In the nucleus, protons and neutrons stay together because of the strong nuclear force (otherwise they will repel each other because of the electromagnetic force). The strong nuclear force is fundamentally different from the gravitational force and electromagnetic force, and it is about 100 times greater than the electromagnetic force. It occurs only when particles like protons and neutrons are at positioned at very small distances like the ones in the nucleus. Because this force is so big the space occupied by the nucleus of the atom is about `10^-15 m` in contrast with the entire atom radius of about `10^-10 m`. (the nucleus is 100,000 times smaller than the atom itself). Therefore protons and neutrons can not move very much (in fact we can say they almost can not move at all).
Now comes into play the Heisenberg principle from quantum mechanics. This principles says that the product of the uncertainty of the product between the position in space and the impulse of a particle needs to be higher than `h/(4*pi)` (here h is the Planck constant)
`delta(x)*delta(p) >=h/(4*pi)` .
`delta(x)*delta(v) >=h/((4*pi)*m)` because the mass of nucleons is fixed for non-relativistic energies.
For nucleons in nuclei, it means that because `delta(x)` is so small, they can achieve very big speeds `delta(v)` . Having such a big speed, means that once in a while, we can find a nucleon (or a group of nucleons) away (very far apart) from nucleus. In fact, this has been observed in radioactive nuclei when they undergo alpha decay. The alpha particle is a nucleus of He (2 protons and 2 neutrons) that has been expelled from one unstable radioactive nucleus.
There is a second way to look at this.
In a nucleus, nucleons are like balls that lie on the bottom of a very deep well, that has its walls made of energy. To exit the well one nucleon will need to have an energy higher that the walls fact that usually does not happen. The Heisenberg principle written for energy and time is
In nature it has been observed that there are some radioactive nuclei for which the life time is extremely small. Because `delta(t)` is so small , the uncertainty in energy `delta(E)` will be big enough to allow one nucleon (or one group of nucleons like the alpha particle) to overcome the energy barrier of the well in which they are located and to be expelled outside the atom, from time to time.
Both of the above situations are similar to one person staying in a room and running through it. Numerous times the person will hit the room walls and bounce back, but once in a while the person could go through the wall like the wall has just vanished. This phenomena is called tunneling an it has been predicted by quantum mechanics.
Thus radioactive decay proves both the Heisenberg principle and the tunneling phenomena in quantum mechanics.