Focal length

Each side is a sphere section. They don't have to have the same radius, they are still simple to understand with different radiuses.

The two things you have to have in mind that this does are:

Theory that atoms exist, i.e. matter is not continuous.

Much before atoms were thought to be "experimentally real", chemists from the 19th century already used "conceptual atoms" as units for the proportions observed in macroscopic chemical reactions, e.g. $2H_2+O_2\leftrightarrow H_{2}O$. The thing is, there was still the possibility that those proportions were made up of something continuous that for some reason could only combine in the given proportions, so the atoms could only be strictly consider calculatory devices pending further evidence.

Subtle is the Lord by Abraham Pais (1982) chapter 5 "The reality of molecules" has some good mentions. Notably, physicists generally came to believe in atoms earlier than chemists, because the phenomena they were most interested in, e.g. pressure in the ideal gas law, and then Maxwell-Boltzmann statistics just scream atoms more loudly than chemical reactions, as they saw that these phenomena could be explained to some degree by traditional mechanics of little balls.

Confusion around the probabilistic nature of the second law of thermodynamics was also used as a physical counterargument by some. Pais mentions that Wilhelm Ostwald notably argued that the time reversibility of classical mechanics + the second law being a fundamental law of physics (and not just probabilistic, which is the correct hypothesis as we now understand) must imply that atoms are not classic billiard balls, otherwise the second law could be broken.

Pais also mentions that a big "chemical" breakthrough was isomers suggest that atoms exist.

Very direct evidence evidence:

Less direct evidence:

Subtle is the Lord by Abraham Pais (1982) page 40 mentions several methods that Einstein used to "prove" that atoms were real. Perhaps the greatest argument of all is that several unrelated methods give the same estimates of atom size/mass:

TODO. Can't find it easily. Anyone?

This is closely linked to the Pauli exclusion principle.

What does a particle even mean, right? Especially in quantum field theory, where two electrons are just vibrations of a single electron field.

Another issue is that if we consider magnetism, things only make sense if we add special relativity, since Maxwell's equations require special relativity, so a non approximate solution for this will necessarily require full quantum electrodynamics.

As mentioned at lecture 1 youtube.com/watch?video=H3AFzbrqH68&t=555, relativistic quantum mechanical theories like the Dirac equation and Klein-Gordon equation make no sense for a "single particle": they must imply that particles can pop in out of existence.

Bibliography:

Ciro's 10 cents: physics.stackexchange.com/questions/32422/qm-without-complex-numbers/557600#557600

Why is there a complex number in the equation? Intuitively and mathematically:

Some ideas: