---
title: "The Finite Well and Wavefunction Penetration"
description: "In a finite square well, a bound-state wavefunction extends beyond each edge and decays in the constant exterior potential. Mass and the gap between exterior potential and state energy set its decay l"
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: 2026-11-12
---

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# The Finite Well and Wavefunction Penetration

In a finite square well, a bound-state wavefunction extends beyond each edge and decays in the constant exterior potential. Mass and the gap between exterior potential and state energy set its decay length. A well of finite width and depth has finitely many 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

- Identifies wavefunction value and slope as the quantities that must match at an ordinary finite potential step with constant particle mass
- Describes the exponential decay in the forbidden region and what sets its length
- Recognises that a square well of fixed finite width and depth has finitely many bound states for a particle of fixed finite mass

## Lesson: Bound states in a finite square well

Recall that an infinite well requires ψ=0 at each wall. Now use a square well of width L with U=0 inside and constant U=U0>0 outside on both sides, extending to infinity. Keep mass m constant across the edges. A bound state has 0<E<U0. Its wavefunction is nonzero at each edge and extends outside in a tail that decays away from the well, giving exterior probability density. A stationary bound state does not escape.

Inside, E>U gives oscillating solutions. Outside, E<U0 gives exponential solutions. A term growing toward either infinity cannot be normalised, so each exterior tail decays away from its edge. At an ordinary finite potential step with constant mass, ψ and its derivative dψ/dx must both match. These conditions select the allowed energies. There are no singular potential spikes in this model.

At outward distance d from an edge, ψ(d)=C exp(-κd), where κ=√[2m(U0-E)]/ℏ. The amplitude decay length is ℓ=1/κ: over ℓ, the magnitude falls by a factor of e, where e≈2.718. Increasing mass or U0-E, with the other fixed, shortens ℓ. Probability density is |C|² exp(-2κd). Its decay length is ℓ/2, and its value at d=ℓ is 1/e² times its edge value.

*(drawing: Samples of |ψ| divided by its edge magnitude, rounded to three decimals. These bars are not a drawing of the continuous wave.)*

A well of fixed finite width, depth and mass has finitely many bound states. Higher states need more interior oscillation, but E<U0 limits k=√(2mE)/ℏ. Only finitely many oscillating solutions can fit that width and match both tails. The count depends on width, depth and mass; it can be large. For E>U0, exterior solutions oscillate and the energies form an unbound continuum.

The ground state has no interior nodes. If a second bound state exists, it has one interior node and decaying tails on both sides. Each higher bound state adds one interior node. Compared with an infinite well of the same width, mass and interior potential, each finite-well bound level lies lower than its corresponding infinite-well level. The shifts need not be small. Use the same potential-energy reference for this comparison.

**Recap.** Bound-state waves match in value and slope at finite edges. Their exterior tails decay. Mass and U0-E set tail length; width, depth and mass affect the finite bound-state count.

*Lesson and questions written for this dataset. OpenStax University Physics Volume 3, sections 7.3, 7.4 and 7.6, were consulted for physics facts and boundary-matching methods.*

## 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)