Comparing quantum to classical physics.

[wanabe]

Classical physics have many rules that the majority of the educated population know(high school), these rules are steadfast.

 

Quantum physics have many rules that the majority of the educated population do not know(college degree), I among them.

 

Are the rules for quantum as steadfast? I have heard that they are random by some accounts; my suspicion is that they are simply more complex, not random.

 

How do the two types compare(in their respective strictness in their rules?)?

[ajb]

The rules are well established and well verified by experiments if that is what you are asking. The difference between classical and quantum physics is that quantum tends to give predictions in terms of probability. This is what you mean by "random".

[wanabe]

Perfect, thank you very much.

[uncool]

The rules are well established and well verified by experiments if that is what you are asking. The difference between classical and quantum physics is that quantum tends to give predictions in terms of probability. This is what you mean by "random".

I'd say that that alone is not enough; statistical mechanics is also like that, and yet still follows all the rules of classical mechanics.

 

The other important point could be considered either to be the idea that there is a lack of commutativity, or the idea that complex operators are necessary.

=Uncool-

[ajb]

I'd say that that alone is not enough; statistical mechanics is also like that, and yet still follows all the rules of classical mechanics.

 

The other important point could be considered either to be the idea that there is a lack of commutativity, or the idea that complex operators are necessary.

 

Sure, the deep thing is that typically operators that correspond to observables do not commute. Given the nature of the opening post I was not sure how much to say. But you are right, the typical lack of commutativity adds "fuzzyness" to the physics.

[wanabe]

Gentlemen, what do you mean by "(complex)operators" and "commute(commutativity)"?

[ajb]

Let me start by saying that the probabilist nature of statistical mechanics is due to treating the "average" properties of a large collection of particles. If one were able to keep track of all the individual particles then they will obey classical mechanics.

 

Quantum mechanics is different. We have to use the language of probability when dealing with a single particle.

 

Ok, so abstractly we say define the commutator between elements of an (associative) algebra as

 

[math][a,b] = ab - ba[/math],

 

with [math]a,b[/math] being elements of our algebra. The algebra is said to be commutative if the commutator between all elements is zero. Otherwise the algebra is noncommutative.

 

A great example of a noncommutative algebra is algebra of linear operators on a finite dimensional space. This is the algebra a nxn matrices, assuming the vector space is of dimension n. Importantly, it is true that the product of any two nxn matrix is an nxn matrix, but in general [math]AB \neq BA[/math] for any two matrices.

 

In quantum mechanics we have a correspondence between classical observables and operators on a special kind of infinite dimensional complex vector space. The vectors correspond to states, and for now just think of operators as infinite dimensional matrices, without worrying what that really means. The algebra of classical observables is commutative (these are just functions in real variables), the algebra of quantum observables (operators on a vector space) is in general noncommutative.

 

This is really the root of the Heisenberg uncertainty principle, which we can talk about more later.

[wanabe]

The uncertainly comes from us not being able to keep track of all the particles in a collection, and being forced to account for average probability instead; if i have understood you correctly. Yes?

 

"n" is essentially the dimensions(number of particles in a collection) "of a large collection"?

 

"[a,b] =ab-ba" the "a" and "b" are representative of different particles in a collection(each with their respective energy?) (Is this how we define types of particles?)? Could we then include [a,b,c]= abc-cba (and of course abc does not equal cba)?

 

It certainly sounds like people on the Internet have no clue(not a surprise, my self included).

[ajb]

The uncertainly comes from us not being able to keep track of all the particles in a collection, and being forced to account for average probability instead; if i have understood you correctly. Yes?

 

For classical statistical mechanics this is essentially the case. For quantum mechanics the single particle has some "fuzziness" or uncertainty associated with it inherently.

 

 

"n" is essentially the dimensions(number of particles in a collection) "of a large collection"?

 

In quantum mechanic the vector spaces are usually infinite dimensional, even for a single particle. Technically, (pure) states are identified with rays in a Hilbert space.

 

 

 

"[a,b] =ab-ba" the "a" and "b" are representative of different particles in a collection(each with their respective energy?) (Is this how we define types of particles?)? Could we then include [a,b,c]= abc-cba (and of course abc does not equal cba)?

 

I was thinking just in terms of operators acting on single particle states.

 

You can describe multi-particle states using creation and annihilation operators, which is closer to what you have described.

[DrRocket]

I'd say that that alone is not enough; statistical mechanics is also like that, and yet still follows all the rules of classical mechanics.

 

The other important point could be considered either to be the idea that there is a lack of commutativity, or the idea that complex operators are necessary.

=Uncool-

 

Classical statistical mechanics not stochastic. I provides deterministic predictions of the aggregate properties of large numbers of particles. There is no inherent randomness to classical statistical mechanics. Neither is there any attempt to predict the behavior of any single particle. Statiostics is used simply because of our inability to either determine the initial conditions of the myriad of particles or to solve in closed form the many-body problem. It is a tool used to model complexity, not randomness.

 

Quantum mechanics is fundamentally different in that the impossibility of a deterministic prediction is accepted, and only probabilities are predicted. In fact what quantum mechanics describes is the evolution of probability measures. Operator theory is secondary and incidental to this fundamental aspect of the theory. Quantum mechanics is fundamentally stochastic.

[Dekan]

Classical statistical mechanics not stochastic. I provides deterministic predictions of the aggregate properties of large numbers of particles.

 

Dr Rocket, could you help me please, with this point - when you say "large numbers of particles", how large does the number have to be? I mean, before Classical Mechanics stops applying, and Quantum Mechanics starts.

 

Suppose we're looking at a lump of lead - in the form of a 1-inch cube. The cube contains billions or trillions of particles as lead atoms. That's obviously a large number. So the cube will always behave in a "Classical" way. It will stay put where it is. We'll never see it suddenly change position by jumping sideways or upwards.

 

And if we make the cube smaller, by shaving lead of its sides. Until it's reduced to a 1-mm cube. Even at that small size, it's still got billions of atoms in it. So it remains Classically static.

 

But what if we continue the reduction, until it's only got a few thousand atoms, or a few hundred. Or just 27 atoms - in a 3 x 3 x 3 cube.

 

Is 27 atoms still a "large" enough number to maintain Classical mechanics. Or will QM start to take over, causing the cube to move about?

[wanabe]

ajb,

For quantum mechanics the single particle has some "fuzziness" or uncertainty associated with it inherently.

Can you explain where this fuzziness comes from? It is because some particles can act as waves and particles? Is there an explanation for that?

 

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Dekan,

 

I probably shouldn't be answering this, but I believe as long as there is a whole atom classical physics applies. I think Its when that single atom is broken(changes from being physical into energy) apart somehow that quantum begins.

[ajb]

Can you explain where this fuzziness comes from? It is because some particles can act as waves and particles? Is there an explanation for that?

 

The fuzziness comes from the fact that the operators corresponding to physical observables do not in general commute. This is the origin of the Heisenberg Uncertainty Principle.

 

The operators corresponding to position and momentum do not commute. This means that we cannot simultaneously measure both position and momentum with absolute certainly. For instance, if you pin down the location exactly, then you cannot know the momentum of the particle.

[Dekan]

Thanks Wanabe for your post#12

 

I think I've got it now. QM applies to single "particles". And a Lead atom isn't strictly a "particle". It's an agglomeration of a large number of particles - lots of protons, neutrons and electrons.

 

Therefore the Lead atom isn't affected by QM. It will always behave according to Classical Physics - ie, stay in the same place, and not suddenly start jumping about in a QM-ish manner.

 

But what about a very simple atom. Such as Hydrogen. That contains only a single proton and a single electron. Just 2 particles. That can't be regarded as a large number. So I wonder whether Hydrogen atoms, might show signs of QM-influenced behaviour?

[ajb]

But what about a very simple atom. Such as Hydrogen. That contains only a single proton and a single electron. Just 2 particles. That can't be regarded as a large number. So I wonder whether Hydrogen atoms, might show signs of QM-influenced behaviour?

 

 

Single particles in some potential, say an electron in the electrostatic potential of a hydrogen nucleus, exhibits quantum behaviour.

[mississippichem]

Therefore the Lead atom isn't affected by QM. It will always behave according to Classical Physics - ie, stay in the same place, and not suddenly start jumping about in a QM-ish manner.

 

The properties of the lead atom are determined by the quantum behavior of the electrons protons and neutrons that comprise the lead atom though. I would say that a single atom, even a large one, is still small enough to exhibit "quantum behavior" because things like the radius of the atom or ion are determined by things like oxidation state, and orbital filling which are quantum phenomena. An atom is mostly empty space containing a few to a couple of hundred quantum objects interacting with each other.

Edited August 1, 2011 by mississippichem

[DrRocket]

Dr Rocket, could you help me please, with this point - when you say "large numbers of particles", how large does the number have to be? I mean, before Classical Mechanics stops applying, and Quantum Mechanics starts.

 

It has nothing to do with the number of particles. Classical statistical mechanics applies to a very large number of interacting particles, the more the better. Problems arise when there are too few particles, not too many. We can solve the two-body problem. Statistically we can solve the very many body problem. The general three body problem is intractable.

 

There is no clear demarcation. Three is a problem. Avogadro's number is tractable.

 

This has little to do with quantum mechanics, though there is also the subject of quantum statistical mechanics to consider. The line between what is recognized as "quantum behavior" and macroscopic "classical behavior" is also fuzzy, and not well understood. This comes under the heading of better understanding of "collapse of the wave function" or "quantum decoherence", or maybe something else. The problem is not when classical mechanics stops working, but how, in detail, classical behavior arises from quantum mechanics.

 

 

A great deal is just not known.

 

At this stage of the game, the role of quantum theory is to try to establish the rules of play at the most fundamental level. It has been fairly successful at doing that, acknowledging that much more research remains to be done before there is a fully mathematically consistent "theory of everything" or even a fully well-defined and rigorous quantum field theory. But even if you accept the success of quantum electrodynamics, the extension of those fundamental rules to everyday phenomena remains beyond our grasp. In theory all of chemistry is just a corollary of QED, but in practice that is not the case. Similarly in theory the forces that bind nucleons, the "residual strong force" are a consequence of quantum chromodynamics, but no one has been able to derive those forces from the underlying field theory.

 

There is a hell of a lot yet to be learned.

[khaled]

im not good in physics, but what I this is that

 

in classical physics we imagine reality as space of forces,

 

but in quantum physics, every point of potential, has probabilistic forces ...

[ajb]

im not good in physics, but what I this is that

 

in classical physics we imagine reality as space of forces,

 

but in quantum physics, every point of potential, has probabilistic forces ...

 

 

In classical physics, if the force is a conservative force then it can be written down in terms of a potential. This is important in formulating particle dynamics in terms of Hamiltonians and Lagrangians, but does not really come up when applying Newton's laws.

 

 

For example, friction is a non-conservative force, and thus cannot be written as a potential. When you balance the forces in a mechanical problem, pulling a massive block up a rough incline for example, you balance the forces and don't care if they are conservative of not.

 

The place that this does matter is when considering the work done. If the force is conservative then the work done in moving a particle through a trajectory that starts and ends in the same point is zero.

 

The standard methods of quantisation of a classical system are based on either the Hamiltonian formulation or Lagrangian formulation. Methods like canonical, deformation and geometric quantisation are founded on the Hamiltonian formulation of mechanics. Feynman's path integral is based on the Lagrangian formulation.

 

Either way the notion of a potential is important and plays a significant role.

 

I confess that I am not 100% clear on what adding forces that depend on the velocity (or momentum) does to the quantum theory. I am sure this has been studied somewhere.

[khaled]

This might be interesting, someone was doing a long research in quantum theory,

 

he modeled reality, which is 4-dim, as a complex 3-dim manifold .. he removed the "time" dimension,

 

but in a confusing way, he made "time" quantum, a super-position in the field, considering a time-clock unit,

 

a uniform fractal to model time state, and considering time in more than one direction .. I couldn't understand

 

how "time" will be interpreted anyway ...

[DrRocket]

This might be interesting, someone was doing a long research in quantum theory,

 

he modeled reality, which is 4-dim, as a complex 3-dim manifold .. he removed the "time" dimension,

 

but in a confusing way, he made "time" quantum, a super-position in the field, considering a time-clock unit,

 

a uniform fractal to model time state, and considering time in more than one direction .. I couldn't understand

 

how "time" will be interpreted anyway ...

 

This would be more enlightening if either:

 

1) You were to provide a reference to the literature

 

or

 

2) It made sense.