---
title: "Quantum tunnelling and the transmission probability"
description: "A quantum particle can be transmitted through a finite barrier even when its energy is below the barrier top. For opaque rectangular barriers, transmission falls approximately exponentially with width"
canonical: https://lightmysky.com/learn/science/quantum-tunnelling-and-the-transmission-probability-mt_tLdPJKx8Id
source: https://lightmysky.com/learn/science/quantum-tunnelling-and-the-transmission-probability-mt_tLdPJKx8Id.md
retrieved: 2026-11-12
---

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# Quantum tunnelling and the transmission probability

A quantum particle can be transmitted through a finite barrier even when its energy is below the barrier top. For opaque rectangular barriers, transmission falls approximately exponentially with width. Mass and the gap between barrier height and particle energy set the attenuation scale. Alpha decay and scanning tunnelling microscopy involve this effect.

Subject: Science · Area: Quantum & Modern Physics · Ages 20 to 21
Page: https://lightmysky.com/learn/science/quantum-tunnelling-and-the-transmission-probability-mt_tLdPJKx8Id

## Ready when they can

- Relates rectangular-barrier transmission to width, particle mass and energy below the barrier top, with other quantities held fixed
- Explains alpha decay as escape through a Coulomb barrier and relates transmission to half-life in a stated escape model
- Explains how a sharp tip and gap-sensitive current give STM atomic-scale resolution, with electronic states also affecting contrast

## Lesson: Through the wall: tunnelling and its exponential rule

Recall the finite well: a stationary bound-state tail extends into U>E without escape. For transmission, a forbidden region has finite width and a classically allowed region lies beyond it, where U<E. An incident particle can have a transmitted outgoing wave there. Transmission probability T is transmitted particle flux divided by incident flux; flux measures particles crossing per unit time.

Take a static rectangular barrier of height U0 and width L, with U=0 outside, constant mass m and 0<E<U0. Define κ=√[2m(U0-E)]/ℏ; ℏ is reduced Planck's constant. For κL much greater than 1, T≈C exp(-2κL). C is a dimensionless boundary-matching factor independent of L. At fixed E, increasing width, height or mass reduces T when the other quantities stay fixed. Mass and U0-E enter κ through a square root.

**Example.** For width comparisons, use the illustrative model T(L)=exp(-aL), with fixed inverse length a>0. This sets C=1 instead of using the rectangular-barrier prefactor. Halving L takes the square root of T; doubling L squares T. If T(L)=0.0004, then T(L/2)=0.02. The size of the increase depends on aL.

*(drawing: Transmission (%): 2 at width L/2 and 0.04 at L, for the illustrative model. These bars compare probabilities, not barrier shapes.)*

An alpha particle contains two protons and two neutrons. In alpha decay, it can escape from a quasibound nuclear state: a state that can decay by emitting the particle. It tunnels through a surrounding Coulomb barrier even though its energy is below the barrier top. Alpha energy and barrier shape affect transmission. Differences in physical barrier width alone do not explain the range of nuclear half-lives.

**Example.** For small transmission, an alpha-escape model gives decay rate ≈ attempt rate × transmission per attempt. Half-life is inversely proportional to decay rate. Hold the attempt rate fixed: if transmission falls to one third, decay rate falls to one third and half-life becomes about three times as long. The bars show this rate change in relative units.

A scanning tunnelling microscope (STM) applies a voltage between a sharp tip and a conducting surface. Electrons tunnel across the vacuum gap. At fixed bias and local electronic states, current falls roughly exponentially as the gap widens. An atomically sharp tip can resolve atomic-scale variations. Current also depends on local electronic states, so an STM image is not a map of geometric height alone.

Elastic scattering from a time-independent potential conserves total energy E. Transmission through a barrier does not require a temporary energy loan. Inside the forbidden region, the wavefunction contains both growing and decaying exponential terms, fixed by matching at the two edges. Beyond the barrier, kinetic energy is E minus the local potential energy. Equal exterior potentials give equal incoming and outgoing kinetic energies.

**Recap.** Width, mass and energy shortfall affect transmission. Alpha decay involves a Coulomb barrier. STM measures gap-sensitive electron current. Static elastic tunnelling conserves total energy.

*Lesson, questions and flashcards written for this dataset. OpenStax University Physics Volume 3, sections 7.4 and 7.6, and College Physics, section 31.7, were consulted for physics facts. No textbook prose is reproduced.*

## Practice

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

## Needs first

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

## Opens up

- [The Quantum Harmonic Oscillator and Zero-Point Energy](https://lightmysky.com/learn/science/the-quantum-harmonic-oscillator-and-zero-point-energy-mt_HyDxmqyReX)
- [The Path Integral and the Sum Over Histories](https://lightmysky.com/learn/science/the-path-integral-and-the-sum-over-histories-mt_QRLheflB1O)