It's possible because it isn't impossible. There is no rule in physics or math that says it is impossible, which means it's possible.

And what it is, is basically saying that some subatomic particles can exist in different states at once.

For example, electrons have a property known as spin. Since they have spin they have direction (kind of like how the Earth spins and because of that, has a north and south pole). Since electrons have a direction, an electron can be "up" or "down."

Superposition says its possible for an electron to be put into a state where it is both up and down at the same time, and will only definitely be one (and not the other) when it interacts with another particle.

It's the other way around - it's not that mathematics proves that quantum superposition is possible, it's that quantum superposition is the best explanation for what we observe in experiments.

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Before we talk about what a superposition is, let's talk about where it comes from namely the field of quantum mechanics. In quantom mechanics, outcomes are probably stick rather than deterministic as they are in Newtonian mechanics.

So what the heck did I just say? In Newtonian mechanics, we know exactly what the outcome of an event will be, we just need to calculate for the variables involved. When I throw a ball up in the air, we can predict exactly how high it will go, where it will land, and how fast it be going when it does.

Quantom mechanics is very different. In a quantum mechanical system, the exact outcome of an event is unknown. Using a model that is commonly discussed known as the double slit experiment as an example, when we fire a particle at the double slit we don't know where the particle will land. Instead, what we know are the probabilities of the particle landing at any particular location. There is no guarantee where the particle will land, and every time we repeat the experiment there will be a different result ( more accurately the results will all fall within one of the expected possible outcomes, not that there will be a unique outcome for each individual experiment). This feature is at the very crux of what it means to be a quantum mechanical system. Now, quantom mechanics is very weird (VERY weird), and certain conditions such as observation can cause "the wave function to collapse" ( this is a reference to the Schrodinger wave function, the equation that is used to calculate the probabilities of certain events in a Quantum system) and a quantom system reverts back to a classical/Newtowian system.

Okay, so back to your question. What is a superposition? Well the simplest answer to that is that a superposition is what we call an event that is still in the probabilistic (ie undetermined) state. A superposition is what exists before a pesky observer gets curious and causes the wave function to collapse. A superposition, by definition, means multiple outcomes are possible and undetermined. How is it possible? Well, it's possible because it makes accurate predictions about the world around us. Superpositions are not just a thought experiment, they're not some theoretical abstract concept that only makes sense on paper. The Schrodinger wave function makes many predictable, measurable, verifiable predictions about particle interactions and is arguably at the very heart if what makes all chemistry work.

Wave particle duality. Everything acts like a wave unless it's being observed (interacted with). Just like normal waves can interfere and overlap creating a superposition, the particle's wave can be split into two states that overlap and create a superposition.

When we describe a particle as a wave, it's basically the probability density of where the particle is.

Schrodingher's cat is the go to example of this. There's a 50% chance the cat is alive, 50% chance it's dead. If we describe an alive cat as a wave function, and a dead cat as a wave function, we can add them together and get the wave function of our superposition cat. We can then take this superposition cat, and do all sorts of math on it. Let's say we heat the box up a few degrees. Instead of taking the wave function of an alive cat and heating it up, taking the wave function of a dead cat and heating it up and then adding the two together, we can take our superposition wave function, heat it up and we get the same result. It's a shortcut that only works because it behaves as a wave.

Once we open the box, we collapse the wave function, and the particle is essentially picking a random point (probably of each point determined by the wave function) and then the particle is there. If it helps you to think of it as a mathematical trick, then sure, you can have that, but it works like this in the real world or else the single photon double slit experiment and quantum tunneling wouldn't work.