Probability to overcome Coulomb repulsion

[andre220]

Homework Statement

The temperature in the interior of the Sun is about 1.5E7 K. Consider one of the reactions in the thermonulcear synthesis chain: p+p->H_2 + e^+ neutrino. In order for this reaction to occur two protons have to be at the distance of about 1 fm (10E-15 m). Estimate the probability that the protons have sufficient energy to overcome the Coulomb repulsion and come close to this distance and compare it to the probability that the protons have the energy of the order of Temperature, but tunnel through the Coulomb barrier.

Homework Equations

Thermal: $E = \frac{3}{2}k_B T$
Coulomb $E = \frac{Z_1 Z_2 e^2}{4\pi\epsilon_0 r^2}$
$P(tunnel) \propto e^{\alpha/r}$

The Attempt at a Solution

I believe there may be a number of approaches one could take here. The main thing being, I do not see how there is a relation to a probability. In class we have been working with mainly the Bohr Quantization and WKB approximation.

So my thought then we have both a potential (coulomb) and a desired energy (thermal) and one could solve and get the quantized energy levels or could plug it into some form of the wavefunction for the WKB approximation. The latter seems like the best approach because the wave function would (hopefully) ultimately lead to a probability. But I'm still not seeing how to incorporate both the thermal and coulomb terms into this picture.

 

[Orodruin]

You are being asked for two different things and to compare them in the end:

1. The probability of the protons having enough energy to overcome the Coulomb potential barrier. This is a purely classical statistical mechanics problem and will be based on the thermal distribution of proton energies.
2. The probability of a proton with the mean thermal energy to tunnel through the Coulomb potential barrier. This is a quantum mechanical phenomenon and you could very well use the WKB approximation to get an estimate.

Once you have these probabilities, you can make a comparison. I assume the exercise is designed to make you think about which of the above is the main reason for fusion reactions occurring in the Sun.