What does superposition mean in quantum mechanics?
When I say $A+B=C$ (forces). I can mean push something with force $A$ + force $B$ together, and that is same as I push it with force $C$.
But when I say wavefunction $A$ + $B$ is also a solution of Schrodinger equation, what do I mean? The physics between them obviously is not same. Is it just something pure mathematical?
If you have an operator $D$ with $$D(\Psi+\Phi)=D(\Psi)+D(\Phi),$$ then if $D(\Psi)=0$ and $D(\Phi)=0$, you can also conclude that $D(\Psi+\Phi)=0$. This is the case for the Schrödinger equation, as it reads
$$D(\Psi):=(i\hbar\tfrac{\partial}{\partial t}-H)\Psi=0,$$
where $H$ is linar. For example you certainly have linearity for the derivatives: $$(f(x)+g(x))'=f'(x)+g'(x)$$ and even more so for multiplicative operators: $$V(x)\cdot (f(x)+g(x))=V(x)\cdot f(x)+V(x)\cdot g(x).$$
The books point out that the superposition is possible like that to emphasise that the probability waves don't affect each other and so this enables you to find solutions of the equation.
If, in contrast, the Schrödinger equation would read
$$D(\Psi):=(i\hbar\tfrac{\partial}{\partial t}-H)\Psi^2=0,$$
which is non-linear because of the $\Psi^2=0$, then you'd have
$$D(\Psi+\Phi)=D(\Psi)+D(\Phi)+D(\sqrt{2\cdot\Psi\cdot\Phi}),$$
and from $\Phi$ and $\Psi$ being a solution ($D(\Psi)=0$ and $D(\Phi)=0$) it would not follow that $\Psi+\Phi$ is a solution too (you only get $D(\Psi+\Phi)=0+0+D(\sqrt{2\cdot\Psi\cdot\Phi})\ne0$).
What do you mean by "the physics between them"?
Anyway, as an illustration, if you have a function like $\Psi(x)=A\text{e}^{-(x-3)^2}$, which is a bump located around the point $x=3$, and you add it with a function $\Phi(x)=B\text{e}^{-(x-7)^2}$, which is a bump located around the point $x=7$, then you get a function $$\chi(x):=\Psi(x)+\Phi(x)=A\text{e}^{-(x-3)^2}+B\text{e}^{-(x-3)^2},$$ which has two bumps.
http://www.wolframalpha.com/input/?i=Plot[Exp[-%28x-3%29^2]%2C{x%2C-1%2C11}]
http://www.wolframalpha.com/input/?i=Plot[Exp[-%28x-3%29^2]%2BExp[-%28x-7%29^2]%2C{x%2C-1%2C11}]
The wave function relate to propability densities, and if you have high probailities at the points $x=3$ for $\Phi$ and at $x=7$ for $\Phi$, then $\Psi+\Phi$ will tend to describe a situation, which has relatively high probabilities on both of these points.
A wavefunction is a fundamentally different concept to anything that exists in "classical" physics. This question deals with what a wavefunction 'looks like'.
You're asking what a superposition of wavefunctions is. You could look at it mathematically as $$|c\rangle = |a\rangle + |b\rangle$$ where $a$, $b$, and $c$ are wavefunctions (if you're unfamiliar with the notation used, take a look at this wikipedia page). But physically this doesn't correspond to the superposition of forces, or electromagnetic fields. The mathematic forms of both are similar, but the physical interpretation is quite different.
For example, if you consider the Hydrogen atom - I'm going to assume you know something about orbitals and energy levels. If not, I can explain further - and you consider an electron in the lowest energy level and call that wavefunction $\psi_{0}(\vec r)$ (or $|\psi_{0}\rangle$), and if the same electron is in the first excited energy level (Let's call is $\psi_{1}(\vec r)$ or $|\psi_{1}\rangle$). Now it is possible to have the electron is a superposed state, where the wavefunction is given by $$|\psi\rangle = |\psi_{0}\rangle + |\psi_{1}\rangle $$ What this seems to mean is that the electron is both in the lowest energy level and the first excited state at the same time. This may seem wrong, because of our intuition, but that's exactly what it means.
Usually, you wouldn't see this superposition if you observed a quantum system. This is because a superposed state will 'collapse' to one of the states that make it up with some finite probability. So you end up seeing the particle sitting definitely in one or the other energy levels. But this year's Nobel has been given to people who've managed to brilliantly circumvent this problem. You could read a little bit about it here, if you haven't already.
So to conclude - A superposition means that you have (mathematically) a sum of two wavefunctions. Physically this corresponds to nothing that you can relate to classically, which is what makes quantum mechanics weird (but awesome).
One way to think of superposition is this: If particles behave to some degree like waves in the sense that they can never be completely "squeezed down" into actual points, then the waves -- the probability functions -- can add together very much like waves on a pond. So, just as on a pond surface you could combine together large waves with crests a foot apart traveling north with small waves whose crests are inch apart traveling east, you can in principle do exactly the same thing with the probability waves of an electron.
Wave addition is surprisingly simple, incidentally, amounting to not much more than simply imposing the smaller wave onto the moving surface of the larger wave. So, while the heights of the waves at any one point will change as the two waves move, the height of the wave at that point will always be nothing more than a simple arithmetic sum of the heights that each wave would have had separately. That nice, simple arithmetic property is called linearity, and (fortunately for physicists seeking simplicity!) it can be found throughout much of physics.
In the case of the electron there is one additional constraint: A single electron can only generate a finite amount of wave action. That wave action can be split up in many different ways and into many different types of waves, but the total sum of all those waves must always add up to one "electron's worth" of wave action. So for example, just as with the pond waves, an electron wave could consist of an equal mix of large waves moving north and small waves moving south, as long as the two sets of waves always add up to "one electron" of total wave action.
Now the fun part is that when electrons are modeled as waves, those waves have a very specific meaning, one that is a bit less than intuitive. The interpretation is this: The big waves traveling north mean that if you poke hard at the wave with something like a photon, you will sometimes (half the time if the two wave types are equal in strength) find an electron moving north, rather slowly. However, the instant you find the electron by using such a poke, all of that wave interpretation "instantly" disappears. (I say "instantly" in quotes because that is a very loaded term in that context; but that's for some other answer!)
However, since there are two types of electron waves added together, that same poke is just as likely to find the electron moving east at a much faster clip, which is what the more tightly spaced eastbound wave means. Once again, if a poke finds the electron moving east, all of the wave interpretations cease to have meaning and you simply have an electron that look a lot more like a particle in terms of where it is located.
Once found, the electron becomes a candidate for creating new waves and starting the process all over again. That is what happens with conduction electrons in metals, for example. Or, alternatively, it could get captured by a heavier object such as an atom, and at that point it would cease to behave like a roaming wave.
However, even then the electron does not stop behaving like a wave. In fact, the entire discipline of chemistry amounts to a detailed mapping out of what happens when the many different waves possible for a charged electron become bound into a tight, cramped, and mostly spherical space, one in which it must argue and negotiate and continually bump into other electrons in an attempt to find its own little bit of turf. From these waves and the intransigence of electrons (called fermion behavior) to pack together tightly comes all of the rich behavior that makes matter and life possible.
Steven,
Mathematically (as I think you already know) superposition means that if I evolve the quantum state $|c\rangle =|a\rangle + |b\rangle$, the result will be the same as separately evolving $|a\rangle$ and $|b\rangle$,and adding the results. That is due to the linearity of the Schroedinger equation.
This means that there esists a simple connection between the states out of which $|c\rangle$ is made and $|c\rangle$ itself. Without this, Quantum mechanics would be infinetely more difficult.
Whether this is purely mathematical is a little bit a matter of semantics. I guess most people would see the same property of the electric field as highly intuitive, but to others it would appear highly mathematical.
