Undergrad Quantum mechanics is random in nature?

[Delta Kilo]

In QM, you can have a particle with an exactly deterministic momentum, but then the position is completely (uniformly, infinitely) random.

The particle does not "have" a random position. Instead, the result of a position measurement, should we choose to perform one, is random. Before the measurement the particle is in well defined state. Randomness only appears as a result of measurement process, which necessarily involves interaction with large number of particles in unknown state.

I don't get why so much fuss is made about randomness in QM. Everyone seems to be OK with classical Brownian motion of a speck of dust being random. But, they say, unlike QM, it is not a "true" randomness, they could predict it if it they knew the positions and velocities of all air molecules in a volume at time t0. But is it really so? Penrose gave this argument in "Road to Reality": moving 1kg by 1 meter somewhere in the vicinity of Alpha Centauri causes enough change in gravitation here on Earth to completely scramble the trajectories of molecules in 1m^3 volume of air within 1 minute. Given that gravitation travels at c and cannot be screened, we'd need to know the state of the entire universe all the way back to Big Bang. So for all intents and purposes the motion of individual molecules of air is truly random.

I feel it is exactly the same way with QM, I don't see much conceptual difference between QM spin measured 50/50 up or down and a classical problem of a ball on a knife's edge falling 50/50 left or right. In theory in the absence of external influences the ball stays on the edge forever and the detector remains in Schrodinger cat state of superposition. In real life the ball eventually falls down and the detector reports a single outcome.

 

[mikeyork]

Vanhees, you say "An observable can never have a determined value that's in the continuous part of the corresponding operator." in response to me. But I said deterministic (meaning in a definite eigenstate) not determined (i.e.observed).

 

[Stephen Tashi]

What do you mean by "realized".

It doesn't say that you cannot measure momentum at any accuracy.

I don't understand the distinction that you are apparently making between knowing a particles momentum was precisely measured to be a specific value at time t and concluding the particle "had" that precise momentum at time t. By a "realized" value , I mean that the value was measured and thus that the random physical quantity is thus know to have taken that specific value. Perhaps erroneously, I think of a theoretical measurement of momentum as a "realization" of a specific value of momentum.

I agree that any practical apparatus does not produce infinitely precise measurements. The conceptual question is whether the thing being measured "had" a exact value when an imperfect measuring apparatus measured it. is the argument that no particle ever had an exact momentum because all practical measuring equipment has limited precision ?[/QUOTE]

 

[Zafa Pi]

Vanhees, you say "An observable can never have a determined value that's in the continuous part of the corresponding operator." in response to me. But I said deterministic (meaning in a definite eigenstate) not determined (i.e.observed).

Confusion regularly results when people blur the lines between theory and reality. QM is a mathematical theory and in that theory a measurement by an obsversable (Hermitian operator) of a state (unit vector in a Hilbret space, or ray, or L² "wave" function) is a random variable (see Feller and Nielsen & Chuang). If the variance of that r.v. is 0 then the state is an eigenvector of the operator. What vanhees71 is saying (my interpretation) is that the momentum operator has no eigenvector in the state (Hilbert) space (same for the position operator). Thus Δp (of a state) is > 0 (Δp is the s.d. = sqrt of the variance)

Now in reality the experimental physicist selects a momentum measuring apparatus (hopefully modeled by QM) and prepares a large number of entities in the same state and procedes to measure them. The resulting measurements are not all the same in spite of the fact that each individual measurement is a single precise value. Thus the collection of all the measurements has a non-zero (statistical) variance. And as vanhees71 says QM and reality agree.

 

[DrChinese]

The particle does not "have" a random position. Instead, the result of a position measurement, should we choose to perform one, is random. Before the measurement the particle is in well defined state. Randomness only appears as a result of measurement process, which necessarily involves interaction with large number of particles in unknown state.

I don't get why so much fuss is made about randomness in QM. Everyone seems to be OK with classical Brownian motion of a speck of dust being random. But, they say, unlike QM, it is not a "true" randomness, they could predict it if it they knew the positions and velocities of all air molecules in a volume at time t0. But is it really so? Penrose gave this argument in "Road to Reality": moving 1kg by 1 meter somewhere in the vicinity of Alpha Centauri causes enough change in gravitation here on Earth to completely scramble the trajectories of molecules in 1m^3 volume of air within 1 minute. Given that gravitation travels at c and cannot be screened, we'd need to know the state of the entire universe all the way back to Big Bang. So for all intents and purposes the motion of individual molecules of air is truly random.

I feel it is exactly the same way with QM, I don't see much conceptual difference between QM spin measured 50/50 up or down and a classical problem of a ball on a knife's edge falling 50/50 left or right. In theory in the absence of external influences the ball stays on the edge forever and the detector remains in Schrodinger cat state of superposition. In real life the ball eventually falls down and the detector reports a single outcome.

Clearly, there are tremendous differences between the classical examples you give and the quantum ones. Classical systems do not feature non-commuting observables. Non-commuting observables not only have specific limits in their precision, those limits can be seen in experiments on entangled pairs. So if you don't see the conceptual difference between these, you need to consider more experiments.

 

[David Lewis]

Given that gravitation travels at c and cannot be screened, we'd need to know the state of the entire universe all the way back to Big Bang. So for all intents and purposes the motion of individual molecules of air is truly random.

My understanding of classical mechanics there are no hypothetically random processes, even if we can't collect enough information to make a prediction.

 

[drschools]

Would not truly random fit the Aristotelian definition of God i.e. the "unmoved mover" ? We might therefore attribute all genuinely random or uncaused phenomenon to God. We would therefore be following a similar argument as the "God of the Gaps". It might be argued also that while "Newtonian phenomenon" increasingly shrinks the unexplained gaps and therefore evidence of God in the universe, "Quantum phenomenon" re-expands the mysterious unexplainable gaps of science and therefore evidence of God (of the gaps).

 

[DrChinese]

Would not truly random fit the Aristotelian definition of God i.e. the "unmoved mover" ? We might therefore attribute all genuinely random or uncaused phenomenon to God. We would therefore be following a similar argument as the "God of the Gaps". It might be argued also that while "Newtonian phenomenon" increasingly shrinks the unexplained gaps and therefore evidence of God in the universe, "Quantum phenomenon" re-expands the mysterious unexplainable gaps of science and therefore evidence of God (of the gaps).

Sure. Although it does beg the question: why are there any physical laws at all if god is reserving her efforts only to fill in those gaps?

 

[Zafa Pi]

The particle does not "have" a random position. Instead, the result of a position measurement, should we choose to perform one, is random. Before the measurement the particle is in well defined state. Randomness only appears as a result of measurement process, which necessarily involves interaction with large number of particles in unknown state.

I don't get why so much fuss is made about randomness in QM. Everyone seems to be OK with classical Brownian motion of a speck of dust being random. But, they say, unlike QM, it is not a "true" randomness, they could predict it if it they knew the positions and velocities of all air molecules in a volume at time t0. But is it really so? Penrose gave this argument in "Road to Reality": moving 1kg by 1 meter somewhere in the vicinity of Alpha Centauri causes enough change in gravitation here on Earth to completely scramble the trajectories of molecules in 1m^3 volume of air within 1 minute. Given that gravitation travels at c and cannot be screened, we'd need to know the state of the entire universe all the way back to Big Bang. So for all intents and purposes the motion of individual molecules of air is truly random.

I feel it is exactly the same way with QM, I don't see much conceptual difference between QM spin measured 50/50 up or down and a classical problem of a ball on a knife's edge falling 50/50 left or right. In theory in the absence of external influences the ball stays on the edge forever and the detector remains in Schrodinger cat state of superposition. In real life the ball eventually falls down and the detector reports a single outcome.

I believe yours' is the valid answer to the original question of this thread. The key is in the 2nd line of your 2nd paragraph : "if they knew the positions ... " But they don't know and cannot give any reasonable way to go about knowing. Those which want to make a distinction between random due to lack of knowledge and "pure random" will come down to events that have a cause and those that don't. A philosophical quagmire of the 1st order in which I have squandered my youth.

In knowing the state of the universe all the way back to the Big Bang is also the ultimate loop hole in the disproving of realism using the measurements of entangled entities (super-determinism). And nobody cares, nor should they.

 

[Zafa Pi]

Sure. Although it does beg the question: why are there any physical laws at all if god is reserving her efforts only to fill in those gaps?

God moves in mysterious ways just like Simone Biles.

 

[drschools]

I'm glad you made that point as the physical laws are in fact also uncaused or truly unexplainable phenomenon... the speed of light for instance . Another argument for God -in this light (no pun intended)- would be the "Fine-tuned" or "Goldilocks principle" of many universal constants and phenomenon.

 

[Zafa Pi]

Clearly, there are tremendous differences between the classical examples you give and the quantum ones. Classical systems do not feature non-commuting observables. Non-commuting observables not only have specific limits in their precision, those limits can be seen in experiments on entangled pairs. So if you don't see the conceptual difference between these, you need to consider more experiments.

This is an example of what I consider the blurring of lines between theory and reality. There is no question that classical and quantum theories are different, as you and David Lewis (post #96) point out. The experiments on entangled pairs show that classical determinism (= realism) is wrong (assuming locality). But in reality if I flip a coin in a wind tunnel I'll get different and random looking results. The classicist says the initial conditions changed, but is incapable of measuring or controlling them.

Delta Kilo says that "in reality" one can't tell the difference between quantum randomness and the hypothetical classical (lack of knowledge) randomness.

 

[Delta Kilo]

Clearly, there are tremendous differences between the classical examples you give and the quantum ones.

Of course they are hugely different. But they are both examples of spontaneous symmetry breaking and I was only referring to the conceptual source of randomness in both cases. I just don't see the need to look any further than the unknown state of the environment. So the fact that some measurement are inherently random does not surprise me at all. It it actually the other way around: it is surprising that some measurements are less random than they should have been according to classical view.

 

[vanhees71]

Vanhees, you say "An observable can never have a determined value that's in the continuous part of the corresponding operator." in response to me. But I said deterministic (meaning in a definite eigenstate) not determined (i.e.observed).

Quantum theory is not determinstic. Some observables may be determined by preparation the system in a corresponding state. This is possible only for true eigenvalues of the self-adjoint operator, i.e., such eigenvalues for which normalizable eigenvectors exist, and these eigenvectors are in the discrete part of the spectrum.

 

[bhobba]

I'm glad you made that point as the physical laws are in fact also uncaused or truly unexplainable phenomenon... the speed of light for instance .

Hmmmm.

See the following:
http://www2.physics.umd.edu/~yakovenk/teaching/Lorentz.pdf

My view is symmetry. Its almost, but not quite magic.

If this is your first exposure I highly recommend Landau:
https://www.amazon.com/dp/0750628960/?tag=pfamazon01-20
'If physicists could weep, they would weep over this book. The book is devastingly brief whilst deriving, in its few pages, all the great results of classical mechanics. Results that in other books take take up many more pages. I first came across Landau's mechanics many years ago as a brash undergrad. My prof at the time had given me this book but warned me that it's the kind of book that ages like wine. I've read this book several times since and I have found that indeed, each time is more rewarding than the last. The reason for the brevity is that, as pointed out by previous reviewers, Landau derives mechanics from symmetry. Historically, it was long after the main bulk of mechanics was developed that Emmy Noether proved that symmetries underly every important quantity in physics. So instead of starting from concrete mechanical case-studies and generalising to the formal machinery of the Hamilton equations, Landau starts out from the most generic symmetry and dervies the mechanics. The 2nd laws of mechanics, for example, is derived as a consequence of the uniqueness of trajectories in the Lagragian. For some, this may seem too "mathematical" but in reality, it is a sign of sophisitication in physics if one can identify the underlying symmetries in a mechanical system. Thus this book represents the height of theoretical sophistication in that symmetries are used to derive so many physical results.'

Magic - no - like I said it seems like it but isn't. However sorting out the real physical assumptions is both rewarding and illuminating.

Start a new thread if interested.

Thanks
Bill

 

[mikeyork]

Quantum theory is not determinstic. Some observables may be determined by preparation the system in a corresponding state.

You should know that a state vector that is a superposition in one basis can be an eigenstate in another. In the case of momentum and position the relationship between bases is a Fourier transform.

 

[Nugatory]

Quantum theory is not determinstic. Some observables may be determined by preparation the system in a corresponding state. This is possible only for true eigenvalues of the self-adjoint operator, i.e., such eigenvalues for which normalizable eigenvectors exist, and these eigenvectors are in the discrete part of the spectrum.

You should know that a state vector that is a superposition in one basis can be an eigenstate in another

It is a <understatement>safe bet</understatement> that vanhees knows this. He's stressing the "discrete part of the spectrum" because applying the same principle to the continuous spectrum, as in

In the case of momentum and position the relationship between bases is a Fourier transform.

is a bit trickier because the "eigenstates" are not physically realizable. First-year QM texts oversimplify the mathematical subtleties here, but if you google for "rigged Hilbert space" you'll get more of the story.

 

[mikeyork]

↑
Quantum theory is not determinstic. Some observables may be determined by preparation the system in a corresponding state. This is possible only for true eigenvalues of the self-adjoint operator, i.e., such eigenvalues for which normalizable eigenvectors exist, and these eigenvectors are in the discrete part of the spectrum.

↑
You should know that a state vector that is a superposition in one basis can be an eigenstate in another

It is a <understatement>safe bet</understatement> that vanhees knows this. He's stressing the "discrete part of the spectrum" because applying the same principle to the continuous spectrum, as in

In the case of momentum and position the relationship between bases is a Fourier transform.

is a bit trickier because the "eigenstates" are not physically realizable.

Ok. I get that. But when you write "physically realizable" are you not confounding an underlying fundamental reality with observability? That is,the fundamental reality may be a definite eigenstate, but the information of an "observer" (either in preparing or detecting a state) is realizable only to a specific precision.

 

[Zafa Pi]

it is surprising that some measurements are less random than they should have been according to classical view.

Could you give a simple example of what you are talking about here?

 

[vanhees71]

Ok. I get that. But when you write "physically realizable" are you not confounding an underlying fundamental reality with observability? That is,the fundamental reality may be a definite eigenstate, but the information of an "observer" (either in preparing or detecting a state) is realizable only to a specific precision.

Within quantum theory generalized eigenstates, which are not in the Hilbert space (but in the dual of the nuclear space, where the unbound self-adjoint operators are defined), do not represent physical states. This is immediately clear also from the usually used heuristic point of view since the states are not normalizable. Take the momentum eigenstates. In position representation they are the plane waves,
$$u_{\vec{p}}(\vec{x})=\frac{1}{(2 \pi)^{3/2}} \exp(\mathrm{i} \vec{x} \cdot \vec{p}).$$
They are obviously not normalizable since the integral over their modulus squared is infinity. They are rather "normalized to a $\delta$ distribution":
$$\int_{\mathbb{R}^3} \mathrm{d}^3 \vec{x} u_{\vec{p}'}^*(\vec{x}) u_{\vec{p}}(\vec{x})=\delta^{(3)}(\vec{p}-\vec{p}'),$$
which clearly underlines the fact that these generalized eigenfunctions are to be interpreted as distributions (in the sense of generalized functions) rather than functions.

 

[Pythagorean]

If my understanding of the issue is up to date, this basically comes down to proving a negative. If you could find a (mathematically) deterministic framework that predicted QM experiments, you could rule out randomness (for which I'm assuming non-determinisic is the operating definition in this context).

The hidden variable was one attempt at demonstrating determinism (and seems to have failed); I don't know if that rules out all possibility of determinism or not, my intuition is to doubt it does.

 

[rootone]

Things that are more or less probable such as the decay of a fissile atom around it's measured half life, are not the same as 'random';
There is in fact a non randomness that makes the half life what it is measured to be.
If events were completely random then no meaningful measurement of anything is possible.

 

[Dr. Courtney]

I prefer the term "probabilistic" to random.

The probabilities are well defined. The outcome of a single event is not.

 

[Delta Kilo]

Could you give a simple example of what you are talking about here?

https://en.wikipedia.org/wiki/Bell's_theorem

 

[bhobba]

If you could find a (mathematically) deterministic framework that predicted QM experiments, you could rule out randomness (for which I'm assuming non-determinisic is the operating definition in this context).

Its more subtle than that.

Bohmian Mechanics (BM) is deterministic. Randomness comes from lack of knowledge - not because its inherently random.

One of the big advantages of studying interpretations is you learn exactly what the formalism says which often is not what is at first thought.

Again it must be emphasized no interpretation is better than any other. This does not mean I am a proponent of BM (I am not - my view is pretty much the same as Vanhees - but that means 4/5ths of bugger all ie precisely nothing) it simply means what appeals to my sense of 'beauty'.

Thanks
Bill

 

[mikeyork]

They are obviously not normalizable since the integral over their modulus squared is infinity. They are rather "normalized to a [delta distribution]
which clearly underlines the fact that these generalized eigenfunctions are to be interpreted as distributions (in the sense of generalized functions) rather than functions.

A limiting distribution with a unique value for which it is non-zero and a vanishing standard deviation. This would not normally be considered "random" although I see your mathematical point. Why do you consider it important to a physicist (rather than a mathematician)?

Also, my original point regarding superpositions being eigenstates in another basis still stands for discrete variables even if you consider the momentum/position example to be a bad one.

 

[bhobba]

Why do you consider it important to a physicist (rather than a mathematician)?

I am pretty sure Vanhees doesn't.

Rigged Hilbert Spaces are just as important to applied mathematicians as physics (without delving into the difference - that requires another thread) eg:
http://society.math.ntu.edu.tw/~journal/tjm/V7N4/0312_2.pdf

And that is just applied math - in pure math it has involved some of the greatest mathematicians of all time eg Grothendieck

Thanks
Bill

 

[Zafa Pi]

https://en.wikipedia.org/wiki/Bell's_theorem

Are you saying that two correlated random variables are less random than independent random variables?

 

[ChrisVer]

You might find this interesting https://en.wikipedia.org/wiki/Hidden_variable_theory

I didn't know that hidden variables had returned to the mainstream quantum-mechanics discussion...

 

[vanhees71]

If my understanding of the issue is up to date, this basically comes down to proving a negative. If you could find a (mathematically) deterministic framework that predicted QM experiments, you could rule out randomness (for which I'm assuming non-determinisic is the operating definition in this context).

The hidden variable was one attempt at demonstrating determinism (and seems to have failed); I don't know if that rules out all possibility of determinism or not, my intuition is to doubt it does.

Of course, it doesn't rule out all deterministic models, but all the ones that are local in the interactions (in the sense of relativistic QFT). Since there is no consistent non-local theory of relativistic QT today and also no convincing no-go theorem either, it's totally open, whether one day one might find a non-local deterministic theory in accordance with all observations today described by QT.