This post is meant to follow my explanation of "Why is Quantum discrete?"
As I explained before, Quantum Mechanics implies that bound particles have discrete energy levels. If a particle is on one of these energy levels, we would say it is in a pure quantum state (or "eigenstate"). But there are more than just pure states in quantum mechanics. There are also mixed states, where a particle exists with two or more different energy levels all at once! We would call this a superposition of different energy states.
Superposition may sound really weird, but mathematically, it's not weird at all. Schrodinger's equation, the equation that defines quantum mechanics, has the property that if you take any two wavefunctions and add them together, you'll get another wavefunction. So if you take one wavefunction with energy E1 and another wavefunction with energy E2, and add them together, you get a mixed state which has a superposition of energy levels E1 and E2 simultaneously!
Everybody loves to hear about how weird quantum mechanics is, with particles being able to occupy multiple locations and have multiple energies, but do they realize that, mathematically, it's all just addition? Sometimes I think people who miss the math miss out.
If you actually try to measure the energy of a mixed state, you are guaranteed to observe one and only one energy. If you have a particle that is partly in the E1 state and partly in the E2 state, the way we interpret that is that there is a certain probability of measuring E1 and a certain probability of measuring E2. As soon as we measure it, the particle changes into a pure state again. This is called wavefunction collapse, which is the doorway to many of the philosophical questions that loom around quantum mechanics. Wait, I've already talked about this before.
A bit of math (only a little, I promise)
Let's say we have a mixed state. That is, a particle is partly in state 1, and partly in state 2. We might call this state (0.5, 0.5), since if we measure the particle, there is a 50% chance of finding state 1, and a 50% chance of finding state 2. But this is the incorrect way of doing it!
The more correct way to characterizing the mixed state is with the square roots of the probabilities. Rather than writing (0.5, 0.5), we should write (0.707, 0.707), since 0.707 is about the square root of 0.5. But recall that every number has two square roots, one positive and one negative. Therefore, we can also have particle in the state (0.707, -0.707). If we measure this particle, we will still have a 50% chance of finding it in state 1, and a 50% chance of finding it in state 2.
But there is a difference between (0.707, -0.707) and (0.707, +0.707)! A difference that can be experimentally tested! I can demonstrate it with a simple diagram:
One of the mixed states is mostly on the left side, while the other is mostly on the right. This is perhaps an oversimplified situation, but the point is that they are indeed different. When I said that superposition is just a matter of addition, I fibbed a bit. Superposition is a matter of addition and subtraction.
When people hear about mixed states, the natural reaction is to say, "Doesn't it make more sense to say the particle is in one state or the other, and we just don't know which it is?" Yes, that would make more sense, wouldn't it? But that's not the reality. If we were merely ignorant of the particle's true state, we would be able to fully characterize it with probabilities (0.5, 0.5). But to fully characterize a particle in a mixed state, we need the square roots of the probabilities. That makes all the difference in the world. A difference that has been experimentally tested and confirmed.
Multiple energies, Multiple locations
What I called a pure state before, that was actually just a pure energy state. That was a state in which the particle has exactly one energy, where we have no probability of observing any other energy. But there are other kinds of pure states. For example, we might have a pure position state. It looks like this:
This is a wavefunction that is zero everywhere, except in the middle, where there is an infinitely high spike. The spike indicates that there is a 100% probability that the particle is in exactly that position. Such a pure position state is not actually possible in the real world; because of the uncertainty principle, the position is always guaranteed to be uncertain. But nonetheless, it is still instructive to consider this mathematical construction.
As I said before, if you add two pure states together, you get a mixed state. If you add an infinite number of pure states together, you get a mixed state. Imagine adding lots of these infinte spikes together, side-by-side. Individually, they are really spiky, but added together, they can make a smooth curve like this one:
Any wavefunction can be constructed this way. Every wavefunction is in a mixed position state. Every particle is in multiple locations at once. It's not weird, it's absolutely commonplace. We just never see it because it's so darn small, the size of an atom. But it acts on every atom. Have I mentioned that it's what keeps atoms from collapsing?
Similarly, we can consider other kinds of states. For example, if we have a cat in a box, it might be in a "dead" state, an "alive" state, or a superposition of the two. That's the idea behind the Schrodinger's cat thought experiment. Of course, a simultaneously dead and alive cat sounds completely ridiculous, and it is! I mean, under certain interpretations, it theoretically makes sense, but there's absolutely no way to test it. For all practical purposes, the wavefunction of the cat collapses instantaneously, and thus the cat can be said to be in a pure dead state or in a pure living state.
I am just a humble physics undergrad, but I'm sure I could answer at least a few questions. I'm probably terribly unclear and incomplete in my explanations, but if no one asks, I never find out. [Warning! I am not an appropriate source for any sort formal research or learning. This much should be obvious.]