# Are newton's laws also an approximation?

by Avichal
Tags: approximation, laws, newton
 P: 278 So are newton's laws also an approximation to quantum phenomena. Can it be derived from quantum laws?
 P: 204 You may be interested in reading about the Correspondence Principle: http://en.wikipedia.org/wiki/Correspondence_principle
 P: 204 To point to a few specifics, there seems to be a few possible interpretations, though I imagine someone else here could tell you more about the current consensus: "Once the Schrödinger equation was given a probabilistic interpretation, Ehrenfest showed that Newton's laws hold on average: the quantum statistical expectation value of the position and momentum obey Newton's laws." "Because quantum mechanics only reproduces classical mechanics in a statistical interpretation, and because the statistical interpretation only gives the probabilities of different classical outcomes, Bohr has argued that classical physics does not emerge from quantum physics in the same way that classical mechanics emerges as an approximation of special relativity at small velocities. He argued that classical physics exists independently of quantum theory and cannot be derived from it. His position is that it is inappropriate to understand the experiences of observers using purely quantum mechanical notions such as wavefunctions because the different states of experience of an observer are defined classically, and do not have a quantum mechanical analog."
Thanks
P: 2,093

## Are newton's laws also an approximation?

Ehrenfest's theorem is not saying that the average values obey Newton's laws, which is wrong! This is only the case for the motion in a harmonic-oscillator potential or in a constant force field. Ehrenfest's theorem says
$$\frac{\mathrm{d}}{\mathrm{d} t} \langle A \rangle =\frac{1}{\mathrm{i} \hbar}\langle [\hat{A},\hat{H}] \rangle,$$
where $A$ is a not explicitly time dependent observable. For momentum you find
$$\frac{\mathrm{d}}{\mathrm{d} t} \langle \vec{p} \rangle =-\langle \vec{\nabla} V(\hat{\vec{x}}) \rangle.$$
Except for a constant force or a force that is linear in $\vec{x}$ the expectation value on the right-hand side is not the same as $-\vec{\nabla} V(\langle x \rangle)$!
 P: 204 Ahh, I stand corrected. The wikipedia quote was a bit misleading, so thanks for the insight vanhees!
 P: 278 Quite a late reply but anyways ... I did not understand. Why can't newton laws be derived from quantum mechanics? Also what do you exactly mean by force in the context of quantum mechanics?
 P: 46 Maybe WKB approximation could derive Newton's law.
P: 278
 Quote by bobydbcn Maybe WKB approximation could derive Newton's law.
Why "maybe"?
P: 46
 Quote by Avichal Why "maybe"?
I am not sure about that. The WKB approximation will be studied in advanced quantum mechanics (gratuate level). I haven't learnt that part.
 P: 787 First of all, all physics laws are approximations. Secondly, quantum mechanics is closely related to Hamiltonian mechanics, which is a formulation of classical mechanics which is equivalent to Newtonian mechanics, but looks very different. The concept of force is not usually used in Hamiltonian mechanics. Nevertheless, force is dp/dt. WKB could easily be covered in undergraduate quantum.
 P: 278 So is this conclusion made by me right? 1) All the macro-laws are approximation of the underlying micro-laws. 2) In theory we could derive the macro laws accurately from the underlying quantum laws but it would be too complicated.
 P: 787 Well, not all macro-laws can be derived from quantum mechanics. For example, gravity. There's plenty of macro-laws which we don't know how to derive from micro-laws, and there's probably a lot of missing stuff from the micro-laws.

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