--- title: "The Particle in a Box: Where Quantised Levels Come From" description: "The simplest bound system shows that quantisation is not an extra assumption. Fitting a wave into a fixed length forces whole numbers of half-waves, and the allowed energies follow from that alone." canonical: https://lightmysky.com/learn/science/the-particle-in-a-box-where-quantised-levels-come-from-mt_rOCXlEi324 source: https://lightmysky.com/learn/science/the-particle-in-a-box-where-quantised-levels-come-from-mt_rOCXlEi324.md retrieved: 2026-09-12 --- > **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`. # The Particle in a Box: Where Quantised Levels Come From The simplest bound system shows that quantisation is not an extra assumption. Fitting a wave into a fixed length forces whole numbers of half-waves, and the allowed energies follow from that alone. Subject: Science · Area: Chemistry · Ages 18 to 19 Page: https://lightmysky.com/learn/science/the-particle-in-a-box-where-quantised-levels-come-from-mt_rOCXlEi324 ## Ready when they can - Applies the boundary conditions at the walls and shows why they select integer quantum numbers - Writes the energy expression and predicts how the spacing changes when the box is made longer or the particle heavier - Counts nodes for a given level and links node count to energy - Estimates an absorption wavelength for a conjugated chain treated as a box, and says why the estimate is rough ## Lesson: Trapped waves can only sing certain notes Picture an electron stuck between two walls it cannot cross, like the electrons that give a dye its colour. Its wave must fall to zero exactly at each wall, like a string pinned at both ends. Only the allowed waves from the last stop survive here, the ones that fit a whole number of half waves between the walls. Each survivor gets a label n = 1, 2, 3, and up. Zero is not allowed: an n = 0 wave would be flat zero everywhere, with no chance of finding the electron anywhere. Each allowed wave has its own energy, and the energies grow with n squared. Level 2 sits at 4 times the ground energy, and level 3 at 9 times. A longer box or a heavier particle squeezes all the gaps smaller. Notice the gaps grow as you climb, which is the fingerprint of this trap. *(drawing: Higher levels carry more nodes. Level 1 has none, level 2 has one, level 3 has two.)* **Example.** Count the nodes to read a level at a glance. Level n has n minus 1 interior nodes, and more nodes always mean higher energy. A dye chain works the same way: take the chain length as the box, and the jump between the highest filled level and the next empty one as the absorbed light. A longer chain gives a smaller jump and hence longer wavelength light, but uneven spacing and repulsion keep the estimate rough. **Tip.** When you meet a new trap, look at the spacing pattern first. Evenly spaced levels mean a spring-like trap, while widening gaps mean box-like walls. Spectra are fingerprints of the forces inside, so let the gaps tell you the trap shape. **Recap.** Walls pin the wave to zero, whole half waves pick the levels, and the gaps grow with n. ## Practice 8 questions on this page, each with its working shown. ## Needs first - [Superposition and Stationary Waves](https://lightmysky.com/learn/science/superposition-and-stationary-waves-mt_AIxA2TYGJN) - [The Schrodinger Equation and What a Wavefunction Says](https://lightmysky.com/learn/science/the-schrodinger-equation-and-what-a-wavefunction-says-mt_WCwbaO9rpR) ## Opens up - [The Hydrogen Atom: Quantum Numbers and Orbital Shapes](https://lightmysky.com/learn/science/the-hydrogen-atom-quantum-numbers-and-orbital-shapes-mt_zt9_eR1884)