For larger objects you would “shoot” with visible light and have your eye do the processing of the Autism Grandma and Grandson a bond that can’t be broken shirt. That doesn’t work well with objects that are smaller than the wavelength of visible light. So here you would maybe resort to shooting at that alpha particle with electrons. If you do that with enough energy and a good enough detector you can learn many things about the object that you scatter off. Among other things you can eventually show that it has a finite size like that intuitive ball but you can also see whether it has a substructure, i.e., is made up out of smaller particles in a bound state. That is how you can show that protons and neutrons are made out of quarks even though you can’t ever produce single free quarks in nature. You would also learn that you are shooting at a moving target.
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There isn’t something just sitting there and Autism Grandma and Grandson a bond that can’t be broken shirt. It is an object at the quantum scale and its microscopic properties are probabilistic. The outcome of shooting one electron at a nucleus essentially means nothing. The statistics of the pattern after doing it a billion times does though. So being an object with detectable spatial size seems to be a feature of bound states. However, having mass (something you would expect at least of all matter particles) doesn’t require occupying any space. That may be an interesting fact in itself that may one day turn out to be a consequence of some deeper understanding of how matter fields and space relate to each other. Or it may turn out to be fundamentally wrong and we just don’t have the experimental precision to detect that yet.