# Randomness in qm

is it true that they say an electron's behavior is random?

is this conclusion based on their finding that it's position is best described as some kind of probability distribution?

if so, that doesn't conclusively prove to me that it's random. for example, a coin's behavior is accurately given by the probability distribution
{(heads, 0.5), (tails, 0.5)} but a coin's behavior is not random. it is just very sensitive to initial conditions. but if the exact same hand position and forces are applied, the outcome will always be the same.

this is also related to the distribution of prime numbers. there is that formula which estimates how many primes are less than n.

randomness in a sequence means that if one tries to compress the sequence into a formulated function, the formula is not significantly smaller than the sequence itself. (i don't know the precise definition but that's the gist of it.)

let's say you're given a sequence (of data representing an electron's position, for example). how does one go about deciding if that sequence can be compressed into a formulated function that is significantly smaller than the sequence itself? even if you can decide that it is possible, how do you find the formula? after that, how do you find the shortest formula?

for example, since there are as many primes as there are natural numbers, there are many functions from N to P (the set of primes) that are bijections. is it possible to firgure out if any of those bijections are representable by formulas that are finite in length? furthermore, is it possible to determine whether or not you've got the shortest formula?

this all seems to be related to an article analog57 sent me and i think the answers are all negative. as far as i know, the state of things is that it can be proven that there are sequences that are random but always impossible to prove a particular sequence is random. in fact, "most" sequences are random, from what i understand. therefore, a sequence of numbers representing data of an electron's position cannot be proven to be random (or not random), if that is the case.

are physicists using a different definition of random than mathematicians (if they indeed think electron behavior is random)?

how was it established that there are no "hidden variables" that make the behavior appear random?

chroot
Staff Emeritus
Gold Member
Originally posted by phoenixthoth
if so, that doesn't conclusively prove to me that it's random.
The postulates of quantum mechanics assume, somewhat axiomatically, that there is a fundamental randomness at microscopic scales. Examples abound -- radioactive decay is a good one. There is no way to determine, in any manner, exactly when a given atom will decay. On the other hand, it is quite possible to make a statistical conclusion about how many atoms of an ensemble will decay in a given period of time.

This question"is the microscopic world truly random?" occurs so often here that it should probably have its own sticky thread. It is by far the most difficult part of quantum mechanics to swallow, since it flies in the face of thousands of years of human thought and experience, culminating in the deterministic ideas of Descartes. Einstein himself, who contributed enormously to the development of quantum mechanics, was not able to swallow the idea that the universe is truly random.

It was believed for a very long time that the particles could have some internal gears or mechanisms by which they decided deterministically when they should decay -- and that with enough investigation, we should understand those gears, too. The gears were called 'local hidden variables,' in the sense that the variables belonged to one particle, and were not seen by us.

Later, Bell's theorem showed that there is no way for a local hidden variable theory to produce the same predictions as a theory based on probability. In fact, one can perform experiments (Aspect et al. did them) in which it can be shown that this universe is probabilistic. The particles in our universe can't contain local hidden variables. Case closed.

Here's a great thread on the topic:

for example, a coin's behavior is accurately given by the probability distribution
{(heads, 0.5), (tails, 0.5)} but a coin's behavior is not random. it is just very sensitive to initial conditions. but if the exact same hand position and forces are applied, the outcome will always be the same.
This is all very true -- but who said the microscopic world MUST be like the macroscopic world? It certainly seems, via experiment, that it is not.
randomness in a sequence means that if one tries to compress the sequence into a formulated function, the formula is not significantly smaller than the sequence itself. (i don't know the precise definition but that's the gist of it.)
This is the concept of information entropy -- you can't compress a random string of symbols, because every symbol is truly independent of all others. There is no redundancy, and thus the shortest representation of a truly random string of symbols is the string of symbols itself.
let's say you're given a sequence (of data representing an electron's position, for example). how does one go about deciding if that sequence can be compressed into a formulated function that is significantly smaller than the sequence itself? even if you can decide that it is possible, how do you find the formula? after that, how do you find the shortest formula?
It's an interesting idea to apply information theoretic arguments to quantum mechanical systems, but I'm not sure how useful is the marriage. You can know the position of an electron to any desired precision -- but you cannot know the momentum as precisely at the same time. I'm not sure how you would attack this concept with information theory.

On the other hand, you can let your sample of radioactive material decay, and write down the lengths of the intervals between each decay event. The resulting string of times should have zero redundancy, and, if each time is already represented in its most compact form, the entire string of times will not be further compressible.
but always impossible to prove a particular sequence is random.
This is correct -- if I give you just one bit -- a lonely zero -- you can say absolutely nothing about its origin, or whether or not it was generated randomly. You have a confidence of 0% in your conclusion of randomness. As I give you larger and larger strings of numbers, you can grow more and more confident that the string is incomressible, and displays all the other features of randomness, and your confidence will increase -- but it will never reach 100%. In fact, it is entirely possible for a random source to generate any length of consecutive zeros. When you're confronted with a gigabyte of zeros, you'd be hard pressed to declare the source random, even though it very well might have been.
are physicists using a different definition of random than mathematicians (if they indeed think electron behavior is random)?
Probably a less rigorous definition, but with the same intent.
how was it established that there are no "hidden variables" that make the behavior appear random?

- Warren

what does "â‰¥ 0" mean?

jcsd
Gold Member
Originally posted by phoenixthoth
what does "â‰¥ 0" mean?

It means the font set has changed, I've corrected my post now tho'

what does NP(&theta;12) mean? (btw, how can you make things subscripts?)

so in QM without LHVT, is it the case that sets like
n(1+,2+,3+) cannot be measured (and thus, whether they are mutually exclusive is unknowable (unless by definition spin cannot be two things at once))? does (or can) fuzzy set theory come into play wherein membership to a set is not given as "yes" or "no" but as a real number in [0,1]? even if so, i would imagine it would be hard to even measure set membership as a number in [0,1].

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In response to Bell, you can't rule out non-local hidden variables, ftl communication, "the universe as one particle," and other such theories.

where is the flaw in the argument as posted by jcsd as posted in the thread linked to by the second post in this thread?

it's not clear to me how my original question, "is it true that they say an electron's behavior is random," was answered for the jcsd article is about atoms.

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chroot
Staff Emeritus
Gold Member
Originally posted by phoenixthoth

it's not clear to me how my original question, "is it true that they say an electron's behavior is random," was answered for the jcsd article is about atoms.
His explanation actually applies to all quantum-mechanical entities, not just atoms. He was just using atoms as an example.

- Warren

jcsd
Gold Member
Originally posted by phoenixthoth
what does NP(&theta;12) mean? (btw, how can you make things subscripts?)

so in QM without LHVT, is it the case that sets like
n(1+,2+,3+) cannot be measured (and thus, whether they are mutually exclusive is unknowable (unless by definition spin cannot be two things at once))? does (or can) fuzzy set theory come into play wherein membership to a set is not given as "yes" or "no" but as a real number in [0,1]? even if so, i would imagine it would be hard to even measure set membership as a number in [0,1].

Sorry it should of been NP+-&theta;12: N is the number of particles, P+- is the probailty that the first measurment is postive and the second negative and &theta;12 is the angle between directions 1 and 2.

Bell's Theorem+Aspect's type experiments seem to rule out hidden variables that are real and local (they do not rule out however non local real hidden variables and more generally quantum realism).I say seems because things are not completely clarified.There is still criticism regarding the validity of Aspect's and further experiments in spite of the fact that a majority of scientists consider them compeling enough.Leting aside the criticism regarding the fact that only a small part of the emited electrons is actually measured by the polarizers (and in general all technical problems related with the sensitivity of polarizers and so on) there is a basic objection (as far as I know no one has now a compeling argument against) regarding the fact that we use different samples of electrons when measuring different combinations of spin.Indeed in order to consider Aspect's experiment valid we must assume that classical inductive logic,classical logic in general,is valid.

In fact this assumption (that classical logic in general-deductive logic in special-is valid when applied at quantum level) is a premise in Bell's work too.Practically he considered as being valid the 'modus tollens' rule from clasical logic in order to establish the conditions which would rule out a local quantum theory:if A=a set of premises assumed true implies the consequence B (Bell's inequalities in this case) then as much as B is also supported by all correctly devised experiments it is assumed TRUE at least provisionally;if experimentally we observe non B (the violation of Bell's inequalities in this case) results that one of the premises is false.

The question is which one?The existence of local hidden variables,quantum realism,the premise that classical logic is valid or BOTH the existence of local hidden variables AND the existence of a hidden ontological reality (the premise that hidden variables exist) at quantum level?From the beginning it's clear that scientists cannot reject the validity of logic,if we reject the validity of classical logic Bell's inequalities are useless,Aspect's experiment proves nothing.

Usually physicists assume that the premise of local hidden variables is false (this is all we have the right to derive from here though many copenhagenists reject also the premise of an ontological quantum reality altoghether-including non local hidden variables) since all experiments,so far,seem to back the supposition that classical logic (at least the principles involved in this case) is a valid tool.Anyway I think we should remain open enough to the possibility that clasical logic might be wrong in fact in this case,there is no argument proving beyond all reasonable doubt that it holds in absolute.At most we can accept it as being,provisionally,based on all observed facts so far,valid.This mean,implicitly,that we do not have the right to discard the local hidden variables hypothesis altoghether.

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