# Does classical mechanics apply to the quantum world at all?

I know we can't use classical mechanics to describe or measure the quantum. That is not what I'm asking. I am asking whether particles still follow the same rules like action/reaction if there is a force involved.

If electron A interacts with electron B, is Newton's 3rd law still being applied to both particles, even though there's no way to predict or measure this? Like, is it just still "happening"? Because how else can *anything* change it's velocity without a previous force involved?

Dale
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Quantum mechanics makes heavy use of the Lagrangian and Hamiltonian formulations of physics, which started in classical mechanics.

bhobba and Galactic explosion
DennisN
2020 Award
"Does classical mechanics apply to the quantum world at all?"

@Galactic explosion:

Two examples: conservation of energy and conservation of momentum applies to both the "classical" and "quantum world". A particle or system interacting with another particle or system will conserve the total energy and momentum of the entire combined system.

Galactic explosion
Quantum mechanics makes heavy use of the Lagrangian and Hamiltonian formulations of physics, which started in classical mechanics.

Interesting. I've always heard that you can't exactly combine quantum with classical, but it's good to know that they are tied together.

Two examples: conservation of energy and conservation of momentum applies to both the "classical" and "quantum world". A particle or system interacting with another particle or system will conserve the total energy and momentum of the entire combined system.

It brings me light to know that both are compatible. Because I can't count how many times I've heard "forget about Newton's laws in the quantum realm" and the such.

atyy
It brings me light to know that both are compatible. Because I can't count how many times I've heard "forget about Newton's laws in the quantum realm" and the such.

It is true that one does not have a direct analogue of Newtons' third law in QM. Also, in the standard interpretation of quantum mechanics, it is not too useful to imagine particles having positions that change with time. One deals instead with the generalization of the third law, which is momentum conservation. Classical mechanics can be recovered as an approximation to quantum mechanics, ie. classical mechanics is the classical limit of quantum mechanics. As @Dale mentioned above, the Hamiltonian and Lagrangian formalisms provide the most direct connections between quantum and classical mechanics.

In the Hamiltonian formalism, the commutator of quantum mechanics becomes the Poisson bracket of classical mechanics. It's a tricky derivation, but @samalkhaiat has a marvellous insight on it: https://www.physicsforums.com/insights/the-classical-limit-of-quantum-mechanical-commutator/.

In the Lagrangian formalism, evaluating the quantum path integral (also known as Feynman's sum-over-all-paths) with a saddlepoint approximation gives the classical path obtained by extremizing the action: http://www.columbia.edu/~xx2146/On Field Theory/path integral and WKB.pdf.

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Galactic explosion and Dale
PeroK
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It brings me light to know that both are compatible. Because I can't count how many times I've heard "forget about Newton's laws in the quantum realm" and the such.

You have Ehrenfest's theorem: that the expectation values of observables obey classical laws. The theorem for position, momentum and potential is here:

https://en.wikipedia.org/wiki/Ehrenfest_theorem

Also, for example, you have the Larmor precession. The spin of charged particle in a magnetic field precesses at a certain frequency. In QM if you measure the spin of a particle you always get, for a spin 1/2 particle for example, the values ##\pm\frac{\hbar}{2}##. Any single measurement of the spin about any axis can only give one of these two values. This is archetypal quantisation. However, if you repeat the experiment many times and measure the spin (about the x-axis say) at different times, you find the expectation value changes with time, with a factor of ##\cos(\omega t)##. Where the angular frequency ##\omega## depends on the particle's magnetic dipole moment in accordance with classical electrodynamics

To explain how this works consider a quantity (spin) that can take only the values ##\pm \frac{\hbar}{2}##. And, consider that at a given time the probability of getting a measurement of ##+ \frac{\hbar}{2}## is ##p##. The expected value of the spin measurement at that time is:

##\langle S_x \rangle = p(+\frac{\hbar}{2}) + (1-p)(+\frac{\hbar}{2}) = (2p - 1)\frac{\hbar}{2}##

For example, if ##p = \frac 1 2##, then ##\langle S_x \rangle = 0##. It's impossible to measure a spin of zero, but if you get ##\pm## with equal probability, then the expected value is ##0##

Now you can see that the expected value can take any value between the two extremes. What's happening in QM is that the probability is changing with time, and this can produce the results with classical analogues.

Galactic explosion
Classical mechanics is an approximation of QFT. Still useful in most contexts but never fully correct. You must use averages to get 'classical' mechanics.

Galactic explosion, vanhees71 and PeroK
vanhees71
Gold Member
There's no classical world. According to what we know today, a classical world is not a consistent description of matter. Already the hydrogen atom is not consistently describable by classical mechanics and electrodynamics since it couldn't be stable to begin with.

Classical mechanics (as well as classical electrodynamics) is an effective coarse-grained description of macroscopic bodies describing the relevant macroscopic degrees of freedom as an average over many irrelevant microscopic degrees of freedom as, e.g., in quantum kinetic theory, which can be derived from quantum many-body theory.

In this sense there's no contradiction between quantum and classical theory, the latter being an effective description of macroscopic bodies. It's analogous to relativity: Newtonian physics is also an effective approximate description in situations where the corresponding non-relativistic approximations are close to the full relativistic descriptions.

Galactic explosion
ZapperZ
Staff Emeritus
It brings me light to know that both are compatible. Because I can't count how many times I've heard "forget about Newton's laws in the quantum realm" and the such.

Actually, you DO have to forget Newton's laws in the quantum realm. The principle of conservation of momentum, conservation of energy, etc. are not "classical physics", and are not the sole properties of classical physics.

Now it doesn't mean that Newton's laws are no longer valid. In fact, whenever there are any QM description of anything, that description needs to merge to the classical description in the classical limit. But the QM approach to describing something is certainly different than the classical approach. All you need to do to convince yourself of this is to find the classical equation of motion of a system, and then do the same thing in QM where you arrive at the QM wavefunction. How are they the same-looking beast?

The FACT that they look different, and the fact that we teach QM in a separate class from classical physics means that they ARE different! It just doesn't mean that they have no connection to each other.

Zz.

Galactic explosion
vanhees71
Gold Member
The question is, what's the essence of "Newtonian physics"! From a modern point of view it's the Gailei-Newton spacetime model, and it's very unwise to forget about it to begin with QM. To the contrary, you should learn group theory, its representation in terms of canonical transformations of Hamiltonian analytical mechanics and then apply it to understand, why non-relativistic QM looks the way it looks. It's the only chance to really understand non-relativistic QM!

bhobba, Galactic explosion and weirdoguy
bhobba
Mentor
Interesting. I've always heard that you can't exactly combine quantum with classical, but it's good to know that they are tied together.

You really need to see a more advanced treatment of Classical Mechanics that also uses Noether's Theorem. Interestingly if you have done calculus then their is a book used first year at Harvard - Morin - Introduction to Classical Mechanics that covers it:
https://www.amazon.com/dp/0521876222/?tag=pfamazon01-20

Many of the problems are really really hard, but heaps of them are all worked out in full detail - the text itself however is explained very well. In fact, and I do not know if any High School actually does it, but the author states it can be used at High School where he thinks students will - in his words - find it a hoot. I must also mention however after reading that you are primed for what was to me a magical experience that turned me from math to physics - Landau - Mechanics:
https://www.amazon.com/dp/0750628960/?tag=pfamazon01-20
You simply do not see reviews like this for other physics books with the added advantage - its true:

'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 underly 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 Lagragian. 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.'

Now we know why classical mechanics has Lagrangian's - it follows for Feynman's Path Integral approach - but why Quantum Field Theories do is a bit of a mystery - they do - but exactly why I do not think anyone knows. But since they do, and we have Noether's Theorem many of the things in Classical Mechanics follows over to QM.

Dirac was particularly intrigued by this:
http://physics.bu.edu/~youssef/quantum/DiracRMP1945.pdf

Thanks
Bill

vanhees71 and Galactic explosion
bhobba
Mentor
The question is, what's the essence of "Newtonian physics"! From a modern point of view it's the Gailei-Newton spacetime model, and it's very unwise to forget about it to begin with QM.

Landau, with his usual terseness explains this. Ballentine - QM - A Modern Development derives - not
postulates or gives a hand-wavy justification, but derives Schrodinger's equation from this symmetry. Not the book to start QM with IMHO - I would recommend Susskind for that - but that can lead to Sakurai : Modern Quantum Mechanics which is a good preparation for Ballentine. Depending on how quick you are in picking things up maybe read Griffiths between Susskind and Sakurai. Unfortunately QM is one of those subjects where they tell you little 'fibs' like the wave particle duality at the start, that are removed from more advanced treatments like Ballentine. The master teacher, Feynman, wasn't happy with this, but try as he might could not really see a way out of it.

Thanks
Bill

vanhees71
How QM gives a classical limit is still an open question I would say. Although we can often show we have approximately diagonal mixed states for macroscopic degrees of freedom, this doesn't really provide a proper merging with classical mechanics in the appropriate limit for several reasons such as the ambiguity of decomposition of mixed states. This is still a problem with ongoing work so we'll just have to wait and see.

vanhees71
Gold Member
Isn't this taught in any many-body/statistical physics lecture nowadays (in Germany it's usually the final lecture in the general theory course)?

Do you mean is Decoherence taught in stat mech?

vanhees71
Gold Member
Why do you think a derivation of the Boltzmann equation from quantum many-body theory is not the solution of the problem to understand the classical behavior of macroscopic systems? After all it's a classical transport equation, which you can of "coarse-grain" further to, e.g., hydrodynamics.

Dale
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2020 Award
We remind the recent posters that this is not the interpretations section of the forum. Please keep interpretations-specific comments in the appropriate section. The interpretations-specific argument has been deleted.

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Why do you think a derivation of the Boltzmann equation from quantum many-body theory is not the solution of the problem to understand the classical behavior of macroscopic systems? After all it's a classical transport equation, which you can of "coarse-grain" further to, e.g., hydrodynamics.
My previous posts on this topic made it sound like I was discussing interpretations. For clarity I am not.

So the succinct main points are that the derivation of the Quantum Boltzman Equation is confined to certain limited scenarios. See this review paper:
https://arxiv.org/abs/0904.3911

A particular case is highly non-Markovian situations. So we are a long way off from showing the classical limit in many cases. Of course one might view this as a "dot the i's and cross the t's" situation. If that's the case then I know what you mean that it demonstrates the classical limit.

Another issue is decomposing the density matrix one gets after decoherence effects. Normally the density matrix is essentially diagonal, so we can decompose like:
$$\rho = \sum_{q \in \chi}\rho_{q}$$
where the index set ##\chi## is one of classical properties like the macroscopic position etc and the ##\rho_q## have specific values for these.

The problem is this decomposition is not unique. There are many other index sets/bases which correspond to highly non-classical quantities. To complete a demonstration of the classical limit we would need to show these bases are invalid in some way. There is some work along these lines, but it's not complete. I'm not sure if you're aware of it. @bhobba above is I suspect as Omnès has some papers where he gives plausible arguments for why some of these "weird" bases are irrelevant. V.P. Belavkin gave a more mathematical treatment of the same. There are many others who work on this. Again you might view it as a "dot the i's and cross the t's" situation.

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vanhees71
Gold Member
Of course, if the macroscopic system shows memory effects you have to keep the non-Markovian evolution when deriving the (semi-)classical transport equation. I still don't understand the fuss about the explanation of classical behavior from quantum dynamics. Of course, any approximation scheme has to be carefully checked to be valid for a given concrete situation.

I still don't understand the fuss about the explanation of classical behavior from quantum dynamics.
So first for the Quantum Boltzmann equations it's that it has only been developed for a very small subset of possible classical dynamics.

For the second point regarding the non-unique decomposition of the density matrix after decoherence I would have thought the "fuss" was fairly obvious. The lack of a unique way to decompose the density matrix, including decompositions that are non-classical, prevents it from being viewed as a proper classical limit. Can you explain why you think the lack of a unique decomposition is not a problem as it's not a view I've heard expressed before?

vanhees71
Gold Member
For me almost any of the success of many-body physics (particularly in condensed matter physics which is what's behind also ll kinds of measurement devices) is well understood from the concept of coarse graining, usually going in the way to derive transport equations for appropriately choosen "quasiparticle degrees of freedom", usually making some more approximations around the thermal-equilibrium state, leading to a theory of the constitutive relations within classical or semi-classical models (in the most simple case in terms of linear-response theory and transport coefficients).

I'm not disputing the successes of many body physics. Do you have any reason for thinking one can ignore the ambiguity in decomposing the density matrix. This is well known problem discussed for example in Schlosshauer's book. I'm a bit confused by how you are just ignoring it, is there a reason? Do you believe it to be solved?

vanhees71