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Ghostwall

Quantum Tunnelling Physics

\[T \approx e^{-2\kappa L}\]
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Can a particle pass through a wall it cannot climb? Ghostwall can, sometimes. In quantum mechanics a particle’s wavefunction leaks into a barrier, so it has a small chance of appearing beyond it, roughly T ≈ e^(−2κL). Tunnelling powers alpha decay, the Sun’s fusion and scanning tunnelling microscopes. Pixel-art black ceramic mug in 11 or 15 oz; equation, symbols and sources in the physics panel below.

Shipping calculated at checkout by destinationMade to order, ships in ~10 days

Details

Size & material11oz · 15oz

Black glossy ceramic mug with a C-handle. Lead- and BPA-free.

  • 11oz: 0.33 l
  • 15oz: 0.44 l
CareDishwasher & microwave safe

Dishwasher safe (top rack) or hand wash. Microwave safe.

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Made to order: about 10 days of production and handling before it ships.

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Returns & replacements21 days

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The physics

A quantum particle has a small chance of passing through a barrier it lacks the energy to climb. That chance falls off exponentially with the barrier's thickness and with how far the energy falls short of the top.

Ghostwall walks up to walls she cannot climb, and sometimes turns up on the other side.

In classical physics a ball without enough energy to get over a hill rolls back, every time. In quantum mechanics the particle's wavefunction does not stop dead at the barrier. It dies away exponentially inside, and if the barrier is thin enough, a little is left on the far side. The particle has a small chance of appearing there.

The chance falls off fast. For an electron one electronvolt short of the top of a barrier one nanometre thick, it is about 4 in 100,000. Double the thickness to two nanometres and it drops to about one in a billion. For a person and a brick wall it is zero for every practical purpose. WALLS ARE SUGGESTIONS, but only very thin ones.

Tunnelling is everywhere in physics. George Gamow, and separately Ronald Gurney and Edward Condon, explained alpha decay with it in 1928. It lets protons in the Sun's core fuse, and the scanning tunnelling microscope, which won Gerd Binnig and Heinrich Rohrer the 1986 Nobel Prize, images single atoms by it.

Ghostwall never breaks a wall. She just does not take it as final.

Equations

\[T \approx e^{-2\kappa L}\]
\[\kappa = \frac{\sqrt{2m(V_0 - E)}}{\hbar}\]
SymbolMeaningUnit
\(T\) probability of passing through the barrier (thick-barrier approximation) \(\mathrm{1}\)
\(\kappa\) decay constant of the wavefunction inside the barrier \(\mathrm{m⁻¹}\)
\(L\) barrier width \(\mathrm{m}\)
\(m\) particle mass \(\mathrm{kg}\)
\(V_0\) barrier height \(\mathrm{J}\)
\(E\) particle energy (less than V_0) \(\mathrm{J}\)
\(\hbar\) reduced Planck constant \(\mathrm{J·s}\)

Sources

  1. HyperPhysics — Barrier Penetration
  2. Gamow (1928) Zur Quantentheorie des Atomkernes, Z. Phys. 51, 204
  3. Gurney & Condon (1928) Wave Mechanics and Radioactive Disintegration, Nature 122, 439
  4. NobelPrize.org — The Nobel Prize in Physics 1986 (Binnig and Rohrer, scanning tunneling microscope)

Source Gamow, G. (1928). Zur Quantentheorie des Atomkernes. Zeitschrift für Physik 51, 204–212.doi.org/10.1007/BF01343196

Field notes

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