Is quantum superposition the same as regular superposition in physics?

[Bianca Meske]

Is the superposition of sound waves comparable to the superposition of a particle or more specifically (for an explanation's sake) an electron? OR are quantum superposition and "regular" superposition two unrelated things?

 

[Nugatory]

They are somewhat related.

The state of a quantum system is found by solving Schrodinger's equation, a differential equation. The state of a sound wave in air is found by solving a similar differential equation. Both differential equations are linear, which means that they have the property that if $A$ is a solution and $B$ is a solution then $C=\alpha{A}+\beta{B}$ (where $\alpha$ and $\beta$ are arbitrary constants) will also be a solution; we then say that the solution C is a superposition of A and B. So they are different physical problems governed by different equations, but the mathematical concept of superposition happens to apply to both equations.

(And be careful here. It's easy to misunderstand and put too much emphasis on this idea that some solutions are superpositions and some aren't. If I have two solutions $A$ and $B$ for my differential equation, I'll know that $C=A+B$ and $D=A-B$ are also solutions, and I'll say they are superpositions of $A$ and $B$. But a bit of algebra will also tell me that $A=(C+D)/2$ and $B=(C-D)/2$ - and now it's $A$ and $B$ that look like superpositions).

I already mentioned photon polarization as a good example of superposition. If $A$ above is the state "polarized vertically" and $B$ is the state "polarized horizontally", then $C$ and $D$ would be the states "polarized 45 degrees left" and "polarized 45 degrees right".