon has a wavelength of approximately 657 nm (red light), belonging to the Balmer series (H-alpha line).
Origins · 1 minNiels Bohr and how this was discovered
Bohr proposed his model of the hydrogen atom in 1913 while working under Ernest Rutherford in Manchester. By combining Rutherford's nuclear model with Planck's quantum hypothesis, he derived the quantised energy levels of hydrogen and explained the discrete spectral lines observed by Balmer and others. He won the Nobel Prize in Physics in 1922 for this work. Later, at his institute in Copenhagen, he became the intellectual centre of the developing quantum mechanics, championing the Copenhagen interpretation of quantum theory. Bohr was an enthusiastic footballer (soccer player) in his youth and his brother Harald was an Olympic silver medallist at the 1908 London Olympics — making the Bohr family uniquely distinguished in both science and sport.
Related Concepts
- Rutherford's Nuclear Model — Bohr built directly on Rutherford's discovery of the small, dense nucleus; understanding Rutherford's model shows exactly what problem Bohr was solving.
- Photons and Energy Quantisation — Bohr's third postulate relies on photon emission and absorption; E = hf connects energy level differences to observable spectral wavelengths.
- Hydrogen Emission Spectrum — The Balmer series and Rydberg formula are the experimental data that Bohr's model explains; seeing the actual spectrum reinforces why quantisation was necessary.
- De Broglie Wavelength — De Broglie later showed that Bohr's quantisation condition mvr = nħ arises naturally when an electron's matter wave forms a standing wave around its orbit — providing a physical reason for Bohr's postulate.