Sommerfeld's model asks what happens if the electron orbit is not forced to be circular. An ellipse has radial and angular motion, so the semi-classical rule must be applied twice: \(\oint p_\theta d\theta=n_\theta h\) and \(\oint p_r dr=n_r h\).
The geometry then connects the quantum labels to orbital shape through eccentricity. The energy remains \(E_n=-E_0/n^2\) in the nonrelativistic treatment, but the model introduces a key idea that survives in a different form: a state may need more than one quantum number.
Sommerfeld generalized Bohr's model by allowing elliptical orbits. In an ellipse, the electron has angular motion and radial motion, so one quantum condition is no longer enough.
The model is still orbital, but it introduces a lesson that survives: one state may require more than one quantum label.
The angular cycle and radial oscillation are quantized separately:
The circular Bohr orbit is recovered when the radial motion is absent, so \(n_r=0\). Elliptical orbits appear when radial motion contributes to the total principal quantum number.
For fixed \(n\), different values of the angular label correspond to different orbital shapes. More angular motion means a less elongated orbit; more radial motion means a more eccentric one.
This is not the modern picture of an electron path, but it anticipates the need for multiple quantum numbers in atomic states.
The nonrelativistic Sommerfeld energy has the same \(n\)-dependence as Bohr's model. The new value is conceptual: it shows that additional quantum labels can describe different orbital shapes at the same energy.
This page is designed to support short guided exercises on: A complementary bridge between Bohr-Sommerfeld quantization and geometric properties of elliptical motion.
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Use these anchors to design compact exercises. The exercise should be answerable from this page plus standard algebra, while longer derivations, full worked examples and broader context should point back to the original book.
- Focus: A complementary bridge between Bohr-Sommerfeld quantization and geometric properties of elliptical motion.
- Conceptual check: state what the main result says physically before using it algebraically.
- Equation: \[\oint p_\theta\,d\theta=n_\theta h,\qquad \oint p_r\,dr=n_r h\]
- Equation: \[n=n_r+n_\theta\]
- Equation: \[E_n=-\frac{E_0}{n^2}\]
- Boundary: use this page for setup and first-step reasoning; cite the book for longer derivations, complete experimental history or solved-problem detail.
- Typical task: derive, interpret, or apply the relation above to a simple case without introducing topics outside this page.