Can Newton's laws be explained by Quantum Mechanics?

Factual Questions
1

I searched the archives for this, and hit Google pretty hard. I found plenty of “historical” sites explaining how QM succeeded Newton’s Laws Of Motion (NLOM), how NLOM are merely an approximation at the macroscopic level, etc.

However, I couldn’t find anything that explains how NLOM (which I believe only apply at the macro- level) can be explained by Quantum Mechanics (which I believe only applies at the micro (understatement) level, except for weird things like Bose-Einstein condensates?)

Are NLOM merely statistical, reflecting (approximately) what happens when a macroscopic body’s numerous component particles all follow QM? If you had a supremely intelligent scientist, that had never heard of NLOM but knew QM inside out, would he be able to derive them? (I guess I’m thinking of an analogous process to the way you can derive the “ideal gas” laws by starting with simple assumptions about the molecules in an ideal gas.)

I’ll reproduce NLOM here for everyone’s reference:

  1. “Every object persists in its state of rest or uniform motion in a straight line unless it is compelled to change that state by forces impressed on it.”

  2. “Force is equal to the change in momentum (mV) per change in time.
    (For a constant mass, force equals mass times acceleration. F=ma)”

  3. “For every action, there is an equal and opposite re-action.”

2

I am not a quantum mechanic or chemist, although I have learned a little bit of quantum chem at an introductory level as part of my BSc.

To my understanding, no, quantum mechanics cannot explain newtonian mechanics, and that is exactly where the problem lies - we cannot say where quantum “ends” and newtonian “begins”.

Take the first law, for example. At the macro level (and even then, you can go quite small before things start to change) this law is true, hence it is a law. Your desk isn’t going to suddenly move of its own free will tothe otehr side of the room. If you push on it, it will move, or pull on it, etc - then it will change its location. An electron, as an example, does not behave this way.

Consider your room to be the “location” of your desk in the sense that an atom is the location of an electron. As I said, the desk won’t move around the room - you know with 100% certainty that your desk is [insert location here, eg. in the corner]. You cannot know with 100% certainty the location of the electron inside its “room” of an atom. In the case of hydrogen, which has a single electron, that electron is likely to be found at any location within a sphere around the nucleus, with decreasing probability as the distance increases. To make things more annoying, electrons in certain orbitals have locations where you can know 100% that it is NOT there, but you cannot say where it is. I have never had this issue with my desk (although homework assignments seem to have quantum behaviour in this sense too).

So what I’m saying is that these electrons, and other quantum units, DO move around, and refuse to stay still, even without any outside forces to cause them. Quantum things WILL randlomly change directions without an applied force.

I think the second and third laws holds true, at least as far as I have studied this stuff (Which, as I said, isn’t much). I don’t remember physics enough to think about whether there might be causes for them not to hold true.

The problem with QM is that its all about probabilities, and things can only be half known at any given time, since “the act of observing disturbs the observed” - Heisenburgs principle comes into this - where you cannot know location and velocity simultaneously. You can know both of these for your desk at any given time.

I remember we calculated the wavelength of a soccerball and compared it to that of a photon in one of my classes (Assuming a soccerball was a wave, just like a photon). We concluded that the wavelength was of such a magnitude that NOTHING could actually measure it in the real world - the quantum calculations just didn’t apply AT ALL.

Read this if you haven’t already: http://www.straightdope.com/classics/a1_122.html
I am sure that as I hit Submit, someone who actually knows what they’re talking about will have slid in a response before me and shown everything I have said to be wrong. At the least, its 100% irrelevent to your question. At least I tried.

:slight_smile:

3

Yeah, I hate to say it, but you’re pretty much wrong here. The principal problem is that if you couldn’t get classical mechanics out of quantum mechanics, you’d know that quantum mechanics has to be wrong, since in its realm of applicability classical mechanics works wonderfully. Similarly, we wouldn’t believe general relativity if you couldn’t get classical gravitation out of it.

I’m not even going to begin to try to explain in detail how you get CM out of QM, mind you. But what you’ll want to do is to look at something called the path integral formulation, due to Feynman.

Basically, what happens is that for classical particles, all the quantum effects cancel out, and you’re left with the classical equations of motion. Once you’ve got those, you’ve essentially got Newton’s laws. All particles wind up being classical particles in the limit of Planck’s constant going to zero.

4

Forgive this former (by 25 years) biological science major, but do the two need to be reconciled? Classical Newtonian mechanics are merely approximations useful in the larger scale universe when not dealing with relativistic effects, IIRC. I didn’t think they were fundamental postulates about the nature of reality, as are quantum mechanics and relativity. But I could be mistaken, I often am.

I’ll have to ask my kid. She’s majoring in physics.

QtM

5

Yes, Qadgop, they really do need to be reconciled. Classical physics, as you noted, is just a non-relativistic approximation to quantum mechanics, but in the realm where it’s applicable, it seems to be right. That means that if we were to do the full quantum mechanical and/or relativistic treatment of a problem that classical physics does right, we darn well better get the classical result since the classical result is the right answer.

This is often known in the trade as something along the lines of “checking the correspondence limit,” and it’s very important that one does this when presenting a new theory; the predictions of the theory have to not only explain the new phenomena but also the already explained phenomena that it ought to cover.

6

g8rguy is right, I believe. Quantum mechanics must reduce to classical mechanics when you’re observing a macroscopic effect. It’s called the classical limit. As the quantum number gets very large, quantum solutions become arbitrarily close to classical solutions. It’s called the correspondence principle.

Consider a simple harmonic oscillator (like a mass on a spring). Check out the first two diagrams on the page. If you’re not familiar with classroom quantum physics, then let me explain. We’re plotting probability versus position. So where the graph is high, the particle is likely to be, and where it’s low, it’s unlikely to be.

Classically, the particle is moving fastest in the center and slowest at the ends, so it’s more likely to be found at the ends, where it’s moving slowest. This is represented by the dashed line in the two graphs. Quantum mechanically, it’s a very different picture at quantum number n = 0, the first graph. The particle is most likely to be found at the center, and not likely to be found at the ends. But that’s okay, this is n = 0. They don’t have to correspond.

Look at the second graph, for n = 10. It’s still different, but now it’s looking more like the classical limit. It’s more likely to be found near the ends. Now if you can imagine what n = 10[sup]16[/sup] must look like - 10[sup]16[/sup] peaks. They run together such that a peak is so close to its corresponding valley that they average out. Right to the classical limit.

7

Thanks for the responses. What I understand people are saying is that there is as yet no comprehensive theory reconciling Newtonian mechanics with Quantum Mechanics? I’m suprised - I expected someone to post something like, “It’s simple - F = ma because the total F is the sum of all the little Fs acting on each fundamental particle”. I then realized that this is merely postponing the problem to, Why does F = ma for fundamental particles?

I guess a related question is, why does F = ma? And why should an object at rest stay at rest, and an object in motion remain in motion (etc.) Are these questions more philosophical than physical, like wondering why the ratio of the circumference of a circle to its diameter is constant? Is there some anthropic argument for why Newton’s laws should hold - that we wouldn’t be around to ask the question if the laws for macroscopic objects weren’t just so ?

8

who cares? they are both incomplete theories.

9

DarrenS: What I understand people are saying is that there is as yet no comprehensive theory reconciling Newtonian mechanics with Quantum Mechanics?

No, that’s not at all what’s being said, I don’t think. Quantum Mechanics includes classical Mechanics, so there’s no need to reconcile them. I don’t know if you’re familiar with fuzzy logic, but that’s a good analogue. Fuzzy logic includes and reduces to boolean logic, in the right limits.

Also, your philosophical-type questions are fine, but there’s a problem. There’s nothing wrong per se with asking why does F = ma, but I think you’ll realize that if you follow this line of reasoning, you’ll never be satisfied. You’ll just ask why ad infinitum like some blindly inquisitive 3-year-old. Seriously I’m asking, what kind of answer would you be satisfied with?

10

No, they don’t need to be reconciled. What needs to be done is to combine GR and QP. In other words, provide a quantum theory of gravity.

So far, GR combines gravity and electromagnetism, while QP combines EM, weak force, and strong force.

So QP cannot explain Newtonian physics because of gravity.

11

Ugh. No, no. QM not only gives the classical answer in the limit of high quantum number/small Planck’s constant (as Achernar noted). And QM not only gives the classical answer, it gives the classical equations, in the proper limit; as Achernar also noted, QM reduces to CM.

So the reason that F=ma is that F=ma yields the classical equations of motion and is implied by the Euler-Lagrange equations (the way a real physicist does classical mechanics… :wink: ) and the classical Lagrangian formalism pops out of the quantum mechanical path integral formalism in the right limit. F=ma is included in QM in a highly highly non-trivial way.

Now, what QM cannot do is provide Newtonian gravity; you don’t get F[sub]grav[/sub] = -GmM/r[sup]2[/sup]. This is of course a distinct issue from getting Newton’s laws.

And Phallacy, no one is claiming that QM is the be all and end all of theories. Nevertheless, it happens to be, as far as we can tell, right in the regimes for which it is expected to apply. Similarly, CM happens to be, as far as we can tell, right in the regimes for which it is expected to apply. So long as we believe that QM is right and that it must in principle also be right for macroscopic systems, it damn well has to reduce to CM. If it doesn’t, not only is it incomplete, it’s utterly wrong and cannot even be an approximation to the complete theory.

12

I should add that Newton’s 1st law is really more of a definition of what we mean by “intertial frame of reference” than anything else.

Nevertheless, given that you’re in such a frame, the 1st law is a simple consequence of the second; if there’s no force acting on an object, its momentum is constant, and if its momentum is constant then plainly its velocity is also constant. Thus, Newton’s second law comes from QM and the other laws follow. So if you want to know why F=ma, you really want to know why the equations of QM are what they are. And of course, no one can tell you.

13

I don’t think this comment is reasonable in the least. Quantum mechanics describes every observation ever made on the quantum level. Classical mechanics describes (nearly) every observation made on the classical level. They are both incredibly important and useful models. No one could do any meaningful work in physics without being conversant in these models.

The classical limit of quantum mechanics comes into play because we are dealing with objects in classical mechanics that act preferentially as waves or particles but not both (as in quantum mechanics). There is an extreme attenuation that happens as you enter the classical regime. While it is true that a baseball has a deBroglie wave associated with it, the attenuation of this wave is so ridiculous that it is practically impossible to measure the wavelike properties of the baseball.

What quantum mechanics and the Schrodinger Equation ultimately do is explain how the quantum world interacts. These formulae are actually extensions of classical theories involving concepts of energies. The extension of quantum mechanics into fields are also analogous to their classical counterparts. Basically, force, which is the gradient of the potential is still equal to the time derivative of momentum (Newton’s second law) in quantum mechanics if you’re dealing with the right fields and the right kinds of waveparticles. If you take the gradient of the potential acted on your wavefunction and then you take the time derivative of your momentum operator acted on the wavefunction, there is basically an equivalence there. It’s a bit more complicated than that, but it is an excellent approximation when your wavefunctions are as attenuated as they are for, say, a baseball. The first and third laws happen to be direct consequences of energy conservation which quantum mechanics basically holds to be true, although “energy” is slightly different in quantum mechanics potentials don’t act as walls but rather borders between different regimes allowing for such bizarre behavior as quantum tunneling. Again, in the limit of the baseball, these two laws are for all practical purposes absolutely true.