If you apply enough force, yes. Fortunately, a human with a sharp object can not apply enough force.
Atoms are made up of electrons, protons, and neutrons.
Atoms connect to other atoms by bonds in the electrons. When you cut, for example paper, you are breaking the bonds between the electrons.
Electrons can not be cut in half because they are elementary particles. There is nothing that makes them up.
Protons and neutrons on the other hand, CAN be “cut”. They are made up of quarks. However, to do so, requires an enormous amount of energy.
This is why we have particle accelerators such as the large hadron collider. They get a particle moving extremely fast and smash it into something like a proton to “cut it in half”.
Atoms hate to touch, they’re highly charged and repel like magnets with incredible strength.
When you cut something, the repulsive force overpowers the binding strength and forces the weaker materal to shear apart.
Actually touching two atomic nuclei together requires incredible amounts of energy, found only in thermonuclear detonations and stellar cores.
Even if you did, there wouldn’t be an explosion. Splitting atoms lighter than iron takes more energy than gets released. Even the fission of a single Uranium or Plutonium atom releases only a very small amount of energy. A quick estimate is that the fission of roughly 100 billion Uranium atoms would release enough energy to heat a gram of water by 1 degree Celsius
Even if you could split an atom this way, it would not cause an explosion. Only nuclei heavier than iron release energy when split, and the amount of energy released from splitting a single atom is minuscule.
To get an explosion you need a material that is fissile. An atom of fissile material will split when a neutron hits it, and release more neutrons in the process. If you have a large enough mass of material, you have a chain reaction where each atom that splits will cause more atoms to split, causing a very rapid release of energy.
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