Let's talk about the classical limit of QM

[andresB]

The classical limit of QM that have always puzzled me. There are common statement saying that you can recover classical mechanics by taking the limit of h->0 or by taking large quantum numbers. Other times times the Erhenfest theorem or the Madelung/hydrodynamics version of the Schroringer equation are claimed to be the key of the classical limit. However, neither of them seems without problems

https://journals.aps.org/pra/abstract/10.1103/PhysRevA.50.2854https://aapt.scitation.org/doi/full/10.1119/1.4751274(Also, Ballentine "quantum mechanics" book)

Ballentine claims that " generally speaking, the classical limit of a quantum state is not a single classical trajectory, but an ensemble of trajectories". This matches the result that the classical limit of a Wigner function is a version of classical statistical mechanics https://arxiv.org/pdf/1202.3628.pdf

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So, is there a consensus on the issue? Does QM go to classical mechanics or classical statistical mechanics (or perhaps neither)?

 

[Demystifier]

https://arxiv.org/abs/quant-ph/0112005

 

[A. Neumaier]

Does QM go to classical mechanics or classical statistical mechanics

It goes to classical mechanics when typical distances and momenta are such that their products are large compared to Planck's constant $\hbar$. This is called the Ehrenfest theorem.

 

[vanhees71]

...and this is most easily understood when you use the path-integral formalism to see that the main contribution to the propagator comes from the classical path, given by the stationarity of the classical action.

Another more formal way is to use "singular perturbation theory" (in the context of QT known as WKB method, which is pretty similar to the derivation of ray optics from classical electrodynamics through the eikonal approximation) to carefully define the limit $\hbar \rightarrow 0$.

 

[andresB]

It goes to classical mechanics when typical distances and momenta are such that their products are large compared to Planck's constant $\hbar$. This is called the Ehrenfest theorem.

I linked a paper about the Inadequacy of the Ehrenfest theorem to characterize the classical limit.

 

[A. Neumaier]

I linked a paper about the Inadequacy of the Ehrenfest theorem to characterize the classical limit.

Well, one needs additional assumptions to get a true limit : the uncertainty in position must be small enough to replace the expectations of the potenial gradient by the potential gradient of the expected position. Decoherence works with other assumptions.

All assumptions can be criticized, but without assumptions no classicality...

 

[vanhees71]

I think, it's pretty hard to achieve classicality for a single particle. What makes our world looking classical is rather the fact that it consists of many-body systems, and we are looking and pretty much coarse-grained macroscopic observables. Averaging over a lot of microscopic degrees of freedom to define these effective macroscopic observables leads to classical (continuum-mechanics) equations of motion.

Even for a single particle this view works in some cases either. One of the earliest papers on the subject is Mott's analysis of the fact that in a cloud chamber one sees "classical trajectories" of particles originating from a radioactive probe (in his case $\alpha$ particles from nuclear decay). Though the quantum state of the single $\alpha$ particle in this case can be described as a spherical wave, an $\alpha$ particle from a specific decay starts randomly in any direction but then follows a straight line through the cloud chamber. The reason of course is that in a sense the particle is continuously watched through its interactions with the vapour in the chamber, i.e., through the interaction of the microscopic single-particle degrees of freedom (like position and momentum) with the surrounding macroscopic gas. Of course the "trajectory" observed as tracks in the cloud chamber is a pretty coarse grained macroscopic observable.

https://doi.org/10.1098/rspa.1929.0205

 

[lucas_]

I think, it's pretty hard to achieve classicality for a single particle. What makes our world looking classical is rather the fact that it consists of many-body systems, and we are looking and pretty much coarse-grained macroscopic observables. Averaging over a lot of microscopic degrees of freedom to define these effective macroscopic observables leads to classical (continuum-mechanics) equations of motion.

First, I like single particle system and pondering what happens between measurement. Been reflecting it for decades. I just want to extend the concept to macroscopic object. And I want to see what kind of quantumness can still be retained. Because while it is logical to treat macroscopic system as nuts and bolts where Brownian motion or thermal agitations prevail over the entire system (because of many degrees of freedom that make it classical). It seems there is a hidden principle where molecule wide system can have some kind of coherence over ridden the thermal motions. Imagine puppet strings over the molecules. Is this totally impossible? Has anyone read any concept about this?

Even for a single particle this view works in some cases either. One of the earliest papers on the subject is Mott's analysis of the fact that in a cloud chamber one sees "classical trajectories" of particles originating from a radioactive probe (in his case $\alpha$ particles from nuclear decay). Though the quantum state of the single $\alpha$ particle in this case can be described as a spherical wave, an $\alpha$ particle from a specific decay starts randomly in any direction but then follows a straight line through the cloud chamber. The reason of course is that in a sense the particle is continuously watched through its interactions with the vapour in the chamber, i.e., through the interaction of the microscopic single-particle degrees of freedom (like position and momentum) with the surrounding macroscopic gas. Of course the "trajectory" observed as tracks in the cloud chamber is a pretty coarse grained macroscopic observable.

https://doi.org/10.1098/rspa.1929.0205

 

[1977ub]

Though the quantum state of the single αα particle in this case can be described as a spherical wave, an αα particle from a specific decay starts randomly in any direction but then follows a straight line through the cloud chamber. The reason of course is that in a sense the particle is continuously watched through its interactions with the vapour in the chamber, i.e., through the interaction of the microscopic single-particle degrees of freedom (like position and momentum) with the surrounding macroscopic gas. Of course the "trajectory" observed as tracks in the cloud chamber is a pretty coarse grained macroscopic observable.

If the vapour is less dense, and the particle is interacting with vapour less often, is there less of a linear appearance to the trajectory - more erratic?

 

[A. Neumaier]

I linked a paper about the Inadequacy of the Ehrenfest theorem to characterize the classical limit.

Each particular approach to classicality needs additional assumptions; the Ehrenfest theorem is just one of these.

So, is there a consensus on the issue? Does QM go to classical mechanics or classical statistical mechanics (or perhaps neither)?

All three are possible, depending on the system and on the limit taken.

There are multiple ways a quantum system can become approximately classical, each under different conditions, and the paper you linked to points this out and gives some examples.

 

[Ian J Miller]

I think one simple way of looking at this is to consider the wave phase as exp(2πiS/h) S the action which increases with time. If S>>h, then the wavelength shrinks to zero, and even if you continue with the wave representation you have Hamilton's waves, which arise from classical dynamics. If the wavelength is irrelevant, you have classical dynamics.

 

[bhobba]

...and this is most easily understood when you use the path-integral formalism to see that the main contribution to the propagator comes from the classical path, given by the stationarity of the classical action.

The easiest way to see how it comes about is using the Lagrangian formalism (the Principle Of Least Action) which follows directly from the path-integral formalism so close paths cancel except at a stationary point where it reinforces. You may not have seen the whole of classical mechanics derived basically from that alone and symmetry - see Landau - Mechanics if that is of interest. But its a good idea to get that book anyway - you simply do not see reviews on physics books like Amzaon has for that book eg:
https://www.amazon.com/dp/0750628960/?tag=pfamazon01-20

'If physicists could weep, they would weep over this book. The book is devastatingly 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 underlay every important quantity in physics. So instead of starting from concrete mechanical case-studies and generalizing to the formal machinery of the Hamilton equations, Landau starts out from the most generic symmetry and derives the mechanics. The 2nd laws of mechanics, for example, is derived as a consequence of the uniqueness of trajectories in the Lagrangian. For some, this may seem too "mathematical" but in reality, it is a sign of sophistication 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.'

Once you have done multi-variable calculus or studied a book on classical mechanics that includes what you need you have enough background. Such a book if you have done single variable calculus such as Quick Calulus is Morin:
https://www.amazon.com/dp/0521876222/?tag=pfamazon01-20
https://www.amazon.com/dp/0471827223/?tag=pfamazon01-20

Some reviews say its advanced. I think yes and no - its not hard to read and understand, but with each reading you get something new. It is advanced if you want to deeply understand it.

Thanks
Bill

 

[Heikki Tuuri]

Let us think about a crystal of a classical size: 6 * 10^23 atoms (= 1 mol).

The atoms are in a lattice and it is very improbable that a single atom is able to jump out of the lattice. Each atom is almost classical itself.

If we throw the crystal in the air, it will move like a classical object. How do we describe this in terms of quantum mechanics?

Maybe it is best to change to co-moving coordinates of the crystal. Then we have a crystal hovering still in space, and we should show in quantum mechanics that it stays there still like a classical object. The uncertainty of its position and momentum are negligibly small.

Each atom is bound in the collective potential of neighboring atoms. The uncertainty in the position of each atom is very small.

Quantum mechanics conserves momentum and energy. The thermal energy in the crystal is enough to break it in two parts and shoot the parts at a large speed to opposite directions. This is not a phenomenon of quantum mechanics but classical statistical physics. One may ask why we only see the main branch of the Many Worlds and not the minor branches where the crystal explodes?

The classicality problem turns out to be the familiar problem why I, as an observing subject, feel that I live in a "typical" branch of the Many Worlds. Why I do not somehow see the very improbable branches simultaneously with the main branch?

Bohmian mechanics assigns a marker to each particle in the Schrödinger equation. The "real" branch where I live is the branch singled out by these markers. All the other branches do exist and interfere with my branch, and affect the trajectories of the particle markers. Those other branches just are not the branch where I live.

Bohmian mechanics solves the classicality problem in this case. The branch singled out by the markers behaves almost always like a classical system.

We know that extending Bohmian mechanics to relativistic quantum mechanics is hard or impossible. We do not then have a fixed set of particles.

Let us again return to the question why I do not see the improbable branches simultaneously with the main branch. The improbable branches do exist and affect history. What would it mean to "see" them? We do see them to the very small extent that they affect our branch. Since the interaction is minuscule, the markers of the atoms in my body are not much affected by the improbable branches.

In the double slit experiment we "see" two branches of the Many Worlds and their interference pattern on the screen. My mental image of the experiment contains two alternative histories.

 

[bhobba]

If we throw the crystal in the air, it will move like a classical object. How do we describe this in terms of quantum mechanics?

Its simple. If follows directly from the path integral formalism. Here is how its done.

You consider the crystal as a single onect. Start out with <x'|x> then you insert a ton of ∫|xi><xi|dxi = 1 in the middle to get ∫...∫<x|x1><x1|...|xn><xn|x> dx1...dxn. Now <xi|xi+1> = ci e^iSi so rearranging you get
∫...∫c1...cn e^ i∑Si.

Focus in on ∑Si. Define Li = Si/Δti, Δti is the time between the xi along the jagged path they trace out. ∑ Si = ∑Li Δti. As Δti goes to zero the reasonable physical assumption is made that Li is well behaved and goes over to a continuum so you get ∫L dt.

Now Si depends on xi and Δxi. But for a path Δxi depends on the velocity vi = Δxi/Δti so its very reasonable to assume when it goes to the continuum L is a function of x and the velocity v.

In this way you see the origin of the Lagrangian. For macro objects by considering close paths we see most cancel and you are only left with the paths of stationary action from which all of classical mechanics follows.

There are no issues of interpretation here - its clear cut. There are issues about explaining a classical world ie how does solidity come about (first figured out by Dyson) and defining quantum mechanically what an observation is (not completely worked out yet - but considerable progress has been made), These sometimes involve interpretational issues, but accepting classical objects exist, which we know they most certainly do, how classical mechanics emerges is clear cut.

Thanks
Bill