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Are newton's laws also an approximation? 
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#1
Feb1813, 12:11 AM

P: 283

So are newton's laws also an approximation to quantum phenomena. Can it be derived from quantum laws?



#2
Feb1813, 12:57 AM

P: 204

You may be interested in reading about the Correspondence Principle: http://en.wikipedia.org/wiki/Correspondence_principle



#3
Feb1813, 12:59 AM

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." 


#4
Feb1813, 02:27 AM

Sci Advisor
Thanks
P: 2,497

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 harmonicoscillator potential or in a constant force field. Ehrenfest's theorem says
[tex]\frac{\mathrm{d}}{\mathrm{d} t} \langle A \rangle =\frac{1}{\mathrm{i} \hbar}\langle [\hat{A},\hat{H}] \rangle,[/tex] where [itex]A[/itex] is a not explicitly time dependent observable. For momentum you find [tex]\frac{\mathrm{d}}{\mathrm{d} t} \langle \vec{p} \rangle =\langle \vec{\nabla} V(\hat{\vec{x}}) \rangle.[/tex] Except for a constant force or a force that is linear in [itex]\vec{x}[/itex] the expectation value on the righthand side is not the same as [itex]\vec{\nabla} V(\langle x \rangle)[/itex]! 


#5
Feb1813, 02:49 AM

P: 204

Ahh, I stand corrected. The wikipedia quote was a bit misleading, so thanks for the insight vanhees!



#6
Aug2113, 10:20 AM

P: 283

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? 


#7
Aug2213, 06:43 AM

P: 46

Maybe WKB approximation could derive Newton's law.



#8
Aug2213, 10:15 AM

P: 283




#9
Aug2513, 08:25 PM

P: 46




#10
Aug2513, 08:46 PM

P: 887

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. 


#11
Aug2613, 02:07 AM

P: 283

So is this conclusion made by me right?
1) All the macrolaws are approximation of the underlying microlaws. 2) In theory we could derive the macro laws accurately from the underlying quantum laws but it would be too complicated. 


#12
Aug2613, 01:45 PM

P: 887

Well, not all macrolaws can be derived from quantum mechanics. For example, gravity.
There's plenty of macrolaws which we don't know how to derive from microlaws, and there's probably a lot of missing stuff from the microlaws. 


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