E.1.11 (HL)—Bohr angular momentum

Apply the angular-momentum condition

The Bohr model permits only integer values of $n=1,2,3,\ldots$. The electron's orbital angular momentum is therefore quantized rather than continuously variable.

Worked example — $n=4$

For $n=4$, $L=4h/(2\pi)=4(6.63\times10^{-34})/(2\pi)=4.22\times10^{-34}\,\mathrm{kg\,m^2\,s^{-1}}$. An intermediate value is not an allowed Bohr-orbit angular momentum.

Connect quantization to energy

Only selected radii, speeds and total energies are allowed. The electron cannot occupy an intermediate orbit energy in this model, which explains discrete atomic levels and line spectra.

Use ratios efficiently

For hydrogen, combining the quantization condition with the electrostatic circular-orbit model gives $r_n\propto n^2$ and $v_n\propto 1/n$. If $r_2/r_1=4$, then $v_2/v_1=1/2$.

Common trap

Do not say that quantization fixes only the radius. The condition restricts angular momentum and leads to discrete allowed energies; do not treat $mvr$ as an arbitrary continuous value.