Quantum physics can sometimes seem so arcane that even humans don't need to worry about it, let alone dogs. It's actually tremendously important to our modern world. In fact, if you're reading this on a computer (and how else would you be getting it?), you have quantum physics to thank for it.
Computers are based on millions of tiny transistors manufactured on chips of silicon. These transistors are combined together to make "bits" that can be in one of two states, which we call "0" and "1." Manipulating these bits lets us do mathematical operations, write books about dogs, and watch videos of humiliating things happening to cats. And none of that would be possible without a detailed understanding of how electrons behave in a solid material.
As every dog knows, atoms are made up of electrons bound to positively charged nuclei. When you bring two atoms close together, you find that some of the electrons in those atoms become uncertain about which atom they belong to, and sort of spread out over the two atoms together. As you add more atoms, you add more confused electrons, until you have a solid chunk of material in which many of the electrons are sort of spread out through the whole solid.
The properties of a given material-- whether it's a conductor or an insulator, how it conducts heat, etc.-- depend on the details of how the atoms making it up go together-- how many electrons they have, what sort of crystal they make, etc. If you change the composition of a material in just the right way, you can make big changes in how it behaves. Pure silicon is a pretty mediocre conductor of electricity, but if you "dope" it by adding a few atoms of arsenic-- maybe one for every 10,000 silicon atoms-- you can make a material that is a better conductor, because it has some extra electrons that can move around and carry current. Add a few gallium atoms instead, and you take away some of the electrons that would normally be in silicon. This also turns out to be a better conductor, because the missing electrons give room for other electrons to shift around, carrying current.
If you put two pieces of silicon together, one doped with arsenic, and one doped with gallium, you can make a diode, which will allow current to flow in only one direction. If you put three pieces together in the right way, you can make a transistor, which can act like a switch turning current on and off. If you know exactly what you're doing, you can make lots of transistors in a single block of silicon, by adding different amounts of dopant to different parts of the block, and when you do that, you can string those transistors together to make a computer chip.
None of that-- diodes, transistors, or computer chips-- would be possible without quantum physics. So, when you fire up YouTube to watch some silly videos, you're really making use of quantum physics, along with all the other cool stuff xkcd talked about.
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