Quantum Tunneling in the Sun and Conservation of Energy

[Jimmy87]

Hi,

In my textbook it says that if you consider the electrostatic repulsive barrier that protons in the Sun need to overcome in order to get into the range of the strong nuclear force to fuse together then it fails to fully account for the measured power output of the Sun.

It says that the observed power output of the Sun can be explained if you take into account the QM effects of the protons wave function. Some protons that don't quite overcome the electrostatic barrier and are not in range for the strong nuclear force can still fuse because their wavefunction has a non-zero probability of being within range of the strong nuclear force i.e. quantum tunnelling.

I am fine with this but I just wondered how energy is conserved. Work needs to be done in order to overcome the electrostatic repulsive force (force x distance). So if protons can fuse at shorter distances does this not violate conservation of energy in some way? Surely energy still has to be conserved and I don't see how it can be.

Thanks in advance.

 

[PeroK]

Work needs to be done in order to overcome the electrostatic repulsive force (force x distance).

This is Newtonian mechanics. Interactions between particles in QM take place according to Quantum Electrodynamics and Quantum Chromodynamics. These dynamics are not based on Newton's laws, from which $W = Fd$ is derived.

 

[mfb]

Classically, the process would look like this:

The protons start with a given energy from thermal motion. Something like 1 MeV of that energy is converted to potential energy as they approach each other. From there on they fuse, releasing 2.5 MeV. Net gain: 1.5 MeV. Classically you need an initial energy of at least 1 MeV. In quantum mechanics you do not. The net gain is still the same, however - 1.5 MeV. You just don't have this big potential hill to overcome as protons can "tunnel through".