---
title: "The Finite Well and Wavefunction Penetration"
description: "When the walls are of finite height the wavefunction does not stop at the edge but decays into the classically forbidden region. The well then holds only a limited number of bound states."
canonical: https://lightmysky.com/learn/science/the-finite-well-and-wavefunction-penetration-mt_Vs4pWnXW7l
source: https://lightmysky.com/learn/science/the-finite-well-and-wavefunction-penetration-mt_Vs4pWnXW7l.md
retrieved: 2026-09-12
---

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# The Finite Well and Wavefunction Penetration

When the walls are of finite height the wavefunction does not stop at the edge but decays into the classically forbidden region. The well then holds only a limited number of bound states.

Subject: Science · Area: Quantum & Modern Physics · Ages 20 to 21
Page: https://lightmysky.com/learn/science/the-finite-well-and-wavefunction-penetration-mt_Vs4pWnXW7l

## Ready when they can

- Matches the wavefunction and its slope at the boundary of a finite well
- Describes the exponential decay in the forbidden region and what sets its length
- Explains why a finite well has a finite number of bound states

## Lesson: Leaky walls: the finite well and its fading tails

The infinite well is the warm-up act: perfectly trapped waves, clean nodes, and energies that climb with confinement. Lower the walls to a finite height and two things change. Bound states sink below the rim, and the waves no longer stop dead at the edge. The node counting and level ordering carry over with small shifts, so keep that picture and bend it.

**Example.** Zoom into one wall. Inside, the wave oscillates; outside, in the forbidden zone, it fades exponentially with distance and never oscillates. At the boundary both the wave and its slope must match, and that matching picks the allowed energies. The fade length is set by the gap between the particle energy and the wall height: a taller shortfall means a faster fade. This leakage is real, binding finite-well states slightly more loosely than infinite ones.

A finite well holds only a limited number of bound states. Each state needs its oscillating fit inside plus its tails, and higher states climb toward the rim. Once a state would rise past the rim, it stops being bound at all. So the ladder ends: fewer and lower levels than the infinite case, with the count set by how deep the well is.

**Tip.** Sketch any finite-well question the same way. Draw the well, mark the energy below the rim, draw oscillation inside and exponential tails outside. Count nodes to order the levels: ground state none, then one, then two. Check that every drawn state stays below the rim.

**Recap.** Finite walls leak exponential tails, boundary matching picks fewer and lower levels, and node counting still orders the ladder.

## Practice

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

## Needs first

- [The Infinite Square Well and Energy Quantisation](https://lightmysky.com/learn/science/the-infinite-square-well-and-energy-quantisation-mt_VV9ItoW7ew)

## Opens up

- [Quantum Tunnelling and the Transmission Probability](https://lightmysky.com/learn/science/quantum-tunnelling-and-the-transmission-probability-mt_tLdPJKx8Id)