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Are all superposition principles related? Is there a relationship between the microscopic superposition principle and the macroscopic superposition principle? Does the microscopic one lend to the macroscopic one?

Clarification:

  • By "microscopic superposition principle," I mean the ability for quantum states to superimpose ($\tfrac{1}{\sqrt 2} (|\psi_1\rangle + |\psi_2 \rangle )$ and all that jazz).
  • By "macroscopic superposition principle," I mean the way we add up multiple vectors to compute a final result, such as in Newton's 2nd Law $\mathbf a = \frac{1}{m} \sum_i \mathbf F_i$ or with Electric fields $\mathbf E = \frac{1}{4 \pi \epsilon_0} \iiint \frac{dq}{|\mathbf r - \mathbf r'|^2} \hat r$.
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Physically I would say they're not really related. Quantum mechanically when a system is in a superposition one way of thinking about it is that it is in multiple states at once. Classically when something is composed of a superposition it is just in a single state but that state is composed of multiple components. The difference is subtle, and somewhat semantic but I'll firmly stand by my statement that quantum and classical superpositions are physically different phenomena.

Mathematically they're simply related in that vectors obey the superposition principle and everything you've listed ($|\psi\rangle, \mathbf{E}, \mathbf{a}$ etc.) are all vectors.

Of course, it can be useful to think in these terms. One can build analogies for quantum superpositions when one understands it is simply a statement about vectors that describe the state. For example, what does it mean for a qubit to be both a 0 and 1 at the same time? Well, it's something like how travelling Northeast is sort of like travelling north and east at the same time. There are some similarities and differences.

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If you're comparing $\psi=\alpha\psi_1+\beta\psi_2$ to an integral of a gravitational force or electric field, you're comparing apples and oranges: the quantum mechanics superposition state is a fundamental property of the particle; it must exist if there are various allowed states. But for the 'macroscopic superposition', it's just because you're examining the forces caused by several different objects, hence the 'superposition'.

We use the term 'superposition' a lot (for instance in the context of mechanical waves and their interference too), but that does not imply a deep relationship between the different cases. However, you can frequently use similar mathematical tools, which is convenient.

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