# Interpretation of quantum superposition and classical Brownian motion

In the standard, Copenhagen interpretation of quantum mechanics, the usual ontology assigned to the phenomenon that repeated measurements of a quantum mechanical observable yielding different results with various probabilities is that the system, due to its being in a quantum superposition before measurement, must "exist in many classical states at once".

To be concrete, suppose the system I have in mind is a single particle (atom/ion/molecule) propagating in a double-well potential and the observable I'm measuring is the position of the particle, but that I really only care about which well (right or left) the particle shows up in upon measurement. Before measurement, the standard interpretation tells me that the particle is literally occupying both wells at once.

Now suppose I take the classical thermal analog of the above system. That is, I confine a classical particle to propagate in a double-well potential and immerse it in a thermal bath so that its trajectory is stochastic/Brownian due to the random bombardment by molecules/degrees of freedom comprising the bath. If I record the position of such a classical particle at times separated by intervals much larger than the oscillation period of the wells, then, due to the meta-stability of potential energy maximum between the wells, I will either find the particle in the left well or the right well with various probabilities (these probabilities will be equal for the symmetric double-well, as will also hold for the quantum case above).

In this classical thermal situation, we would never say that the particle is occupying both wells at once. This is due to the fact that we can use light waves to observe the particle continuously without affecting its dynamical state. In the quantum setting, however, any attempt to observe the system causes an abrupt disturbance to the dynamical state. I like to call this issue in the quantum case "intrusive measurement".

My question has two parts:

1. why must we say that a quantum particle is in two places/states at once just because we observe different measurement outcomes with various probabilities?

2. Considering the classical thermal analog above as a starting point, can we not say that there exist fluctuating degrees of freedom that cause quantum systems to exhibit random outcomes upon measurement?

• In what sense is a particle in a double-well potential + a thermal bath a classical thermal analog? A thermal bath has a randomness which arises as an approximation of very complicated, non-integrable/thermal interactions of its components, while a quantum system is just random "in itself". Also, the quantum randomness is there even at absolute zero temperature. There are of course even more differences. – plan Dec 22 '17 at 22:12
• This sounds like semantics or just a communication breakdown. I agree with everything you just said. Yes, quantum systems are random "in themselves". Is it not possible to explore this further and figure out what gives rise to such fluctuations? Yes, the quantum randomness exists at absolute zero temperature so it's obviously of a non-thermal nature; hence my using the word "analog". Sorry if the original statement was unclear. – subquantum Dec 22 '17 at 22:24
• Ok, I did not at first understand why you wanted this precise analogy. So what you are interested in is comparing a stochastic process with a quantum mechanical process, and see if there can be microscopic explanation to quantum fluctuations? – plan Dec 22 '17 at 22:30
• I don't have an explanation but I guess you are already familiar with t'Hoofts ideas on this? arxiv.org/abs/1405.1548 – plan Dec 22 '17 at 22:31
• I think that in the end, this is going to come down to Bell's inequality. Are you familiar with that? The root of your confusion is in your question "Why must we say that a quantum particle is in two places/states at once just because we observe different measurement outcomes with various probabilities?" The answer is: we don't. We insist on the superposition picture because experimental violations of Bell inequalities prove that local hidden variable theories don't describe Nature. – DanielSank Dec 23 '17 at 5:13