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I have some phenomenological problems with understanding probabilities in quantum mechanics, and I suspect that the reason for the confusion is that scientists themselves have not yet fully decided.

There are several options:

  1. The first option, which goes back to the classics. In quantum theory itself and its dynamics, there are no probabilities. A quantum system evolves unitarily, if it is a system of two interacting particles, then it is a single system, the particles are in a state of interaction constantly and continuously, that is, it is impossible to factorize the state. The probability appears only when an external macroscopic observer, who does not have a complete description of the system, makes a measurement. That is, the very concept of probability concerns the relationship between a quantum system and a macroscopic observer.

  2. Everything is probabilistic. Even without any measurement, two quantum particles interact probabilistically, that is, after passing at a certain distance from each other, they will either interact or not (depending on the probability).

Which point of view is the most correct?

Арман Гаспарян
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QM is intrinsically a probabilistic theory: It says nothing about single systems as it deals with ensembles of systems. Therefore the notion of probability is pervasive. The notion of state itself has a probabilistic nature. A state is nothing but the assignment of the probability of every elementary YES-NO proposition testable on the systems of the ensemble. Such an assignment is equivalent to a density matrix or a state vector as proved by Gleason with his celebrated theorem. Pervasivity is also evident, for instance, in the use of unitary operators to describe symmetries but also time evolution is a consequence of the requirement that probabilities are preserved under the action of the symmetry or the time evolution.

Classical physics can be stated in a completely probabilistic fashion as well. A state is a Liouville probability density in the space of phases and sharp states are described by Dirac deltas, which are probability measures as well. From this perspective QM and CM are quite close to each other. The difference stays in the space of events where the notion of probability is given which is different in classical and quantum theory to take account, in the second case, of the existence of incompatible (in quantum sense) observables. This answer of mine tackles these issues from a general perspective

Valter Moretti
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  • Let's say there are two electrons moving towards each other. There is no observer, no measuring device. Will these electrons interact probabilistically? That is, with some probability they will scatter on each other, but with some will not notice each other and fly away? – Арман Гаспарян Jun 08 '21 at 14:12
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    That is not the type of experiments treated by QM and there is no answer to that question at all. QM considers an ensemble of such couples and describe how the probability of this or that outcome evolves in time even if I do not perform an experiment. That is the content of a quantum state. – Valter Moretti Jun 08 '21 at 14:13
  • I agree. Because quantum mechanics describes precisely the measurement of a quantum system using a macroscopic device. And the results of such a measurement are always probabilistic (the square of the modulus of the wave function value). But this does not mean that processes at the quantum level are also probabilistic. – Арман Гаспарян Jun 08 '21 at 14:16
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    Yes, indeed, but that is not the subject of QM. Up to now there is no accredited physical theory capable to explain what happens at the level of single systems. Hidden variable theories (of a single system) are possible but they move along a very impervious route in view of the numerous no-go results at our disposal. – Valter Moretti Jun 08 '21 at 14:16
  • On the other hand, in quantum field theory, interacting fields are described as constantly and continuously interacting. Although this is more of a philosophical question, factorization and division into subsystems seems to be rather an artifact of the human approach and an approximate description of phenomena. – Арман Гаспарян Jun 08 '21 at 14:22
  • In addition, the notion that individual particles interact with each other probabilistically is equivalent to the notion that a quantum particle probabilistically changes the value of the momentum or coordinate. – Арман Гаспарян Jun 08 '21 at 15:09
  • Valter, of course, QM is intrinsically a probabilistic theory. However, I think that the consequence "It says nothing about single systems as it deals with ensembles of systems." should be kept as a separate statement, depending on the interpretation of the underlying probability theory. For example, a subjective interpretation would say which event to put a bet on, even for a single system, thus saying something on it. – GiorgioP-DoomsdayClockIsAt-90 Jun 09 '21 at 06:13
  • Clearly, calculating 2-2 cross sections, factoring the number 15 and explaining why a single Hydrogen atom is stable are all things that QM can do. Choosing to work in a regime where its predictions can be reconciled with frequentist probability is one thing. Claiming that the theory doesn't apply to small systems is quite another. – Connor Behan Jun 09 '21 at 10:44
  • Yes Giorgio, I assumed a frequentistic interpretation of the probability. I know that there are different interpretations, e.g., K. Popper discussed these interpretations of probability in famous books on the very interpretation of the formalism of QM. However, the only interpretation (of probability) I find satisfying (concerning QM) is the frequentistic one. On the other hand I completely agree with you on your remark. My statement that QM says nothing about single systems is biased by this assumption of mine. – Valter Moretti Jun 09 '21 at 11:45
  • It's false to say QM says nothing about single systems. Probability statements apply to single systems too; just consider the results having P=0 or P=1. They obviously apply to single systems, and so must other values. – Brent Meeker Jan 04 '22 at 01:41
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It all depends on your interpretation of QM. Your case 1. seems to come close to relational quantum mechanics. In RQM we cannot ask “what state is Schrödinger’s cat in ?” (well, we can ask, but the question has no answer). All we can ask is “how can we best describe our knowledge about Schrodinger's cat ?”. Before the box is opened our best description is a superposition of “alive” and “dead” states, and we can create a wave-function that represents this superposition. After the box is opened, we know that the cat is alive (or maybe dead) but this is a change in our knowledge about the cat, not the mysterious physical collapse of some wave-function.

gandalf61
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  • In relational quantum mechanics, the state of a system depends on the observer, but there everyone can be considered an “observer,” including other quantum systems. But this is the problem. An individual quantum particle does not interact with another particle probabilistically. – Арман Гаспарян Jun 08 '21 at 15:17
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You are not alone in worrying that the Born rule seems like an ad hoc part of quantum mechanics. The part that physicists should agree on is that option 2 is not correct because it's very similar to the premise of a hidden variable theory. At least when there is no observation occurring, a system's wavefunction should evolve unitarily.

Within option 1, however, there are conflicting interpretations because performing a measurement on any system inevitably means entangling it with our measurement apparatus. So the question is what exactly happens during this decoherence. I hope I'm not being unfair to them when I say that Copenhagen adherents believe there is a process somewhere along the line which is not unitary evolution. This is a collapse where one outcome in the support of the wavefunction has been "chosen" and the rest "forgotten".

Personally, I find the Many Worlds interpretation more appealing which holds that the rest of the wavefunction still exists. This means that there is no collapse when we open the box containing Schroedinger's cat. Instead, opening it just ensures that the state of our eyes and the state of the cat can no longer be factored and we have to talk about a joint wavefunction where they are highly correlated. It's still unclear why we only perceive a part of this wavefunction instead of the whole thing. But in Many Worlds, that becomes a question about consciousness instead of physics.

Connor Behan
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  • The Born rule isn't really ad hoc. Given the Hilbert space construction of QM, it's the only possible consistent rule for inferring probabilities. – Brent Meeker Jun 08 '21 at 20:04
  • Initially, not everyone liked the idea of a separate axiom saying that the state is more than just an abstract vector. But indeed, once you accept that it encodes probabilities, they have to be given by the Born rule. – Connor Behan Jun 08 '21 at 21:00