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I've been told that, according to QM, when Hydrogen atoms are left together there is a non-zero probability that they spontaneously fuse (I accept this bit). I've been told further that, because of huge amount of Hydrogen atoms present in the Sun, a substantial proportion of the Sun's energy is released due to this phenomenon - is this true?

EDIT: On second thoughts, this question may be unclear. I think I would answer myself that the conditions of the sun make fusion more probable than elsewhere, so it happens with much greater frequency.

What I was meaning to ask was, with a small amount of hydrogen (neglecting the reduced gravitational effect) would we have to apply a greater pressure and heat in order to produce fusion than with a much, much greater amount - since the greater amount has the benefit of the higher probability of some of the atoms spontaneously fusing anyway?

EDIT #2: in this video there is a claim made regarding quantum tunnelling in the sun, can someone please explain this phenomenon to me in more detail and tell me whether the video is correct?

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Presumably "when Hydrogen atoms are left together there is a non-zero probability that they spontaneously fuse" is meant to specify "at low temperature" where fusion would be impossible without tunneling effects. As for "a substantial proportion of the Sun's energy is released due to this phenomenon", I don't have a reference, but I believe that this is not the case. – dmckee Sep 6 '12 at 18:57

Yes, basically we'd need very high temperatures to achieve fusion.

And yes, this temperature could be rendered less substantial with tunneling. I.e. the nuclei could have almost the required energy and tunnel through remaining barrier.

Tunneling analogy would be a ball resting on the ground on one side of the mountain. To transport it to the other side (let's assume 2-dimensional mountain you can't walk around) you will need quite a bit of energy to carry it all the way up, and then probably let it slide on the other side. Turns out quantum objects can tunnel right through the mountain, without energy required to go all the way up.

And since there's (very, very) large amount of stuff on the Sun, tunneling is much more likely to happen. I think I read somewhere we'd need like 10 Suns energy to achieve it here on Earth.

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Yes, about 96% of the energy released by fusion in the Sun is due to the proton-proton cycle. The first step of this cycle is p-p fusion. The energy required to bring two protons into contact with each other is about 600 keV (Calculate the potential energy of a system of two protons with a center-to-center separation of about 2.4 fm.) As quick estimate of the temperature, assume that each proton has half the energy (300 keV) and assume the this is the mean energy in a distribution of a proton gas. The mean energy is about 3/2$kT$, where $k$ is the Boltzmann constant and $T$ is the absolute temperature. Wiht this estimate you get about $2\times 10^9$ kelvins, but the Sun's core is only $15\times 10^6$ kelvins.

Most of the p-p fusion happens with protons on the high energy tail of the Maxwell-Boltzmann distribution, high about the 3/2$kT$ mean. Even on that tail, the energy is still below the Coulomb barrier for most, but they have reached a point where the tunnelling through the coulomb barrier becomes reasonably probable.

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