Angular momentum quantization allows only certain electron orbits in the Bohr model. By imposing , the model produces discrete orbital radii and therefore discrete energy levels for hydrogen.
Angular momentum quantization allows only certain electron orbits in the Bohr model. By imposing , the model produces discrete orbital radii and therefore discrete energy levels for hydrogen.
For an electron of mass moving in a circular orbit around a proton, the electrostatic attraction supplies the centripetal force:
Classical physics alone permits any radius and energy. Bohr instead proposed angular momentum quantization:
where . Combining the force equation with the quantization condition gives
Thus, the electron cannot occupy an arbitrary circular orbit: only radii proportional to are allowed. The smallest radius, at , is the Bohr radius.
A later wave interpretation uses the electron's de Broglie wavelength. A stable Bohr orbit fits an integer number of wavelengths around its circumference: . Since , this rearranges to . Therefore, other radii are excluded by destructive self-interference, while allowed states correspond to standing matter waves. For example, has four times the radius of and energy , whereas ionization raises the electron from a negative bound energy to at infinite separation.
The electron's total energy is the sum of kinetic and electric potential energy:
Substituting gives
The negative sign means the electron is bound to the nucleus. When it moves between allowed levels, it absorbs or emits a photon satisfying , explaining hydrogen's discrete line spectrum.
A common misconception is that quantization explains why electrons literally travel in fixed circular paths. It does not: the Bohr orbit is a limited model. Modern quantum mechanics describes electrons using wavefunctions and orbitals, but the Bohr model correctly predicts hydrogen's energy levels.
This is Additional Higher Level (AHL) content in E.1 Structure of the atom. For a Show question, write the electrostatic-force equation and , combine them to obtain allowed radii, then connect to . Do not state only the final formula.