--- title: "Quantum Tunnelling and the Transmission Probability" description: "A particle meeting a barrier it cannot classically cross has a small chance of appearing on the far side, falling off exponentially with barrier width and height. Alpha decay and the scanning tunnelli" 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: 2026-09-12 --- > **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`. # Quantum Tunnelling and the Transmission Probability A particle meeting a barrier it cannot classically cross has a small chance of appearing on the far side, falling off exponentially with barrier width and height. Alpha decay and the scanning tunnelling microscope both live on that exponential. 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 transmission probability to barrier width, height and particle mass - Explains alpha decay as tunnelling out of the nuclear potential - Says why the tunnelling microscope resolves single atoms ## Lesson: Through the wall: tunnelling and its exponential rule Tunnelling lets a particle cross a barrier it classically cannot climb. The wave fades exponentially inside the wall, yet a small transmitted wave emerges on the far side. The crossing chance drops exponentially with barrier width, height, and the particle mass: thin low barriers leak, while thick high ones barely do. Run the triple test on any barrier question first: wider means weaker, taller means weaker, heavier means weaker. **Example.** Alpha decay is tunnelling at work. An alpha particle sits trapped by the nuclear wall without the energy to climb over, yet it escapes by tunnelling out. Because the rate is exponential in width, a slightly thicker wall means a vastly longer half-life, which explains the huge observed range. The same exponential rule governs devices and decays alike. The scanning tunnelling microscope turns the exponential into vision. Electrons jump a vacuum gap between tip and surface, and the current swings wildly with tip distance, so scanning maps single atoms. Heavier particles tunnel far less readily than electrons under identical barriers, which is why the microscope uses electrons and not ions. **Tip.** Predict before you compute. Apply the triple test to rank barriers, and remember that a small width change multiplies the rate by orders of magnitude. Total energy stays the same throughout: the particle never owns extra energy inside the wall, it simply appears on the far side with unchanged E. **Recap.** Tunnelling falls exponentially with width, height, and mass, which explains alpha decay spreads and lets a microscope current feel single atoms. ## 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)