Exploring Randomness and Probability in Quantum Mechanics

[Phrak]

The postulate:

The modulus squared of the inner product of the state vector of the system with an eigenvector of a physical observable represents probability that the value of physical observable is equal to the corresponding eigenvalue of the physical observable in the given state of the system.​

I don't know how to define a probable outcome without referring to a few other devices, physical or mathematical. Which of the following devices is quantum mechanics dependent upon?

The first is a pseudo random number generator. The values obtained sequentially from this generator eventually repeat. Is there any evidence that outcomes of a Stern-Gerlach experiment, or something like it, repeat?

Second on my list is a particular roulette wheel at the Cowboy Cassino in Reno, Nevada. Its outcomes have fit the definition of truly random. Between the hours of 11 PM and 4 AM on march 2, 1962 it produced some very random number by a blind and deaf croupier.

If you will forgive my silly scenario for a moment, we can effect Born's notion of randomness by deferring to classical statistical mechanics. But this seems rather odd to me. We shouldn't have to define quantum mechanics in terms of classical mechanics, should we?

By intent, it should be the other way around, right? This is really my primary question in posting this OP. But, there are more possible ways to arrive at the notion of 'probability' that shouldn't be discarded immediately. So...

Third, the digits of a transcendental number

Are these digits random, or do they eventually repeat? (Do any of the mathematicians on these Physics Forums know?) Assume for a moment that the sequential outcomes of a quantum experiment are the function of the digits of pi in base 2. This constitutes a hidden variable theory. Is this sort of hidden variable precluded?

Finally, the 4th. I have it on rumor that a sequence of random numbers of length L can be produced, as I recall, by a function over a sequence integers. The function consisting of mathematical symbols having length great than L. Could we conjecture that for every experiment of sequential outcomes there is a function of this kind at work?

 

[arkajad]

The first is a pseudo random number generator. The values obtained sequentially from this generator eventually repeat.

What do you mean by "eventually"? If it will take longer than the lifetime of the Universe, will it really matter?

Is there any evidence that outcomes of a Stern-Gerlach experiment, or something like it, repeat?

Repeat? With what accuracy?

 

[Phrak]

I've approached this issue incorrectly. I was trying to be exhaustive in alternative causes of probable outcome according to the Born postulate.

Let me put it this way-

Postulatory quantum mechanics, as well as quantum field theory, and string theory by inheritance, do not displace classical mechanics, but are dependent upon, and therefore super sets of classical statistical mechanics. It's all smoke and mirrors.

 

[arkajad]

Postulatory quantum mechanics ...

Don't you think it is a rather vague term? Can you be more precise? Some particular reference would certainly help, but a precise definition would help more, because a reference would be a reference to an author and not to some "particular kind of quantum mechanics".

Different authors use different postulates, some are not postulating anything, they are just "deriving". As a rule they all come agreement when it comes to solving particular problems, though even then different formalisms usually give the same result.

 

[Fredrik]

Postulatory quantum mechanics, as well as quantum field theory, and string theory by inheritance, do not displace classical mechanics, but are dependent upon, and therefore super sets of classical statistical mechanics. It's all smoke and mirrors.

I can't make sense of what you're saying here. Everything but the word "super" suggests that you think QM is nothing more than classical statistics (that claim would be absolutely false), and the word "super" suggests that you might actually mean the opposite.

 

[Phrak]

I can't make sense of what you're saying here. Everything but the word "super" suggests that you think QM is nothing more than classical statistics (that claim would be absolutely false), and the word "super" suggests that you might actually mean the opposite.

It may make sense if you read the OP. What do you think supplies the sequence of outcomes of measurements such that they appear to mimic statistical mechanics?

 

[lugita15]

Let me discuss each of these possibilities separately:

1. In principle this is a theory that can be proven right. In practice, however, if the period of repetition is sufficiently large it will be impossible to tell one way or another. But the incredible accuracy of quantum mechanics indicates that, if there is such a period, it must be truly astronomical. This is because there's an extremely large number of quantum particles buzzing about all around us. If they were all consulting some pseudorandom generator with any reasonable period, we would very quickly see repetitions. But that would imply that there would be correlations in unconnected quantum mechanical calculations, which is manifestly not observed.
2. If I'm not mistaken this is basically the Bohmian interpretation. Under this view, all quantum uncertainty is just a manifestation of good old classical thermodynamic uncertainty. Based on your subsequent posts in this thread, I'm guessing this is what you subscribe to. Currently it's a non-falsifiable, but I suppose that experimental phenomena where the Bohmian formalism was directly relevant would lend much more credence to this position. Currently, there's no case as far as I know where quantum potentials and Bohmian velocity were the most natural tools to use.
3. On the face of it this seems crazy, but believe it or not there are people (or at least one person) who believe that this is correct. Roger Penrose believes that when a wavefunction collapses, it doesn't just randomly go into some state. Instead, the state the wavefunction collapses into can be determined by pure mathematical calculations. In principle, it could be as simple as calculating the digits of some real number. But there's a catch: Penrose thinks that such a real number would be noncomputable, meaning that no classical computer, or Turing machine, could ever compute its digits. Don't worry if you can't use your iPhone to calculate it, though, because according to Penrose the powers of the human mind surpasses any computer, and it's in fact sufficiently powerful to predict the result of any quantum mechanical measurement. This may all seem a bit far-fetched, but Penrose defends it at length in his books the Emperor's New Mind and Shadows of the Mind (the latter more than the former).
4. I don't understand what this means. Can you clarify?

 

[strangerep]

Postulatory quantum mechanics, as well as quantum field theory, and string
theory by inheritance, do not displace classical mechanics, but are dependent
upon, and therefore super sets of classical statistical mechanics. It's all
smoke and mirrors.

Certainly, QM reduces to CM under certain circumstances (such as if
the relevant observables commute, or if the action of the system is
very large compared to $$\hbar$$).

The maths of QM (operators, eigenvalues, etc, etc) is such that it
satisfies the classical probability axioms -- except for the
one involving "and", i.e., $$P(A \& B | C)$$, which must be handled
more carefully due the noncommuting nature of observables in QM.

The probability axioms then imply a law of large numbers, which
corresponds to the usual intuitively sensible frequentist view of
probability.

I don't see why you say:

It's all smoke and mirrors.

The passage from probability theory and classical dynamics/symmetries
to QM is (IMHO) clearly explained in Ballentine's textbook (section 1.5,
and chapters 2, 3, 8 ,9). If you can obtain a copy, I'm sure it would
help blow away some of the smoke. :-)

[BTW, if the digits in a decimal representation of a transcendental number
repeated, it would instead be a rational number.]

 

[arkajad]

The maths of QM (operators, eigenvalues, etc, etc) is such that it
satisfies the classical probability axioms -- except for the
one involving "and"...

Operator and eigenvalues can live without talking about any probability at all. We just have them, we can study them, we can use them.

Talking about probabilities is an extra added feature. In QM we have standard probability - nothing special, nothing extra comparing to classical physics. It works in a classical way for commuting families of operators. It works in a classical way for Abelian algebras.

Classical probability is based on Boolean algebras satisfying axioms of the classical logic.

In QM wee meet a generalization of this concept, we can discuss "logics" of projection operators. They are non-Boolean and indeed there is one axiom that is different.

Then, once we extended the concept of the underlying Boolean algebra structure, we can discuss "quantum states" and "quantum probabilities" - surely they will not have all the classical properties, because the underlying structure is not that of a Boolean algebra of sets.

That's all fine and dandy, but do we really need these "quantum logics" and "quantum probabilities"? I would like yet too see any application of them in one simple calculation leading to a prediction of an effect that can not be predicted in a simpler way.

 

[strangerep]

[...] but do we really need these "quantum logics" and "quantum probabilities"? I would like yet too see any application of them in one simple calculation leading to a prediction of an effect that can not be predicted in a simpler way.

I'm glad you brought this up -- since I tend to agree. :-)

My impression of all the quantum logic stuff is that it's a way to pass from a
lattice of propositions (non-Boolean in general, as you say), and then establish
an isomorphism to projection operators on a Hilbert space. But the papers I've seen
which give details of the latter always seem to stay within the comfort of bounded
operators on Hilbert space. I haven't seem them address the physically important
but inconvenient cases of unbounded operators and/or generalized eigenstates
in rigged Hilbert space. (Does anyone know of papers that do this, btw?)

Hmmm, ... I guess one would have to set up some sort of topological net of
propositions replacing the usual lattice, and then try to establish a homeomorphism.

 

[arkajad]

Even the spectral measure of unbounded Hermitian operators consists of bounded projections and Nature is afraid of infinities, therefore the quantum logic falk is justified concentrating on bounded operators. All information about unbounded Hermitian operators is contained in their bounded functions, so nothing is lost. X and P operators are convenient for getting quick (or not so quick) results by using dirty methods and physical intuition.

ark

 

[Phrak]

It may make sense if you read the OP. What do you think supplies the sequence of outcomes of measurements such that they appear to mimic statistical mechanics?

Hmm. Sorry, I meant "re-read."

 

[unusualname]

Trying to explain quantum randomness by a deterministic algorithm is pretty much doomed, since experiments are urging us to give up on classical realism:

http://physicsworld.com/cws/article/news/27640
http://www.google.com/search?q=anton+zeilinger+rules+out+realism

even allowing for non-local models probably doesn't save realism:
http://arxiv.org/abs/0708.0584

You can protest about loopholes and other experimental deficiencies but if you do I think you're losing the battle, the evidence is becoming overwhelming that nature just ain't real and can't be explained by an entirely deterministic mechanism.

So I don't think any of your suggested schemes for generating the probabilities are viable.

I believe eventually we will just have to accept that fundamental randomness is a property of nature at microscopic scales. Hopefully it won't take several more decades/centuries of exhaustive search for an alternative deterministic theory no matter how tortuously complex before this is generally accepted by the scientific community.

 

[Phrak]

Let me discuss each of these possibilities separately:

1. In principle this is a theory that can be proven right. In practice, however, if the period of repetition is sufficiently large it will be impossible to tell one way or another. But the incredible accuracy of quantum mechanics indicates that, if there is such a period, it must be truly astronomical. This is because there's an extremely large number of quantum particles buzzing about all around us. If they were all consulting some pseudorandom generator with any reasonable period, we would very quickly see repetitions. But that would imply that there would be correlations in unconnected quantum mechanical calculations, which is manifestly not observed.

Thank you for your thoughtful reply. I'm not real keen on 1, 3 and 4 as serious contenders. But these options dilute any strict statement about #2, so I thought it prudent to include them.

2. If I'm not mistaken this is basically the Bohmian interpretation. Under this view, all quantum uncertainty is just a manifestation of good old classical thermodynamic uncertainty. Based on your subsequent posts in this thread, I'm guessing this is what you subscribe to. Currently it's a non-falsifiable, but I suppose that experimental phenomena where the Bohmian formalism was directly relevant would lend much more credence to this position. Currently, there's no case as far as I know where quantum potentials and Bohmian velocity were the most natural tools to use.

No, I don't subscribe to any interpretation of quantum mechanics that I am aware of (there are so many variants) including Bohmian qm in it's various forms, although I understand there is a diffeomorphism invariant version of bqm. I think diffeomorphism invariance is a necessary condition.

3. On the face of it this seems crazy, but believe it or not there are people (or at least one person) who believe that this is correct. Roger Penrose believes that when a wavefunction collapses, it doesn't just randomly go into some state. Instead, the state the wavefunction collapses into can be determined by pure mathematical calculations. In principle, it could be as simple as calculating the digits of some real number. But there's a catch: Penrose thinks that such a real number would be noncomputable, meaning that no classical computer, or Turing machine, could ever compute its digits. Don't worry if you can't use your iPhone to calculate it, though, because according to Penrose the powers of the human mind surpasses any computer, and it's in fact sufficiently powerful to predict the result of any quantum mechanical measurement. This may all seem a bit far-fetched, but Penrose defends it at length in his books the Emperor's New Mind and Shadows of the Mind (the latter more than the former).

Interesting. I'll look for it. I have Emperor's New Mind on my bookshelves but not Shadows of the Mind.

4. I don't understand what this means. Can you clarify?

I'm not sure I can beyond what I recall from a passing remark. I don't recall the attributed mathematician's name. In my interpretation of this remark: Consider a function, f over integers [1,2,3...], built of ordinary functions. Count the number of symbols in f, including constants and delimiters and call it L. For the length of any part of the domain, modulo L', to meet the conditions of a random sequence it is a necessary condition that L~>L'. In other words, not every function will work, but L must be more-or-less greater than L' for one those that do. In retrospect, this was almost certainly a conjecture.

 

[arkajad]

Trying to explain quantum randomness by a deterministic algorithm is pretty much doomed, since experiments are urging us to give up on classical realism...

Determinism and "classical realism" are two different things. It is good to remember this. Otherwise one may draw wrong conclusions owing to a faulty reasoning.

 

[unusualname]

Determinism and "classical realism" are two different things. It is good to remember this. Otherwise one may draw wrong conclusions owing to a faulty reasoning.

Determinism simply implies classical realism with tinier things (than things that are usually considered classical).

I use the term "classical realism" to mean deterministic realism, since it's not so common to speak of "non-deterministic realism".

In fact, I personally think the non-realism issue in QM is resolved by reinterpreting nature as fundamentally random, but I don't want to get into a philosophical argument now.

 

[arkajad]

The point is that some people who call themselves "realists" may have non-realistic views. Some realists may adopt thre quite realistic view that any classification into determinism and non-determinism is in fact confusing, because we do not have a rigorous definition of what determinism is. It is all words that different people will interpret in different way. The same with realism. Some realists will claim that acausal couplings are non-realistic, some other will claim that they are realistic because Nature is acausal if we study it from a certain angle.

We do not know whether Nature can be described by a deterministic complex algorithm or not. Because people do not know it - they "believe" this or that. But who interested in what some or other person believes or not? We are interested in facts and then we have competing explanations of these facts. That's all that counts, or so I believe :)