Definition of a random variable in quantum mechanics?

[Aidyan]

TL;DR Summary

A brief clarification on the definition of a random variable and probability in quantum physics.

In a line of reasoning that involves measurement outcomes in quantum mechanics, such as spins, photons hitting a detection screen (with discrete positions, like in a CCD), atomic decays (like in a Geiger detector counting at discrete time intervals, etc.), I would like to define rigorously the notion of 'random variable' and 'probability of outcomes' in quantum physics. I defined it as follows.

Let us consider a discrete random variable $X$--that is, a measurable function $X: \Omega \rightarrow K$, with $\Omega$ a finitely countable sample space of possible outcomes and $K$ a measurable space--and $P(X)$ its discrete probability distribution (PD) (here only discrete PDs are assumed) defined as the set of probabilities that $X$ takes on the non-zero probability values $x_{i}$ $(i=1,..,C)$ as: $$P(X=x_{i})=p_{i}=\frac{n_{i}}{N},$$ with $n_{i}$ the number of events relative to the i-th possible outcome, $N$ the overall number of events or measurements, and $C$ the number of all possible outcomes, such that, in the limit of the large numbers ($N \rightarrow \infty$), for a normalized PD, $\sum_{i} p_{i}=1$.

However, I'm told this is not clear mathematical language. Is there something wrong or missing with such a statement?

Moreover, I'm told that one can have a probability space and well-defined random variables without appealing to a frequentest interpretation.
But, while it is true that in a very general context one must not necessarily appeal to a frequentest interpretation, nevertheless, once specified that we are dealing with events in the context of quantum mechanics, don't we always imply a frequentest interpretation of the measurements?

Am I’m missing something and can the definition be made more rigorous?

 

[PeroK]

Quantum mechanics uses complex probability amplitudes. See, for example:

https://www.scottaaronson.com/democritus/lec9.html

 

[Aidyan]

Of course. But the discussion is set in a purely experimental context that discusses measurement outcomes only. Say, on how to define a discrete probability distribution for the interference fringes at a detection screen of the double slit experiment without necessarily recapitulating all the Hilbert space theory, bra-ket formalism, modulus-squared, etc.

 

[PeroK]

Of course. But the discussion is set in a purely experimental context that discusses measurement outcomes only. Say, on how to define a discrete probability distribution for the interference fringes at a detection screen of the double slit experiment without necessarily recapitulating all the Hilbert space theory, bra-ket formalism, modulus-squared, etc.

I don't understand your question. The minimal statistical interpretation of QM is implicity frequentist. But, there is a Bayesian interpretation of QM:

https://en.wikipedia.org/wiki/Quantum_Bayesianism

 

[Aidyan]

That was my point too with those who doubted my exposition. I don't see any reason to dwell into non-frequentist formalism and/or interpretations if we simply want to discuss the statistics, say, of the photon distribution on a detection screen in a double slit experiment, the measurement outcomes of a spin observable, or count atomic decays, etc. But that's the kind of objection I got while trying to define in a semi-rigorous manner the notion of probability as above applied to these kinds of experimental contexts. I have been told it isn't clear mathematical language, but can't see what's wrong and/or missing and/or how it could be stated clearer.

 

[PeroK]

That was my point too with those who doubted my exposition. I don't see any reason to dwell into non-frequentist formalism and/or interpretations if we simply want to discuss the statistics, say, of the photon distribution on a detection screen in a double slit experiment, the measurement outcomes of a spin observable, or count atomic decays, etc. But that's the kind of objection I got while trying to define in a semi-rigorous manner the notion of probability as above applied to these kinds of experimental contexts. I have been told it isn't clear mathematical language, but can't see what's wrong and/or missing and/or how it could be stated clearer.

Who's telling you these things?

 

[Aidyan]

Knowledgeable people.
But I guess they didn't get the point I was trying to make. I will stick to the above definition.

 

[PeterDonis]

I have been told it isn't clear mathematical language

If you're only being "semi-rigorous" (your words), you should expect to be told this, since it's true. A rigorous formulation of probability requires something like the Kolmogorov axioms, which aren't inherently frequentist or inherently Bayesian; they are logically prior to both.

 

[PeterDonis]

Knowledgeable people.

This is not a valid reference. Either give a specific reference to who told you what, with a link to where we can read their words ourselves, or don't refer to it at all. We can't have an argument by proxy with nameless people whose statements you are paraphrasing without attribution.

 

[Aidyan]

It was in a personal conversation. Nothing that can be referenced or linked.
I only want to know whether the above definition is internally consistent or not.

 

[Vanadium 50]

Who's telling you these things?

Knowledgeable people.

Top men. Top men,.

I think you will find that most physicists are interested in the question "does the math predict what we observe?" which is a different question entirely from "can the math be proved rigorously from more basic postulates".

 

[PeterDonis]

It was in a personal conversation. Nothing that can be referenced or linked.

Then, as I said, you shouldn't reference it. You're asking for yourself, not to argue with whoever told you whatever. So you should leave whoever out of it.

I only want to know whether the above definition is internally consistent or not.

Have you looked in the literature to see what the mathematically rigorous definition of probability is? Have you looked up the Kolmogorov axioms?

Have you looked in any QM textbooks to see how they define probability?

 

[Vanadium 50]

Something to think about. If you can find what you have been told in a textbook, you can cite the textbook. If you can't, doesn't that tell you something.