--- title: "Molecular Motion Quantised: The Oscillator and the Rotor" description: "Molecules store energy in vibration and rotation as well as in electrons. Two solvable models, a spring between two masses and a rigid dumbbell, give the level patterns that every molecular spectrum i" canonical: https://lightmysky.com/learn/science/molecular-motion-quantised-the-oscillator-and-the-rotor-mt_dHgi-i9pxc source: https://lightmysky.com/learn/science/molecular-motion-quantised-the-oscillator-and-the-rotor-mt_dHgi-i9pxc.md retrieved: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Molecular Motion Quantised: The Oscillator and the Rotor Molecules store energy in vibration and rotation as well as in electrons. Two solvable models, a spring between two masses and a rigid dumbbell, give the level patterns that every molecular spectrum is read against. Subject: Science · Area: Chemistry · Ages 19 to 20 Page: https://lightmysky.com/learn/science/molecular-motion-quantised-the-oscillator-and-the-rotor-mt_dHgi-i9pxc ## Ready when they can - Writes the vibrational level pattern as evenly spaced and the rotational pattern as widening with quantum number - Relates vibrational spacing to force constant and reduced mass, and predicts the shift on deuterium substitution - Relates rotational spacing to bond length through the moment of inertia - Explains what zero-point energy is and why a bond cannot be still even at absolute zero ## Lesson: Springs and dumbbells: how molecules move A chemical bond vibrates like a spring between two masses. Stiffer springs and lighter masses oscillate faster, with the frequency set by the square root of stiffness over mass. Quantum rules allow only evenly spaced levels, written E = (v + 1/2) h f. Each step up the ladder costs the same packet. **Example.** Swap hydrogen for deuterium in H-Cl and the band moves down. The force constant stays the same, but the reduced mass rises from about 0.97 to 1.89 atomic mass units. Frequency goes as one over the square root of reduced mass, so D-Cl vibrates at about the square root of 0.97 / 1.89, roughly 0.72 times the H-Cl value. A spinning molecule is a rigid dumbbell with levels E = B J(J + 1). Unlike vibration the gaps widen as J climbs, running 2B, then 4B, then 6B. Measure that spacing to get B, convert B to the moment of inertia I, and solve I = mu r squared for the bond length r. An even level ladder means vibration. A widening one means rotation, and since each gap grows by 2B, neighbouring rotational lines sit a steady 2B apart. Wider line spacing means larger B, smaller I, and a shorter bond for equal masses. The half quantum at v = 0 is the zero point energy, so a bond never sits still, even at absolute zero. **Recap.** Vibration climbs an even ladder set by stiffness and mass, while rotation spreads wider with J and reveals the bond length. ## Practice 8 questions on this page, each with its working shown. ## Needs first - [Many-Electron Atoms: Spin, Antisymmetry and Effective Charge](https://lightmysky.com/learn/science/many-electron-atoms-spin-antisymmetry-and-effective-charge-mt_WPesw2f8T3) ## Optional connections - [Taylor and Maclaurin Series](https://lightmysky.com/learn/mathematics/taylor-and-maclaurin-series-mt_SgI9Pn9RIO) ## Opens up - [The Boltzmann Distribution over Molecular Levels](https://lightmysky.com/learn/science/the-boltzmann-distribution-over-molecular-levels-mt_-DHIox6GBm) - [Selection Rules and Why Some Transitions Never Appear](https://lightmysky.com/learn/science/selection-rules-and-why-some-transitions-never-appear-mt_oJDr81W4fW)