Physics HL · Chapter 21: Atomic Physics
21.4 Quantisation of Angular Momentum in the Bohr Model
Derive allowed orbit radii and energies by combining Coulomb force with Bohr's angular-momentum quantisation postulate.
Estimated time: 42 minutes
The Classical Stability Problem
A purely classical orbiting electron is accelerating centripetally, so classical electrodynamics predicts continuous radiation and energy loss. The orbit would collapse rapidly into the nucleus, contradicting stable matter. Bohr's bold move was to postulate special non-radiating orbits selected by a quantisation rule rather than by continuous mechanics alone.
Bohr Postulate and Allowed Angular Momentum
Angular momentum is restricted to integer multiples of hbar, excluding intermediate values.
Combining mvr = n hbar with Coulomb-force centripetal balance gives discrete orbit radii and energies. The derivation is algebra-heavy but conceptually simple: one continuous equation from force balance and one quantisation condition together force a discrete state ladder.
For hydrogen-like ions of nuclear charge Z, radius scales as n^2/Z and energy scales as -Z^2/n^2. This means stronger nuclear charge shrinks orbits and deepens binding, while higher n expands orbit size and moves energy toward zero.
Allowed Radius and Energy Relations
a_0 is the Bohr radius for hydrogen ground state.
These formulas are not separate facts; they are linked by the same quantisation rule. Radius tells you geometric scale, energy tells you spectral scale, and both depend on the same quantum number. That linkage is why Bohr could explain hydrogen spectra with surprising accuracy for such a compact model.
Simulation: Bohr Orbit Quantisation Workspace
Vary n and Z to inspect allowed orbit radius, angular momentum, orbital period, and bound-state energy in one view.
Explore how atomic structure evidence, quantised levels, and spectral lines connect to one another.
Orbit radius
4.763 x10^-10 m
Angular momentum
3.16e-34 J s
Kinetic / potential
1.51 eV / -3.02 eV
Total energy
-1.511 eV
Model notes: Rutherford mode compares angular deflection scaling for concentrated vs spread positive charge; transitions and Bohr modes use hydrogen-like one-electron formulas; spectra mode emphasizes line-position matching between emission and absorption.
Test Yourself
For hydrogen (Z = 1), enter the orbit radius for n = 2 using a0 = 5.29 x 10^-11 m.
Hint: Use r_n = a0 n^2.