---
title: "The Infinite Square Well and Energy Quantisation"
description: "A particle trapped between hard walls can only hold whole numbers of half wavelengths, so its energies form a discrete ladder that rises as the square of the quantum number. Quantisation comes from th"
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source: https://lightmysky.com/learn/science/the-infinite-square-well-and-energy-quantisation-mt_VV9ItoW7ew.md
retrieved: 2026-09-12
---

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# The Infinite Square Well and Energy Quantisation

A particle trapped between hard walls can only hold whole numbers of half wavelengths, so its energies form a discrete ladder that rises as the square of the quantum number. Quantisation comes from the boundary conditions, not from an extra rule.

Subject: Science · Area: Quantum & Modern Physics · Ages 19 to 20
Page: https://lightmysky.com/learn/science/the-infinite-square-well-and-energy-quantisation-mt_VV9ItoW7ew

## Ready when they can

- Solves the equation in the well and applies the boundary conditions to get the allowed energies
- Sketches the first few wavefunctions and their probability densities
- Predicts how the energy levels shift when the well is made narrower

## Lesson: A trapped wave can only sing some notes

Hard walls force psi to zero at both ends, so only whole half-wavelengths fit between them. Each fit is one allowed state with index n equals 1, 2, 3 and on. Energies climb as n squared, giving a ladder of 1, 4 and 9 times the ground energy. No extra rule is pasted on: the walls alone quantise the energies.

**Example.** A proton boxed in a nucleus of width 1.00e-14 m has a ground energy of 2.05 MeV and a first excited energy of 8.20 MeV, exactly four times higher. Dropping between them emits a 6.15 MeV photon that carries the difference away. The shapes go one hump, then two lobes with a central node, then three lobes with two nodes.

Squeeze the well narrower and every level rises while the gaps between levels grow. The n equals 1 state is the ground state, n equals 2 the first excited state, and so on. Each level's chances come from squaring its wave, and blends of levels are mixed-energy states rather than new rungs.

*(drawing: First three well shapes: humps grow and nodes appear between them.)*

**Recap.** Walls fit only whole half-waves, energies rise as n squared, and squeezing the box lifts every rung.

## Practice

8 questions on this page, each with its working shown.

## Needs first

- [The Time-Independent Schrödinger Equation](https://lightmysky.com/learn/science/the-time-independent-schrodinger-equation-mt_zeBVPery5T)

## Opens up

- [The Finite Well and Wavefunction Penetration](https://lightmysky.com/learn/science/the-finite-well-and-wavefunction-penetration-mt_Vs4pWnXW7l)