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Is quantum mechanics stochastic or probabilistic?

Is the universe fundamentally deterministic?

Indeterminism in Quantum Mechanics is given by another "evolution" that the wavefunction may experience: wavefunction collapse. This is the source of indeterminism in Quantum Mechanics, and is a mechanism that is still not well understood at a fundamental level (this is often called as "Measurement Problem").

De Broglie–Bohm theory

The Bohmian trajectories for an electron going through the two-slit experiment. A similar pattern was also extrapolated from weak measurements of single photons

In the quantum world, in the case of quantum mechanics, particles are studied by means of probability, because there nothing is exact. I don't know of any quantum mechanics equation that is not probabilistic but exact...

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    $\begingroup$ You should explain what the difference between "stochastic" and "probabilistic" is for you. $\endgroup$
    – d_b
    Oct 6, 2021 at 21:04
  • $\begingroup$ @d_b I think I would define "stochastic" as that described using probability (that is, calling for limiting the relative frequencies of events). I think I would define "stochastic" as that described using probability (that is, calling for limiting the relative frequencies of events). Probabilistic only implies that the user is convinced that the event can be placed into a theoretically repetitive test sequence. $\endgroup$ Oct 6, 2021 at 21:10
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    $\begingroup$ I’m not sure I understand “ particles are studied by means of probability, because there nothing is exact”… what exactly is not exact? Surely the value of Rydberg constant or Planck constant is exact. The solution to the TISE for hydrogen is exact… $\endgroup$ Oct 6, 2021 at 21:26
  • $\begingroup$ Is hidden variable theory what you're looking for? $\endgroup$
    – g s
    Oct 7, 2021 at 0:29
  • $\begingroup$ @Behemooth the definitions in your comment above are more like those of frequentist vs. Bayesian, than stochastic vs. probabilistic. $\endgroup$ Oct 7, 2021 at 4:44

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Is quantum mechanics stochastic or probabilistic?

One works with probability distributions to model mathematically observations and measurements.

In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon in terms of its sample space and the probabilities of events (subsets of the sample space).2

(italics mine)further:

Examples of random phenomena include the weather condition in a future date, the height of a randomly selected person, the fraction of male students in a school, the results of a survey to be conducted, etc.

A stochastic process

In probability theory and related fields, a stochastic or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appear to vary in a random manner.

In classical physics probability distributions are used in experiments where a random process is modeled. Classical physics is deterministic, and various functions have been found to fit particular experimental observations.

It is the randomness for the observations fitted with probability distributions in classical physics that defines the distributions, given the underlying constraints of the individual particle interactions, that can be called stochastic.

The theory of quantum mechanics is probabilistic, the probability is in the axiomatic postulates that are used to model the data at the microcosm of atoms/molecules etc.

The solutions of the quantum mechanical equations do not determine individual trajectories at the particle level, but probability distributions for accumulated data with the same boundary conditions.

An intuitive understanding can obtained by the double slit single electron at a time experiment. The footprint of a single electron looks random, but the accumulation shows interference effect, which can be completely modeled with the quantum mechanical theory. The wavefunction $Ψ$ predicts the probability distribution $Ψ^*Ψ$. That is what is meant when calling quantum mechanics probabilistic. The classically expected randomness can only be described by a probability distribution that differs form the classical probability distribution expected from classical balls.

Bohmian mechanics is not able at present to describe the experiments in high energy particle physics, but there are a number of theorists working on finding an underlying deterministic theory that could give the same predictions as the existing standard model of particle physics. See for example the contributions of G. 't Hooft on this site. (example)

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