Are Newton's laws also an approximation?

In summary, Newton's laws may be an approximation to quantum phenomena, but they can still be derived accurately from quantum laws.
  • #1
Avichal
295
0
So are Newton's laws also an approximation to quantum phenomena. Can it be derived from quantum laws?
 
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  • #3
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
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
[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 right-hand side is not the same as [itex]-\vec{\nabla} V(\langle x \rangle)[/itex]!
 
  • #5
Ahh, I stand corrected. The wikipedia quote was a bit misleading, so thanks for the insight vanhees!
 
  • #6
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
Maybe WKB approximation could derive Newton's law.
 
  • #8
bobydbcn said:
Maybe WKB approximation could derive Newton's law.

Why "maybe"?
 
  • #9
Avichal said:
Why "maybe"?

I am not sure about that. The WKB approximation will be studied in advanced quantum mechanics (gratuate level). I haven't learned that part.
 
  • #10
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
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.
 
  • #12
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.
 

1. What are Newton's laws?

Newton's laws of motion are three physical laws that describe the relationship between the forces acting on an object and its motion. They were developed by Sir Isaac Newton in the 17th century and are considered fundamental principles in classical mechanics.

2. What does it mean for Newton's laws to be an approximation?

An approximation is a simplified version of a more complex concept or idea. In the case of Newton's laws, they are considered an approximation because they do not hold true in all situations and are limited by certain conditions such as high speeds or extremely small scales.

3. Why are Newton's laws an approximation?

Newton's laws are an approximation because they do not take into account certain factors such as air resistance, friction, and the effects of relativity. Additionally, they are based on classical mechanics and do not accurately describe the behavior of objects at very high speeds or on a quantum level.

4. How accurate are Newton's laws?

The accuracy of Newton's laws depends on the situation and the level of detail being examined. In everyday situations, they are generally accurate enough to make accurate predictions. However, in more extreme conditions, such as at high speeds or on a very small scale, they may not hold true and more advanced theories are needed to accurately describe the behavior of objects.

5. Can Newton's laws be applied to all objects?

No, Newton's laws are limited in their applicability and may not accurately describe the behavior of all objects. They are most accurate when applied to objects at everyday speeds and scales, but may not hold true for objects moving at very high speeds or at a microscopic level.

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