# Dividing line between QM and CM?

• protoplast3r
In summary, the author is discussing how QM applies to macroscopic objects, specifically diamond vibrating. Neither statement is true, and the author gives credit to those who try to circumvent saying that the diamond is both vibrating and not vibrating.
protoplast3r
Hey guys,

I study biophysics, so maybe this is a sophomoric question. I've done a cursory, albeit fruitless search of the forums and the web, I think I don't know how to properly ask the question...

I was wondering if there is some sort of limit -- I would assume a given mass of an object -- where we would expect to see the quantum mechanical effects on the object begin to diminish, and classical mechanics begin to make more sense?

I would imagine the QM laws apply regardless of mass, but I suspect/know this is not true for observed effects. IE: I have to worry about the superposition of a very small particle, but not a massive one.

Can anyone enlighten me? I'm not scared of math.

Cheers,
George

The answer is often given via the formula $\lambda=\frac{h}{p}$ which gives the wavelength associated to a particle with momentum p. As the momentum of the particle gets larger, its wavelength(Maybe its better to say the mid-wavelength of its wavelength spectrum or any other representative of its wavelength spectrum) gets smaller and so the quantum effects become harder to observe and the particle becomes more classical.But its not like Lorentz transformations that you have an equation for which you can take the limit and get the classical formulas.Its somewhat qualitative in this way.
But if you really want to see the transition,you should understand Ehrenfest theorem.

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Makes sense. So in practice, is there a rough dividing between QM and CM in terms of λ?

Obviously if I tried to use a beam of coconuts for the double-slit experiment, their λ would be too small and we would expect classical results.

I guess I'm looking for a ballpark, qualitative sort of idea.

protoplast3r said:
Makes sense. So in practice, is there a rough dividing between QM and CM in terms of λ?

Obviously if I tried to use a beam of coconuts for the double-slit experiment, their λ would be too small and we would expect classical results.

I guess I'm looking for a ballpark, qualitative sort of idea.

The point is that nowhere in physics you can find such "rough dividing"s!
It all depends on the desired accuracy. A problem can be encountered by two groups working in different fields.One of them may need a low enough accuracy to let them use classical mechanics but the other may want a level of accuracy which only using QM can give them.Its the same about other theories and also numerical and analytical approximations.
So the only thing I can say is, in this respect,you can divide physical phenomena into three groups.Group 1:Phenomena for which everyone uses classical physics because quantum effects are practically unobservable for them.Group 2:Phenomena for which everyone uses QM because classical physics is too wrong for them! Group 3:Phenomena which lie between the first two groups and both classical and quantum physics can be used for them and only the application and desired accuracy tells which one should be used.Even the range of such phenomena isn't sharp!

Conceptually it's much more complicated than what's being implied here. I don't know if you care about all the conceptual subtleties involved in the classical limit of QM (think Schrodinger's cat) so I won't mention anything further, in avoidance of undesired complication.

That being said, the general idea is that the center of mass motion of a macroscopic body can be described by a Gaussian wave packet sufficiently localized in both position and momentum (since the spreading of the wave packet due to Schrodinger's equation will be negligible for macroscopic bodies we can on these length scales find approximate simultaneous eigenstates of both position and momentum on the order of said length scales) so that the classical phase space trajectory of the center of mass can then be described by the Ehrenfrest relations for the time evolution of the ensemble average for both the position and momentum operators in formal analogy with Hamilton's equations. But as noted this is not without its conceptual issues.

EDIT: Chapter 6 of Shankar "Principles of Quantum Mechanics" has a very detailed discussion of the classical limit in terms of the Ehrenfest relations for a Gaussian wave packet but again the conceptual issues are completely sidestepped.

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StevieTNZ said:
In principle, QM applies to macroscopic objects as well as microscopic systems.

For example: http://www.nature.com/news/entangled-diamonds-vibrate-together-1.9532

“Neither the statement 'this diamond is vibrating' nor 'this diamond is not vibrating' is true.”

Logic fail.

But i give them credit for trying to circumvent saying that the diamons is both vibrating and not vibrating.

Maui said:
“Neither the statement 'this diamond is vibrating' nor 'this diamond is not vibrating' is true.”

Logic fail.

But i give them credit for trying to circumvent saying that the diamons is both vibrating and not vibrating.

Well I put it to you, seeing you seem to be questioning a lot of Logic, how to describe the diamonds when they're entangled.

protoplast3r said:
I was wondering if there is some sort of limit -- I would assume a given mass of an object -- where we would expect to see the quantum mechanical effects on the object begin to diminish, and classical mechanics begin to make more sense?

Its actually got nothing to do with size - all objects show quantum effects. There is a technological limit on detecting it depending on size - but that is a moving target depending on exactly where we are technologically - however as a matter of principle its there always regardless of size.

What its got to do with is interaction with the environment which via decoherence converts superpositions to improper mixed states giving the appearance of definite properties, and is how the classical world emerges.

Depending on your background the books to get are:
https://www.amazon.com/dp/0691004358/?tag=pfamazon01-20
https://www.amazon.com/dp/3540357734/?tag=pfamazon01-20

Omnes book is at the lay or beginning QM level.
Schlosshauer is THE book on the issue if you have a good background in QM.

Thanks
Bill

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Maui said:
“Neither the statement 'this diamond is vibrating' nor 'this diamond is not vibrating' is true.”

Logic fail.

But i give them credit for trying to circumvent saying that the diamons is both vibrating and not vibrating.

https://en.wikipedia.org/wiki/Three-valued_logic

DrewD said:

Well, it's been customary to assume that quantum states of elementary particles reflect indefinite knowledge on part of the observer(i.e. states are not real in any sense) but this is about states of classical diamonds, i.e. classical matter? While it may make some sense to some that a quantum particle has no definite position outside the measurement context(i.e. does not exist prior to measurement) , it makes no sense to sense to say that a diamond cannot be said to be vibrating while it's vibrating(here we must acknowldge some reality to diamonds being classical - we separate the world into classical and quantum stuff to make the theory work anyway).

So you're saying that there is a divide between QM and CM, that macroscopic objects don't obey Quantum Theory?

I don't think that is the case. It's a bit like trying to make a distinction between relativistic and non relativistic objects. Relativity applies to all of them it's just that the effects are not very noticeable for the non relativistic ones. In QM if the energies (and characteristic frequencies) are very high it is difficult to observe interference effects on a macroscopic scale.

StevieTNZ said:
So you're saying that there is a divide between QM and CM, that macroscopic objects don't obey Quantum Theory?
That's the standard treatment for getting definite outcomes and it obviously leaves a lot to be desired.
Jilang said:
I don't think that is the case. It's a bit like trying to make a distinction between relativistic and non relativistic objects. Relativity applies to all of them it's just that the effects are not very noticeable for the non relativistic ones. In QM if the energies (and characteristic frequencies) are very high it is difficult to observe interference effects on a macroscopic scale.
Mass seems to be a fundamental factor in 'classicality' and it's visible in the relationship between momentum and wavelength.

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Jilang said:
I don't think that is the case. It's a bit like trying to make a distinction between relativistic and non relativistic objects. Relativity applies to all of them it's just that the effects are not very noticeable for the non relativistic ones. In QM if the energies (and characteristic frequencies) are very high it is difficult to observe interference effects on a macroscopic scale.

There's a huge conceptual difference between the two so the analogy isn't really fair. Relativity and Newtonian mechanics are both classical theories so the formal mathematical limit isn't divorced from the conceptual aspects of the limit. The same is obviously not true of the classical limit of QM wherein the classical limit via Ehrenfest's theorem engenders a divide between mathematics and concept. While we can easily set up Gaussian wave packets whose centroids mathematically satisfy an average form of Hamilton's equations through the Ehrenfest relations so as to yield an approximately classical definite phase space trajectory of the centroid or in other words a set of centroid trajectories associated with a classical ensemble, and make sense of this in the limit as characteristic length scales get large, it's clearly not a sufficient limiting scheme because we can easily have macroscopic systems exhibiting quantum behavior by means of entanglement. The degree of entanglement and the potentiality to decohere through environmental interactions provides a better characteristic scale of the quantum/classical tendencies of a system.

Indeed. The smaller and faster objects have a longer wavelength and exhibit more quantum effects than the heavier slower ones. There is probably a really useful formula as to what the relationship has to be between between the rest mass and the momentum so that quantum effects become important.

Any one? ...

It's p=m.v

Wouldn't h need to be in there somewhere too?

Jilang said:
Wouldn't h need to be in there somewhere too?
Yes indeed, p=m.v is useful only for massive particles.

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Jilang said:
Wouldn't h need to be in there somewhere too?

Definitely. My stat mech teacher made a big deal about this, but unfortunately I don't remember the specifics. We were usually interested in ##\lambda## vs particle separation, but he also gave a very nice (not always rigorous) way of testing for "quantumness". It had something to do with a limit as ##\hbar## goes to zero, but I don't really remember since it is usually fairly obvious where the line is at the Undergrad level. Since ##\hbar## is an action, I want to say that it had to do with a Lagrangian limit, but I really don't remember (it made so much sense that I didn't write it down!)

P.S.
https://en.wikipedia.org/wiki/Quantum_logic
logic isn't as easy when the real world is involved.

If there are no takers... I'll give it a go!
Kx -wt > pi.
Px-Et >h (approx.)

In the non-relativistic limit:
Mvx - mc^2 x/v> h
Mc^2 x/v (v^2/c^2 - 1)>h

Where x is the scale on which the effects are being observed. Apologies I don't have Latex.
I'm braced to be shot down in flames, but it looks about right. It can be massively improved of course by making the distance along the axis of motion much longer than the transverse axis.

## 1. What is the difference between Quantum Mechanics (QM) and Classical Mechanics (CM)?

Quantum Mechanics and Classical Mechanics are two different theories used to describe the behavior of matter and energy. Classical Mechanics is used to explain the behavior of large objects, while Quantum Mechanics is used to explain the behavior of subatomic particles.

## 2. What are the key principles of Quantum Mechanics and Classical Mechanics?

The key principles of Quantum Mechanics are superposition, uncertainty, and entanglement. Superposition states that particles can exist in multiple states simultaneously, uncertainty states that it is impossible to know the exact position and momentum of a particle at the same time, and entanglement describes the correlation between particles even when they are separated. The key principles of Classical Mechanics are determinism, causality, and locality. Determinism states that the future state of a system can be predicted based on its present state, causality states that every event has a cause, and locality states that objects can only be influenced by their immediate surroundings.

## 3. How do Quantum Mechanics and Classical Mechanics differ in their predictions?

Quantum Mechanics predicts that particles can exist in multiple states simultaneously, while Classical Mechanics predicts that particles have a definite position and momentum at all times. Additionally, Quantum Mechanics predicts that there is a fundamental limit to the precision of measurements, while Classical Mechanics does not have this limitation.

## 4. Can Quantum Mechanics and Classical Mechanics be reconciled?

Currently, there is no complete theory that can reconcile Quantum Mechanics and Classical Mechanics. However, some theories, such as Quantum Field Theory, attempt to bridge the gap between the two by incorporating elements of both theories.

## 5. How does the division between Quantum Mechanics and Classical Mechanics affect our understanding of the universe?

The division between Quantum Mechanics and Classical Mechanics highlights the limitations of our current understanding of the universe. While Classical Mechanics has been incredibly successful in explaining the behavior of macroscopic objects, it fails to accurately describe the behavior of subatomic particles. This division challenges scientists to continue to explore and develop new theories to better understand the fundamental nature of the universe.

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