Week 8: How do two nuclei that can't touch, touch?
The Sun's core is hot, but not remotely hot enough to force two protons past their mutual repulsion. Quantum mechanics does not get them there by giving them more energy. It gets them there by not requiring a sharp edge at the wall.
28 September 2026
Last week ended with a wall. The Sun burns hydrogen into helium, and that only works if hydrogen nuclei — bare protons — get close enough for the strong nuclear force to grab them. Protons carry the same electric charge. Push two of them together and they push back, harder the closer they get.
Put a number on how hard.
The Sun’s core sits at around (1.5 \times 10^7) kelvin. A proton’s typical thermal energy there is on the order of
$$ k_BT \sim 1\ \text{keV}. $$
Getting two protons close enough to touch — within about a femtometre of each other, where the strong force can finally act — means climbing an electrical hill worth roughly
$$ \text{barrier height} \sim 1\ \text{MeV}. $$
A million electronvolts against a thousand. The core is supplying something like a thousandth of what the wall demands, and that gap does not close by looking at the fastest protons in the crowd instead of the average one. Nothing in the Sun’s thermal population is getting over that wall.
Classically, this reaction should not happen. Not rarely — essentially never, for as long as protons only get past barriers by having more energy than the barrier is tall.
The Sun is shining anyway.
What tunnelling actually is#
Quantum mechanics does not describe a proton by a single position tracing a single path toward the barrier. It describes it by a quantum state rather than by one definite classical position and trajectory. Solve for how that state behaves when it meets a barrier taller than the proton’s energy, and it does not stop dead at the edge, the way a rolled ball would. The state’s amplitude falls away sharply inside the barrier, but it does not fall to zero. If the barrier is thin enough relative to how fast that amplitude decays, some of the state is still there when the barrier ends.
That is what tunnelling is: a quantum state extending into a region a classical trajectory could never reach, coming out the other side with a small but real amplitude — which means a small but real probability of finding the proton on the far side, having never had enough energy to be there.
Here is what tunnelling is not. The proton does not borrow energy from somewhere to clear the wall and quietly pay it back afterward. Its energy is what it always was, below the barrier, the whole time — nothing is borrowed, and nothing needs to be repaid. It is tempting to reach for the energy–time uncertainty relation here, as though it hands out short-term loans against energy conservation. It does not. That relation limits how precisely energy and duration can be pinned down together; it is not a licence for a system to spend energy it does not have. Energy conservation is not suspended for the length of the tunnelling event. What changes is simpler and stranger than a loan: the theory never required the proton’s energy to exceed the barrier in order to find it on the other side of it.
For any single pair of protons, the odds of this happening on a given approach are tiny — the taller and wider the barrier is compared to the proton’s energy, the smaller those odds get, and here they get very small indeed. But the Sun is not offering a single pair a single attempt. It has been running this experiment with something like (10^{57}) protons for billions of years. A vanishingly small probability, multiplied by that many attempts, still adds up to a star.
So the barrier is real, the classical verdict is correct, and fusion happens anyway, because the proton was never obliged to behave like a tiny hard ball following a definite path toward it.
Which is itself the harder question. If a proton’s state can extend through a wall its trajectory could never cross, and detecting it always turns up one proton in one place — not a smear, not a fraction here and a fraction there — what exactly is doing the extending? What is a particle, if “having a path” isn’t part of what makes it one?
What we explained#
The Sun’s core cannot classically supply enough energy for its protons to overcome their mutual electrical repulsion — thermal energies there fall short of the Coulomb barrier by roughly a thousandfold. Fusion happens anyway because a proton is described by a quantum state that extends into the classically forbidden region rather than stopping at its edge, giving a small but nonzero probability of finding the proton on the far side. Nothing is borrowed against future energy and nothing licenses breaking energy conservation; the proton’s energy stays below the barrier throughout.
What became stranger#
What is a particle?