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Bohr Energy Levels
Level 1 - Physics topic page in Matter Waves.
Principle
The Bohr model explains hydrogen lines by quantizing electron angular momentum and energy.
Notation
\(n\)
principal quantum number
\(r_n\)
Bohr orbit radius
\(\mathrm{m}\)
\(E_n\)
hydrogen energy level
\(\mathrm{eV}\)
\(a_0\)
Bohr radius
\(\mathrm{m}\)
\(\hbar\)
reduced Planck constant
\(\mathrm{J\,s}\)
\(m_e\)
electron mass
\(\mathrm{kg}\)
Method
Derivation 1: Quantize angular momentum
The standing-wave condition around a circular orbit gives integer angular momentum.
Standing wave
\[2\pi r_n=n\lambda\]
de Broglie relation
\[\lambda=\frac{h}{m_ev}\]
Angular momentum
\[m_evr_n=n\hbar\]
Derivation 2: Use hydrogen level energies
Combining Coulomb attraction with angular-momentum quantization gives discrete radii and energies.
Radius levels
\[r_n=n^2a_0\]
Energy levels
\[E_n=-\frac{13.6\,\mathrm{eV}}{n^2}\]
Derivation 3: Calculate spectral lines
A photon energy equals the difference between two Bohr energy levels.
Rules
Quantized angular momentum
\[m_evr_n=n\hbar\]
Bohr radius levels
\[r_n=n^2a_0\]
Hydrogen energies
\[E_n=-\frac{13.6\,\mathrm{eV}}{n^2}\]
Transition photon
\[E_\gamma=E_i-E_f\]
Examples
Question
Find the energy of the
\[n=2\]
hydrogen level.Answer
\[E_2=-\frac{13.6}{2^2}=-3.40\,\mathrm{eV}\]
Checks
- Bound hydrogen levels have negative energy.
- Larger \(n\) means less negative energy and larger radius.
- Emission goes from higher \(n\) to lower \(n\).
- The Bohr model is useful for hydrogen-like atoms, not general multi-electron atoms.