I 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

(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)?

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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

Gold Member
...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

Gold Member
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.

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.

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.

"Let's talk about the classical limit of QM"

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