Is Quantum Mechanics Infinitely More Complex than Classical Mechanics?

[bob012345]

TL;DR Summary

Please critique the following text I found in a research article.

"Quantum mechanics is infinitely more complicated than classical mechanics. Quantum dynamics happens within the full Hilbert space of the system while classical dynamics are described by a path through a finite-dimensional phase space."

Please critique this text. It came from a research article* I found but I'm only interested if the sentence is 100% accurate or not and not in the specifics of the article itself. Are they suggesting Hilbert space is always infinite? Thanks.

Quantum mechanics is infinitely more complicated than classical mechanics. Quantum dynamics happens within the full Hilbert space of the system while classical dynamics are described by a path through a finite-dimensional phase space.*https://www.researchgate.net/publication/264708873_A_Review_of_Wave_Packet_Molecular_Dynamics

 

[phinds]

Is Quantum Mechanics Infinitely More Complex than Classical Mechanics?

No. That's ridiculous on the face of it. That would mean that QM is infinitely difficult. I don't think it's THAT hard.

 

[bob012345]

No. That's ridiculous on the face of it. That would mean that QM is infinitely difficult. I don't think it's THAT hard.

I agree especially since much of the formal structure of Quantum Mechanics comes directly from Classical Mechanics.

 

[PeterDonis]

It came from a research article*

The arxiv preprint is here:

https://arxiv.org/abs/1408.2019

Are they suggesting Hilbert space is always infinite?

The paper is saying that the Hilbert space of QM is always infinite dimensional. The reason is that, for a full description of any quantum system, we have to include its configuration space degrees of freedom (position and momentum), and the Hilbert space that describes those is infinite dimensional.

In some cases, where we can ignore the configuration space degrees of freedom and only consider, for example, spin, the Hilbert space is finite dimensional. However, for such cases there is no corresponding classical phase space, so there is no way to compare the dimension of the Hilbert space with that of a classical phase space.

 

[bob012345]

The arxiv preprint is here:

https://arxiv.org/abs/1408.2019
The paper is saying that the Hilbert space of QM is always infinite dimensional. The reason is that, for a full description of any quantum system, we have to include its configuration space degrees of freedom (position and momentum), and the Hilbert space that describes those is infinite dimensional.

In some cases, where we can ignore the configuration space degrees of freedom and only consider, for example, spin, the Hilbert space is finite dimensional. However, for such cases there is no corresponding classical phase space, so there is no way to compare the dimension of the Hilbert space with that of a classical phase space.

I follow your words but the meaning eludes me somewhat. Take a simple example of a Hydrogen atom. How is its Hilbert space infinite dimensional? What does that really mean? In another thread regarding Lithium wave functions we spoke of a nine dimensional space neglecting spin. Thanks.

 

[TeethWhitener]

The dimensionality of the Hilbert space is not the dimensionality of the wavefunction. The hydrogen atom has an infinite set of eigenvectors; that’s where the infinite dimensional Hilbert space comes from.

 

[Nugatory]

Take a simple example of a Hydrogen atom. How is its Hilbert space infinite dimensional? What does that really mean?

Consider an even simpler system: a free particle floating in space all by itself with no outside forces acting on it. Its Hilbert space has one dimension for each possible position and there is an infinite number of possible positions (the position is a continuous variable) so we have an infinite-dimensional Hilbert space even in this simplest possible system.

 

[bob012345]

Consider an even simpler system: a free particle floating in space all by itself with no outside forces acting on it. Its Hilbert space has one dimension for each possible position and there is an infinite number of possible positions (the position is a continuous variable) so we have an infinite-dimensional Hilbert space even in this simplest possible system.

Ok, that's a mathematical definition. There is nothing particularly 'quantum' about that. Couldn't I define the same for a classical particle in space?

 

[bob012345]

The dimensionality of the Hilbert space is not the dimensionality of the wavefunction. The hydrogen atom has an infinite set of eigenvectors; that’s where the infinite dimensional Hilbert space comes from.

In theory n goes to infinity as the electron approaches ionization. But that doesn't make the problem infinity complex as suggested in the original quote.

 

[Nugatory]

Ok, that's a mathematical definition. There is nothing particularly 'quantum' about that.

It is a mathematical definition, yes, but it is the mathematical definition of the fundamental concept of quantum mechanics - the state (often called the wave function) is a vector in an infinite-dimensional Hilbert space. In contrast, states in classical mechanics are points in phase space, which has one dimension for each degree of freedom so is finite-dimensional.

And that's pretty much what the text you quoted in your original post says...

 

[TeethWhitener]

In theory n goes to infinity as the electron approaches ionization. But that doesn't make the problem infinity complex as suggested in the original quote.

"Infinitely complex" is not a scientific term, so until you define it, it's meaningless. I was answering your specific point about how the dimensionality of a Hilbert space can differ from the dimensionality of a wave function.

 

[bob012345]

It is a mathematical definition, yes, but it is the mathematical definition of the fundamental concept of quantum mechanics - the wave function is a vector in an infinite-dimensional Hilbert space. In contrast, states in classical mechanics are points in phase space, which has one dimension for each dgree of freedom so is finite-dimensional.

And that's pretty much what the text you quoted in your original post says...

Can you agree that the quote is at least grossly misleading. Problems in Quantum mechanics are not infinitely complex. And it seems to me that yes, you can define states in classical mechanics as finite dimensional but each finite dimensional has infinite positions. Why can't we define classical state dimensions in the same way as quantum states?

 

[Nugatory]

Why can't we define classical state dimensions in the same way as quantum states?

Because then it wouldn't be classical mechanics, it would be some other theory

 

[bob012345]

"Infinitely complex" is not a scientific term, so until you define it, it's meaningless. I was answering your specific point about how the dimensionality of a Hilbert space can differ from the dimensionality of a wave function.

I thank you for your answer but my point is I'm not making that claim. I'm asking if people think it's correct or not. The author has to define it. If what he means is just a mathematical definition, then perhaps he is being overly dramatic in his introduction.

 

[bob012345]

Because then it wouldn't be classical mechanics, it would be some other theory

Well, there are many classical boundary value problems that are similar in the sense that they have infinite modes which could be considered as a Hilbert space. Also, would you consider Koopman-von _Neumann classical mechanics as a different theory or a restatement of Classical Mechanics in a formal structure similar to Quantum Mechanics?

https://en.wikipedia.org/wiki/Koopman–von_Neumann_classical_mechanics

The Koopman–von Neumann mechanics is a description of classical mechanics in terms of Hilbert space, introduced by Bernard Koopman and John von Neumann in 1931 and 1932.[1][2][3]

As Koopman and von Neumann demonstrated, a Hilbert space of complex, square integrable wavefunctions can be defined in which classical mechanics can be formulated as an operatorial theory similar to quantum mechanics.

 

[TeethWhitener]

I'm asking if people think it's correct or not.

Your OP asked if it was 100% accurate. I mentioned that you'd have to supply a definition to determine that. The author thinks it's correct. @phinds doesn't. So the answer is yes. And no. It's semantics at this point.

 

[bob012345]

Your OP asked if it was 100% accurate. I mentioned that you'd have to supply a definition to determine that. The author thinks it's correct. @phinds doesn't. So the answer is yes. And no. It's semantics at this point.

I concur with that. I think the discussion had value for me because it made me think more about how the different mechanics are defined and more about the definition of Hilbert spaces. Fortunately, in general, solving Quantum Mechanical problems isn't infinitely more difficult that solving problems in Classical Mechanics though some problems might seem like it such as multi-electron atoms as compared to multi-body celestial mechanics.

 

[dextercioby]

I was just going to suggest the Koopmann - von Neumann formulation of classical mechanics, but you did it yourself. This proves that the perceived „complexity” of quantum mechanics coming from using functional analysis is actually common to classical mechanics, as well. We also have a symplectic formulation of Quantum Mechanics, which is another way of saying that fundamentally the mathematics used in both theories are not that different.

To add one more thing: classical mechanics is taught in a university (with exception of graduate programs designed for PhDs and research) traditionally, meaning Newton Laws + Lagrange and Hamilton with not too much (or even not all) emphasis on differential geometry, while Quantum Mechanics courses do not touch the mathematical intricacies of functional analysis. So, we could say that it is a false perception that QM is more complex than CM, because Hilbert spaces are sold superficially as „advanced mathematics”.

 

[PeterDonis]

ake a simple example of a Hydrogen atom. How is its Hilbert space infinite dimensional?

Take an even simpler example, a single spinless particle confined to a one-dimensional line. The Hilbert space for this particle has one dimension for each point on the line. That means an infinite number of dimensions.

 

[EPR]

Classical mechanics is a special case of QM and is only a useful approximation(statistical coarse graining of averages). You should not take it to mean it's comparable to qm as the precision is not even similar. It's okay for some practical purposes but lacks the explanatory power of its more comprehensive treatment - qt. Finding solutions to more complicated quantum systems is often almost infinitely complex.

Position, momentum and energy can take on an infinite number of values. For that you need an infinite Hilbert space. There is no work around.

 

[user30]

Quantum entanglement is arguably the most profound and mind-blowing phenomenon ever. So I would say yes. How could that alone not be viewed as more complex?

 

[Vanadium 50]

Did anyone actually read the paper? It's not about the differences between classical and quantum. It's about running numerical simulations of "warm dense matter systems" (e.g. fusion) on large computers. The author is not expounding on great fundamental truths; he's talkingh about a specific situation.

There is a tendency to try and understand QM like a lawyer understands laws - grabbing the exactly right set of words from some expert somewhere. That has yet to have worked in this case.

 

[user30]

Classical mechanics is a special case of QM and is only a useful approximation(statistical coarse graining of averages). You should not take it to mean it's comparable to qm as the precision is not even similar. It's okay for some practical purposes but lacks the explanatory power of its more comprehensive treatment - qt. Finding solutions to more complicated quantum systems is often almost infinitely complex.

Position, momentum and energy can take on an infinite number of values. For that you need an infinite Hilbert space. There is no work around.

Is it really fair to state that all models of classical mechanics are approximations? That wasn't the view traditionally, was it?

 

[user30]

There is a tendency to try and understand QM like a lawyer understands laws - grabbing the exactly right set of words from some expert somewhere. That has yet to have worked in this case.

How do you expect that a theory with fundamental knowledge restrictions as one if its postulates will ever be fully understood? Understanding the little we do know is better than nothing.

 

[Vanadium 50]

I expect that people who want to understand learn the math. I also expect them to pick relevant sources.

 

[user30]

I expect that people who want to understand learn the math. I also expect them to pick relevant sources.

If the math was sufficent to understand it we wouldn't have competing interpretations. If you reduce QM to math, then it's not neccesarily more complex than classical mechanics.

 

[bob012345]

Classical mechanics is a special case of QM and is only a useful approximation(statistical coarse graining of averages). You should not take it to mean it's comparable to qm as the precision is not even similar. It's okay for some practical purposes but lacks the explanatory power of its more comprehensive treatment - qt. Finding solutions to more complicated quantum systems is often almost infinitely complex.

Position, momentum and energy can take on an infinite number of values. For that you need an infinite Hilbert space. There is no work around.

If by 'almost infinitely complex' you mean virtually intractable, I agree.

 

[bob012345]

Did anyone actually read the paper? It's not about the differences between classical and quantum. It's about running numerical simulations of "warm dense matter systems" (e.g. fusion) on large computers. The author is not expounding on great fundamental truths; he's talkingh about a specific situation.

There is a tendency to try and understand QM like a lawyer understands laws - grabbing the exactly right set of words from some expert somewhere. That has yet to have worked in this case.

But the quote at the beginning of section 2 is a general introductory comment, not limited to just the problem in the paper. And I specified in the OP that was the operating assumption of the question, not the specifics of the problem the paper addresses. And I agree with you, even if I don't like that quote, it in no way invalidates the work.

 

[user30]

But the quote at the beginning of section 2 is a general introductory comment,not limited to just the problem in the paper. And I specified in the OP that was the operating assumption of the question, not the specifics of the problem the paper addresses. And I agree with you, even if I don't like that quote, it in no way invalidates the work.

To answer your original question: Yes, it is an accurate scientific description and comparison of Quantum Mechanics with Classical Mechanics.

 

[user30]

I agree especially since much of the formal structure of Quantum Mechanics comes directly from Classical Mechanics.

I take it you mean partial differential equations?

 

[Botsina]

It's widely believed that BQP (including the simulation of general quantum systems) requires exponential time and classical physics simulations only take polynomial time. If this is true (and it may very well not be), then doesn't the gap between the complexity of quantum and classical systems become infinite (specifically uncountably infinite) as the complexity of the underlying system goes to infinity?

 

[bob012345]

I take it you mean partial differential equations?

No, the formal structure. Especially the Hamilton-Jacobi formulation. There are many mathematical formalisms that are borrowed and modified in Quantum Mechanics. For instance, Sakurai, Revised edition page 50 shows how the classical Poisson brackets are transformed into the quantum Commutators by the multiplication of ih. The math necessary for formal Quantum Theory was there for the tweaking in the 1920's but Sakurai makes the important point that while Classical mechanics can be derived from Quantum Mechanics, the reverse is not true. It was the formal framework to begin with and became the limit as h goes to zero. This was also true for the development of Quantum Field Theory. The Classical concepts of generators and Canonical transformations morphed into concepts like Feynman's propagators.

 

[bob012345]

It's widely believed that BQP (including the simulation of general quantum systems) requires exponential time and classical physics simulations only take polynomial time. If this is true (and it may very well not be), then doesn't the gap between the complexity of quantum and classical systems become infinite (specifically uncountably infinite) as the complexity of the underlying system goes to infinity?

It just means you need a quantum computer to solve quantum systems.

 

[EPR]

Is it really fair to state that all models of classical mechanics are approximations? That wasn't the view traditionally, was it?

With a view of qt, classical mechanics is a useful approximation. Even in qt, classical mechanics emerges as an approximation.

Are you saying that classical mechanics is fundamental and qt emerges from classicality? How would you explain blackbody radiation, stability of matter, quantum tunneling, sharing of electrons in covalent bonds, etc. processes that agree to extraordinary precision given the successes of qt?

 

[Paul Colby]

Well, the statement in the paper isn't "accurate" whatever that means.

If cardinality is a measure of complexity then I would argue QM is simpler. The number of states in a Hilbert space are countable (I think it's called separability?) whereas the points in classical phase space are uncountable.