How often does nuclear fusion occur within the human body?

I'm just curious. I figure atoms fuse occasionally just by chance, like quantum tunneling or rogue waves. Is this true? If so, any idea how often?

Answer

This'll be a very rough order of magnitude estimate, but as you'll see it's good enough.

Suppose that two hydrogen atoms bump into each other. In order to fuse, the nuclei have to tunnel to within about a nuclear distance of $10^{-15}$ m of each other. The tunneling probability is something like $e^{-(2mE)^{1/2}L/\hbar}$, where $E$ is the energy gap, $m$ is the particle mass, and $L$ is the distance. The distance is of order $10^{-10}$ m (a Bohr radius) and the energy is about an MeV (the electrical potential energy of two protons right next to each other. I work out the numbers to get a probability of about $e^{-20000}$.

You'd next have to multiply that by the number of "chances" (number of times two atoms collide with each other). That's a large number by ordinary standards, but it's not exponentially large in the same way that the probability is exponentially small. Say you've got $10^{29}$ atoms in you, and each one collides with something else $10^{10}$ times per second. Then the number of chances per second is a mere $10^{39}$. I made up that number $10^{10}$ out of nowhere, but whatever it is, it's not $10^{10^4}$, which is what it'd have to be for there to be any significant probability.

So it never happens.