Postulates of Bohr Theory

In 1913, Danish physicist Niels Bohr proposed a revolutionary model of the hydrogen atom that resolved a critical contradiction in classical physics: why do electrons orbiting a nucleus not continuously emit radiation and spiral inward? Classical electromagnetism predicted that any accelerating charge — including an electron moving in a circular orbit — must radiate energy, causing the atom to collapse within nanoseconds. Bohr's model introduced the radical idea that only certain orbits are allowed, and that electrons in those orbits do not radiate energy at all. Bohr constructed his model on four carefully chosen postulates, each of which broke with classical physics in a specific way. The most mathematically significant postulate is the quantization of angular momentum: the angular momentum of an orbiting electron can only take values that are integer multiples of h/2π (written ℏ, 'h-bar'). This single constraint, when combined with Coulomb's law and Newton's second law, allows us to derive exact expressions for the orbital radii, velocities, and energy levels of hydrogen — results that matched spectroscopic measurements with stunning precision. Bohr's postulates introduced the concept of stationary states — discrete energy levels in which an electron can exist indefinitely without radiating. Electrons only emit or absorb electromagnetic radiation when they transition between stationary states, releasing or absorbing a photon whose energy exactly equals the difference between the two energy levels. This explained the sharp spectral lines of hydrogen and gave birth to the field of quantum chemistry. Although Bohr's model was later superseded by full quantum mechanics, its postulates remain a crucial conceptual bridge between classical physics and the modern quantum world.

Key takeaways

Postulate 1 — Circular Orbits in Stationary States
Postulate 2 — Quantization of Angular Momentum
Postulate 3 — Quantum Jumps and Photon Emission/Absorption
Postulate 4 — The Ground State and Correspondence Principle

Postulate 1 — Circular Orbits in Stationary States

Electrons revolve around the nucleus in fixed circular orbits called stationary states. While in a stationary state, an electron does not emit or absorb energy, even though it is accelerating centripetally. This directly contradicts classical electrodynamics, which demands that all accelerating charges radiate. Bohr simply asserted this as a postulate, justified by the empirical stability of atoms.

Postulate 2 — Quantization of Angular Momentum

The angular momentum L of an electron in a permitted orbit is restricted to integer multiples of h/2π. Mathematically: mvr = nℏ = nh/(2π), where m is the electron mass, v its orbital speed, r the orbital radius, n is a positive integer (the principal quantum number, n = 1, 2, 3, …), and h is Planck's constant. This is the heart of Bohr's model. It means not every orbit is possible — only those for which the electron's de Broglie wave fits a whole number of wavelengths around the circumference (2πr = nλ), a fact later explained by de Broglie's wave-particle duality.

Postulate 3 — Quantum Jumps and Photon Emission/Absorption

An electron can transition from one stationary state to another. When it falls from a higher energy level E₂ to a lower level E₁, it emits a single photon of frequency ν given by Bohr's frequency rule: hν = E₂ − E₁. Conversely, an electron absorbs a photon of exactly that frequency to jump from E₁ to E₂. This explains why atomic emission spectra consist of sharp, discrete lines rather than a continuous spectrum — each line corresponds to one specific transition between two quantized energy levels.

Postulate 4 — The Ground State and Correspondence Principle

The lowest allowed orbit (n = 1) is called the ground state. The electron cannot lose further energy and does not spiral inward. For very large values of n (highly excited states), the predictions of Bohr's model must agree with classical physics — this is Bohr's own Correspondence Principle. As n → ∞, the energy levels become so closely spaced that quantization becomes imperceptible and classical behaviour is recovered.

The Key Equation

mvr = \frac{nh}{2\pi} = n\hbar

m — Electron mass (kg): Rest mass of the electron, 9.109 × 10⁻³¹ kg
v — Orbital speed (m s⁻¹): Tangential speed of the electron in its circular orbit
r — Orbital radius (m): Radius of the nth allowed circular orbit; r₁ (Bohr radius) = 0.529 Å for hydrogen
n — Principal quantum number (dimensionless): Positive integer (1, 2, 3, …) labelling the allowed orbit; n = 1 is the ground state
h — Planck's constant (J s): Fundamental constant of quantum mechanics, h = 6.626 × 10⁻³⁴ J s
\hbar — Reduced Planck constant (h-bar) (J s): h/(2π) = 1.055 × 10⁻³⁴ J s; frequently used in angular momentum equations

Worked Example

Problem. Calculate the radius of the first Bohr orbit (n = 1) for the hydrogen atom. Given: m = 9.109 × 10⁻³¹ kg, e = 1.602 × 10⁻¹⁹ C, ε₀ = 8.854 × 10⁻¹² F m⁻¹, h = 6.626 × 10⁻³⁴ J s.

r_n = \frac{n^2 h^2 \varepsilon_0}{\pi m e^2} = n^2 a_0, \quad a_0 = 5.29 \times 10^{-11}\text{ m}

Answer. The first Bohr orbit radius (Bohr radius) a₀ = 5.29 × 10⁻¹¹ m = 0.529 Å. For any orbit n, the radius scales as rₙ = n²a₀.

Figure

Diagram

Origins · 1 minNiels Bohr and how this was discovered

Niels Bohr

1885–1962 · Danish

Bohr proposed his quantum model of the hydrogen atom in 1913, introducing the concepts of stationary states and quantized angular momentum. His model successfully predicted the wavelengths of hydrogen's spectral lines (Balmer, Lyman, and Paschen series) and won him the Nobel Prize in Physics in 1922. He later founded the Niels Bohr Institute in Copenhagen, which became a leading centre of quantum mechanics research. Bohr was an avid football (soccer) player in his youth and was good enough that his brother Harald (a mathematician) played for the Danish national team and won a silver medal at the 1908 Olympics. Niels himself played as goalkeeper for a local club. According to legend, when Bohr gave a seminar on his new atomic model, his football reputation initially drew a larger audience than his physics work.

Hydrogen Spectrum and the Balmer Series — Bohr's model was specifically designed to explain the discrete spectral lines of hydrogen. The Balmer series (visible light, n → 2 transitions) was the empirical data against which Bohr tested his postulates.
The Rydberg Equation — Bohr's energy level formula directly derives the Rydberg constant R_H from fundamental constants (m, e, h, ε₀), explaining why the empirical Rydberg equation works.
de Broglie Wave-Particle Duality — de Broglie's hypothesis (λ = h/mv) provides a physical justification for Bohr's quantization rule: only orbits where an integer number of electron wavelengths fit around the circumference are allowed.
Quantum Numbers and Atomic Orbitals — Bohr's principal quantum number n evolved into the full set of quantum numbers (n, l, mₗ, mₛ) in Schrödinger's wave-mechanical model, which replaced Bohr's circular orbits with three-dimensional probability orbitals.
Photoelectric Effect and Planck's Constant — Planck's constant h appears in both Planck's radiation law and Einstein's photoelectric equation. Bohr borrowed E = hν to describe photon emission during electron transitions, linking his model to earlier quantum ideas.