Is it possible to prove the non-existence of "free will"?

Below, I want to demonstrate the non-existence of "free will."

Our brain has a structure, but this structure consists of "events" that happened in the past, and over which no "free will" exists.

When we make a choice, our synapses (neurobiology) spring into action, and quantum-mechanical mechanisms take place. The laws of quantum mechanics dictate that the outcome is completely random, which means "free will" plays no part here either.

In conclusion, "free will" does not exist, and so we are left with determinism with a roll of the dice thrown in. In other words, if I were to murder my brother right now, that is not "free will."

This also implies that if I do this under "conscious thought," I cannot be punished for it, but can only be removed from society and put in prison as a precaution, in order to protect society. However, the fear of punishment is good for society, because people will then commit fewer murders.

I would love to hear your opinion.