Bohr Model: Angular Momentum Quantisation and Photon Emission/Absorption

Angular Momentum Quantisation

Electron in circular orbit has angular momentum quantised in units of Planck's constant divided by 2π (ħ).
Core statement: L = n \hbar \quad \text{with} \quad n = 1,2,3, \dots
This implies only discrete circular orbits are allowed for the electron in the Bohr model.

For a hydrogen-like atom (Z = 1 for hydrogen), the angular momentum quantisation leads to a relation with velocity and radius:
- Angular momentum: L = m_e v r = n \hbar
- Centripetal force equals Coulomb attraction for a circular orbit: \frac{me v^2}{r} = \frac{1}{4\pi\epsilon0} \frac{Z e^2}{r^2} \quad (Z=1\text{ for H})
From these two relations, the orbit radius depends on n:
- rn = a0 n^2, \quad a0 = \frac{4\pi\epsilon0 \hbar^2}{m_e e^2} \approx 5.29 \times 10^{-11} \text{ m}
The energy of level n is:
- En = -\frac{me e^4}{32 \pi^2 \epsilon_0^2 \hbar^2} \frac{1}{n^2} = -\frac{13.6\ \text{eV}}{n^2}
Key consequence: energy becomes less negative (closer to zero) as n increases; higher n means higher (less bound) energy levels.

When the electron transitions between orbits, energy is exchanged with the electromagnetic field in the form of a photon.
Photon energy equals the absolute difference between the initial and final energy levels:
- E{\text{photon}} = h \nu = |E{nf} - E{n_i}|
- For hydrogen energy levels: E{\text{photon}} = 13.6\ \text{eV} \left|\frac{1}{nf^2} - \frac{1}{n_i^2}\right|
Emission vs absorption:
- If ni > nf , the atom emits a photon (transition to a lower energy level).
- If nf > ni , the atom absorbs a photon (promotion to a higher energy level).
Wavelength of the emitted or absorbed light can be expressed via the Rydberg form:
- \frac{1}{\lambda} = RH \left( \frac{1}{nf^2} - \frac{1}{n_i^2} \right)
- Where R_H \approx 1.097 \times 10^7 \text{ m}^{-1}
Worked example: transition from ni = 3 to nf = 2 (Balmer series, H-alpha line)
- Photon energy: E_{\text{photon}} = 13.6\ \text{eV} \left( \frac{1}{2^2} - \frac{1}{3^2} \right) \approx 1.89\ \text{eV}
- Wavelength: \lambda \approx \frac{hc}{E_{\text{photon}}} \approx \frac{(4.1357 \times 10^{-15} \text{ eV s})(3.0 \times 10^8 \text{ m/s})}{1.89\text{ eV}} \approx 656.3 \text{ nm}
- This corresponds to the H-alpha line in the Balmer series.
Specific notes:
- The Bohr model explains hydrogen line spectra but is a simplification; real atoms involve multi-electron interactions, spin, and fine structure.
- Selection rules and spin are more fully treated in quantum mechanics beyond the Bohr model.

The quantisation of angular momentum and discrete energy levels underpin the emission/absorption spectra of atoms.
This framework laid the groundwork for quantum mechanics and quantum electrodynamics.
Limitations of the Bohr model motivate more advanced treatments (Schrödinger equation, Dirac equation) for multi-electron atoms and relativistic corrections.