19 minutes ago, Killtech said:

to keep this short look up the atomic form factor then https://en.wikipedia.org/wiki/Atomic_form_factor - you'll see a rho popping in that formula.

It's similar to what you do in the Hartee-Fock method for many electron atoms where you also calculate effective potentials.

That's the rho for the scatterers!!

Form factors measure the spatial shape of scatterers.

OMG. Please don't ask artificial intelligence again. It's almost indistinguishable form natural stupidity.

Can you set me free now?

Yeah, let's keep this short, please, oh please.

where ρ(r) is the spatial density of the scatterer about its center of mass (r=0), and Q is the momentum transfer.

(quote from https://en.wikipedia.org/wiki/Atomic_form_factor).

If you've done some physics it takes you about half a second to figure out that's what they mean. Even if you don't remember the whole context.

For Pete's sake.

2 hours ago, joigus said:

OMG. Please don't ask artificial intelligence again. It's almost indistinguishable form natural stupidity.

Love it.

+1

9 hours ago, joigus said:

That's the rho for the scatterers!!

Form factors measure the spatial shape of scatterers.

yeah, obviously. i am not even sure what you thought i was talking about?

The original question was that given a wave function of a quantum state, how much arbitrary information does it contain. the idea is to use it as a scatter target to figure that out from the resulting scattering amplitudes of test particles. that this should have been clear from my previous posts. Ultimately, it turns out that almost all of the information embedded it the wave function is physically relevant. that is you cannot drop it.

it's been been quite a few years since i completed particle physics course, but i still remember it quite well. i did make a mistake of not getting help from AI to formulate my posts clearly - and i admit it is a repeated experience that this sometimes leads to misunderstandings when i talk to people.

so where did i lose you? where was my formulation of what i intend to do not clear enough? maybe i should have pronounced it more that the electron bound in an atom is the scattering target - rather then the probing particle like it is used in many experiments. that is maybe quite unusual to begin with, so did that lose you? the reason is that wouldn't work here with using electrons for probing as it would disturb the scatterer wave function too much for the purpose of repeated scattering of the very same scatter target. hence i specifically wrote that i use a theoretical test particle multitudes lighter and less charged then the electron - because then the theory almost allows to effectively measure and track the target wave function.

27 minutes ago, Killtech said:

so where did i lose you?

When and where you said wave functions are random variables. Random variables in QM are the observables. Wave functions have a status of their own.

37 minutes ago, joigus said:

When and where you said wave functions are random variables. Random variables in QM are the observables. Wave functions have a status of their own.

random variables are a concept of probability theory and therefore not part of QM formalism at all. it shouldn't be mixed into it without establishing a clean view how you can model QM via classical probability theory.

you may seem to have a misconception though what the term means as you seem to be driven by a very intuitive interpretation which goes quite against the concepts needed in probability theory. random variables - or measurable functions as they are called in measure theory - are an abstract definition not just applied to observable quantities. it is in fact a crucial technicality that a function is compatible with your sigma algebra and that means that you can do integration over it. if the wave function would not be a random variable, then integrals over it become not well defined... and in this case for no reason, because technically we know quite well how to define them.

Random variables are there so something like Banach Tarsky does not happen. That's all they are.

Edited June 24Jun 24 by Killtech

I don't think the whole motivation for introducing random variables boils down to solving the finer points of topology TBH.

Going back to the main point, it's not clear to me that quantum probabilities represent some kind of brand new concept of probability that Laplacian probabiliy (or Kolmogorov's for that matter) could not already handle.

The actual divide, IMO, is in how quantum probabilities derive from these strange things called "amplitudes" (complex quantities that force us to do the Boolean algebra of YES, NO, OR, AND, etc, on them instead of on the probabilities themselves).

The question could be every bit as cogently posed as "why is it that the basic logic of the world we see is projected on these amplitudes?" What "are" these amplitudes and how do they relate to the things that seem to "be"?

That's the essential difference, not a new concept of probability incompatible with the former.

EDIT: I'm aware (as @swansont has pointed out) that Kolmogorov came later than QM. But his concept is very much a generalisation of the old one AFAIK.

Just now, Killtech said:

if the wave function would not be a random variable,

The wave function is not a variable it is called the 'wave function' because it is .. a function.

Just now, Killtech said:

random variables are a concept of probability theory and therefore not part of QM formalism at all. it shouldn't be mixed into it without establishing a clean view how you can model QM via classical probability theory

I agree and said this earlier in the thread.

Classical probability is a limit to infinity.

As a result we have to make do with the best estimators we can.

41 minutes ago, studiot said:

The wave function is not a variable it is called the 'wave function' because it is .. a function.

The name "random variable" is really misleading. Measure theory calls the same definition a "measurable function" instead.

From Wikipedia: "The term 'random variable' in its mathematical definition refers to neither randomness nor variability^[2] but instead is a mathematical function [...]" (https://en.wikipedia.org/wiki/Random_variable)

1 hour ago, joigus said:

The actual divide, IMO, is in how quantum probabilities derive from these strange things called "amplitudes" (complex quantities that force us to do the Boolean algebra of YES, NO, OR, AND, etc, on them instead of on the probabilities themselves).

The question could be every bit as cogently posed as "why is it that the basic logic of the world we see is projected on these amplitudes?" What "are" these amplitudes and how do they relate to the things that seem to "be"?

indeed. what makes them strange is their interpretation which attempts to force the concept of interreference into probabilities. classic probability does not have a problem to model a process with interferences but those have to go into the state space and be treated akin to some non-observable underlying physical process. This does not even require any change of the calculus of QM but translate just into a change of terminology and interpretation.

The question about amplitudes is why people want to have them treated purely as a probabilistic object if in fact the behave like a function of underlying physical-like and probabilistic aspects. the latter separation allows to model them in classic probability. In fact specific non-linear waves exhibit a lot of the same behavior as the quantum states do.

also in recent years many new experiments were able to conduct types of weak measurements which show more and more that there is indeed an underlying physical aspect to wave functions and the resulting amplitudes that cannot be ignored - and challenge what we though is observable.

Just now, Killtech said:

The name "random variable" is really misleading. Measure theory calls the same definition a "measurable function" instead.

From Wikipedia: "The term 'random variable' in its mathematical definition refers to neither randomness nor variability^[2] but instead is a mathematical function [...]" (https://en.wikipedia.org/wiki/Random_variable)

Yes but this thread is very firmly in the Physics section.

Variables in Physics usually have physical dimensions, and that includes the variables in the 'wave function', which is a physical quantity of interest.

Probability is a dimensionless variable.

Furthermore even non dimensional varaibles in maths may have different domains and certainly different codomains.

This is not trivial since 0< P(x) < 1 at any measurement and the whole measure = 1.

5 hours ago, Killtech said:

Measure theory calls the same definition a "measurable function" instead.

Do you mean "measure theory" as in mathematics? MT is concerned with metric properties of a function (generalisation of volume).

Measurement in physics is completely different.

quantum amplitudes (I prefer to say that over "wave function") must be measurable in a mathematical sense (the famous L²(R³) class of integrable functions, which require the Lebesgue theory of the measure (in particular in order to include distributions).

That doesn't mean they can be measure in a physical sense.

Sorry if I'm coming across as a something of a stickler for precision in the terms. I need precision at every step.

28 minutes ago, joigus said:

Do you mean "measure theory" as in mathematics? MT is concerned with metric properties of a function (generalisation of volume).

Measurement in physics is completely different.

quantum amplitudes (I prefer to say that over "wave function") must be measurable in a mathematical sense (the famous L²(R³) class of integrable functions, which require the Lebesgue theory of the measure (in particular in order to include distributions).

That doesn't mean they can be measure in a physical sense.

yes, the mathematical measure theory. it is indeed both underlying QM and probability theory while the Lebesgue measure is a fundamental tool for both. Kolmogorovs probability theory is in fact mostly a rebranding of MT because apart from adding a bit special terminology, it is really just a specialization of MT to positive finite measures (think: finite volume). Otherwise it is the same with a few renaming given a somewhat different purpose and interpretation. Only going further to stochastic processes like Markov theory come with significant expansions of the framework.

Of course the mathematical meaning of measure has nothing to do with the physical concept of measurement - the mathematical concept is build around avoiding Banach Tarsky because it will break the definition of integrals while measurement in physics and in QM is an entirely different topic altogether. we cannot avoid using both of these terminologies when discussing probability theory and physics. This has indeed created some confusion and misunderstandings so far.

8 hours ago, joigus said:

EDIT: I'm aware (as @swansont has pointed out) that Kolmogorov came later than QM. But his concept is very much a generalisation of the old one AFAIK.

Indeed, Kolmogorov build on the works of former mathematicians dealing with this subject, gave it a clean axiomatic unified framework based on measure theory which was done 30 years before. The building on the latter allowed to handle problems with infinite many events and continuous problems with a solid toolset. And you really feel it: any university course on probability theory does nothing else but measure theory for the 1st semester of it.

43 minutes ago, joigus said:

Sorry if I'm coming across as a something of a stickler for precision in the terms. I need precision at every step.

It is very reasonable to ask for precision especially in context of a discussion where we have two terminologies colliding and creating ambiguities for some words.

I must admit that when i want to quickly respond to a forum post, i often do so too hastily and my answers may lack precision and thus become open for misunderstandings. i am sorry whenever that happens. so please, whenever anything i write is not clear or ambiguous, point it out so i can clarify.

Edited June 24Jun 24 by Killtech