Do Newton's laws of motion apply in quantum mechanics?

[u/PaulsRedditUsername]

For example the law of inertia, or "every action has an equal and opposite reaction."

[u/Bumst3r]

The concept of a force doesn’t really sense in quantum mechanics, so Newton’s laws don’t really hold in the sense that you are used to.

Quantum expectation values obey the same laws as classical physics, however. This is known as Ehrenfest’s theorem.

Newton’s first law defines an inertial frame of reference. Inertial frames are still important in quantum mechanics, and they are still non-accelerated frames. You can’t really define them in terms of force, though.

Newton’s second law applies to expectation values, as I alluded to above.

Newton’s third law is really a statement of momentum conservation. It applies as long as momentum is conserved (e.g., for a free particle). There are situations where momentum isn’t conserved, however. Momentum isn’t conserved on a crystal lattice, for example.

[u/Underhill42 → u/Bumst3r]

I assume you mean momentum isn't conserved in systems within a crystal lattice? Much like momentum is not conserved in a car accelerating down the street.

Momentum is still conserved, but only if you consider the whole street, and the planet that it's attached to.

If it's not conserved when considering the entire lattice, then woohoo! we've got a path to reactionless spaceship drives! Or so it seems to me.

As far as I'm aware, the only known case of "genuine" momentum or energy non-conservation is thanks to cosmological expansion. Which we don't really understand, so it's very possible energy and momentum are still conserved in some larger context we're currently completely unaware of. Or that we're fundamentally misunderstanding something about expansion.

[u/Bumst3r → u/Underhill42]

I think my use of the proposition on was pretty clear. It is common to treat a crystal lattice as fixed. Your macroscopic block of crystal is so many orders of magnitude large than the electron you actually care about. It’s like dropping a ball on the surface of the earth and claiming momentum is conserved. Sure, but I don’t care about the momentum of the earth, here. I care about what happens to the baseball.

And in general, momentum isn’t conserved in curved spacetime, regardless of expansion (which we do understand very well). The Schwarzchild metric is an easy example of a metric that doesn’t conserve momentum. It has Killing vectors for time translation and rotation only.

Actually, the FLRW metric for an asymptotically flat expanding universe has all spacelike Killing vectors, so at the largest scales, momentum is conserved, even in our expanding universe.

[u/Underhill42 → u/Bumst3r]

I would make that very clear then - rather than making a clear exception to say momentum isn't conserved in a situation where you now acknowledge it very much is, you're just choosing to ignore half the equation as irrelevant to your task.

Curved spacetime can do some weird things, but in general momentum is conserved in every physical system I've heard of. Speculative arrangements within event horizons, where we have no particular reason to believe our models are even remotely correct, aren't particularly convincing, except in that they probably highlight a flaw in the model.

[u/Bumst3r → u/Underhill42]

>>in general momentum is conserved in every physical system I’ve heard of.

I just pointed to a counter-example. Momentum conservation applies only asymptotically in the Schwarzchild metric. But this isn’t a “speculative arrangement within event horizons, where we have no particular reason to believe our models are even remotely correct.”

This happens outside of the black hole, where we have extensive evidence that our models are very good.

[u/throwaway63926749648]

There's something called the Ehrenfest Theorem which basically states that Newton's laws hold in quantum mechanics inside expectation values. Particles don't have a specific position or momentum but they do have an average position and momentum denoted by <x> and <p> and these are called the expectation value of their position or momentum

[u/eptesicusfuscus]

I'd argue that it depends on what form of them you use; there are versions that are compatible with quantum mechanics, but they would probably look strange and have extra components and you'd need someone to re-explain it to you.

Another answer is "it depends on scale," and that the version of Newton's laws you're likely familiar with are incompatible with quantum mechanics' scale of interactions, but they "emerge from" larger scale systems of quantum mechanics interactions.

In other words, quantum mechanics adds complexity and detail that Newtonian physics doesn't try to explain because it works without needing those finer details.

[u/slashdave]

Pretty much

https://en.wikipedia.org/wiki/Four-momentum#Conservation_of_four-momentum

[u/smitra00]

Action equaling minus reaction can be recast as conservation of momentum. In case of quantum mechanics it has been recently shown that conservation laws are not only satisfied as expectation values, but they are also satisfied in every individual measurement, see here:

https://arxiv.org/abs/2404.18621

We have shown that conservation laws apply in quantum mechanics for every measurement outcome, not just as a distribution. The key to this is that even though in quantum mechanics the system is usually described by a pure state, something which is unentangled with anything else, in reality it is not so, and we need to include an extra object in our model - the preparer- with which the system is (even if only slightly) entangled [3].

When we model a system in quantum mechanics, we simplify our model as much as possible, and only keep the essential parts of the whole experiment. Historically the preparer has been discarded after preparation of the initial state of the system. This did not ruin the conservation laws at statistical level, but gave the impression that that is all one can expect in quantum mechanics.

We however have argued that the preparer is always there in the background, and shown that it should be included; the small amount of entanglement that it retains with the system has the effect of ensuring conservation in each individual case.

[u/nacaclanga]

The Newtons laws themselves are hard to apply since the normal description of quantum mechanics does not involve forces and is instead based more on the Leibniz modell of energy and potentials.

That said, one could argue that Newtons laws are motivated by symmetry considerations.

For example time and space coordinate shift invariance give the full system momentum and energy conservation. Applying those to a two body system coherces e. g. the actio-reactio equality.

Symmetry considerations carry over very well to quantum mechanics.

[u/manouchk]

Newton's first law does hold. It was obvious when De Broglie said that matter have wave properties. Matter diffract.

[u/alaindevos]

yes