# Newtonian Mechanics and Quantum mechanics

Why isn't Newtonian mechanics valid in Quantum world? Suppose you isolate an alpha particle and accelerate it in absolute vacuum. Why it doesn't follow the equation $F=ma$? If Newtonian mechanics is invalid in quantum world, what is the guarantee that Quantum mechanics is valid in macroscopic world?

• Mar 23 '13 at 14:33

Why isn't Newtonian mechanics valid in Quantum world?

The answers to "why" questions in physics end up in "because it has been observed to be so"

When science progressed into the realm of the microscopic, of dimensions the size of an atom, i.e. less than a nanometer, it was observed that newtonian mechanics and classical electrodynamics were in contradiction with experiments, could not explain them. For example, they could not explain :

2) The table of elements which showed regularities unexpected by a simple atomic (a la Demokritos) nature following newtonian mechanics and classical electricity

3) The light spectra. The existence of the hydrogen atom which forced the quantum mechanical view finally, because a differential equation was found which completely described the energy levels seen in the light spectrum of the atom. There was no explanation using using classical electromagnetism and newtonian mechanics

4) Interference effects seen in particles, like electrons, as if they were waves: individual electrons passing through slits showed an intensity pattern appropriate to waves not to newtonian particles

Suppose you isolate an alpha particle and accelerate it in absolute vacuum. Why it doesn't follow the equation $F=ma$?

At the microscopic level, forces don't have a meaning, because nothing touches directly anything else. There are intermediate force carriers of what is perceived as "force" macroscopically.

If Newtonian mechanics is invalid in quantum world, what is the guarantee that Quantum mechanics is valid in macroscopic world?

The guarantee that macroscopically the newtonian mechanics and classical electrodynamics appear as we have validated them experimentally is that all of quantum mechanical behavior rests on h_bar, a very small number which is irrelevant for the distances and energies we move and observe macroscopically. There is a smooth mathematical transition from the QM regime to the classical regime.

• What is that mathematical transition from the QM regime to the classical regime? I think I misunderstood a point, but I want to mention it, Newtonian mechanics is invalid because it cannot explain photoelectric effect and so on, If the case is that, Can QM give us an expression for time period of simple pendulum leading her to be a perfect science? Mar 25 '13 at 19:28
• Here is an advanced article of how classical fields emerge from the quantum mechanical mathematical regime , but it cannot be understood at the highschool level How classical fields, particles emerge from quantum theory . Generally in the build up of theories of physics, even classical physics, there are regimes of application. In each regime powerful mathematical tools have been developed that describe the experiments for that regime and thus are validated. Example: Newtonian Mechanics for many particles leads to statistical mechanics and thermodynamics, in a mathematically smooth fashion. Mar 26 '13 at 5:08
• Each regime is described with different axiomatically consistent mathematical theories. Quantum mechanics is the regime of the very small, the relative sizes defined by the Heisenberg Uncertainty Principle. Macroscopically it makes little sense to use quantum mechanical mathematical tools to describe macroscopic physics . It is like looking at an elephant with a microscope and expect to learn about its family habits. Here is the article that describes how one goes from quantum mechanical to classical fields motls.blogspot.gr/2011/11/… . Mar 26 '13 at 5:12
• QM is the underlying layer of Nature on which everything else is built. It is not necessary to use the QM mathematics for macroscopic physics, except for special cases (as superconductivity for example), cases that classical physics fails to predict/explain. Once a proof exists of a transition between regimes the appropriate classical mathematical formulations are valid. Mar 26 '13 at 5:19
• Thank you. I understood your thoughts.It is highly appreciable. Mar 27 '13 at 12:24

Why isn't Newtonian mechanics valid in Quantum world?

Because Newtonian mechanics only describes a subset of classical systems.

First, the classical $F=ma$ is only valid in special cases. A more general classical equation of motion is $F= dp/dt$: one of Hamilton equations of classical mechaniccs. This reduces to the former only when $p=mv$, with $m$ being constant.

An alpha particle is not a quantum particle and thus does not follow the classical laws. Contrary to what the other answer says, the concept of force continues being valid in the microscopic domain. The classical equation $F= dp/dt$ is generalized by the quantum $\mathbf{F}= d\mathbf{p}/dt$: one of Heisenberg equations of the matrix formulation of quantum mechanics. The bold face denote matrices.

It is possible to find an alternative non-matricial expression $F_Q=dp/dt$ using the Bohmian formulation of quantum mechanics. Here $F_Q$ is the quantum force, which consists of the usual classical force plus a purely quantum term which is function of the so-called quantum potential $Q$.

If Newtonian mechanics is invalid in quantum world, what is the guarantee that Quantum mechanics is valid in macroscopic world?

Precisely the Bohmian formulation is very useful to study the classical limit. When the quantum potential Q is negligible one recovers the classical expressions.

• I think what I understood from your post is classical mechanics and quantum mechanics are not step brother and can be explained as a same if we modify theorems. Mar 25 '13 at 19:06
• @kiranadhikari Classical mechanics (CM) can be considered a subset of quantum mechanics (QM). You can formulate QM like CM plus a set of quantum corrections: Quantum Hamilton-Jacobi equation, Quantum force, Phase space quantum mechanics Mar 26 '13 at 19:41

The formal answer is a long story, one requires many lectures on quantum theory to deeply understand how to get "$F=m a$" from the quantum formalism. I will just mention here the most relevant points.

Quantum mechanics can be seen as a generalization of classical mechanics in order to include small (and slow) particles. You can see an abstract in the Wikipedia page about the classical limit, and check the links therein. As you correctly inferred, quantum theory cannot be completely inconsistent with classical theory.

The first way to get "classical results" (e.g. $F = \frac{d p}{d t} = m a$) from the quantum world is called the Ehrenfest theorem.

Another idea is to "expand" wave functions in $\hbar$ as in the solution of the Schroedinger equation using the WKB approximation. This is called the semiclassical approximation.

One more way to go to the classical regime that deals with the dynamics of the evolution can be found when you learn that the Liouville equation can be obtained from the Moyal equation in the limit when $\hbar$ goes to 0. From the Liouville equation, again, you can get "$F = m a$".

See this post to understand better how the transition quantum -> classical physics could be made.

For more information you can search the terms I marked for you in bold.

It's not that the position of the particle won't change the way that Newtonian mechanics predicts. It's that particles don't have well-defined positions in the first place. The uncertainty in position times the uncertainty in momentum must always be greater than a constant. The constant is very small, so for anything that's not microscopic Newtonian physics is a good enough approximation, but it's not perfect.

We know quantum mechanics is valid in the macroscopic world because we have experimentally shown Newtonian mechanics to be valid in the macroscopic world, and that quantum mechanics makes indistinguishable predictions in the macroscopic world.

The answer is very simple. Newtonian mechanics can't explain observations in the quantum world. Why? - because Classical physics (which contains Newtonian mechanics) predicts the phenomenon of UV catastrophe - it can't explain photoelectric effect - it can't explain black body radiation - it can't explain stability of atoms - it can't explain interference of electrons - the list increased as time passed. You should know that,

no theory could be proved to be 100% true unless you know everything about the universe. So its better we have theories which explains observations the best. No matter how successful a theory just one counter observation could prove it wrong. This the principle of falsifiability. This is exactly why we have discarded Newtonian theory for explaining quantum phenomena.

It seems to me that the answers given to the question that has been posed are not correct at all and do not address the question: Is quantum mechanics valid in in the classical limit? The answer is NO. The example of sending an electron in a vacuumtube is actually a very good example. It turns out that the differential equations of Quantum Mechanics are NOT able to correctly describe the experiment where you would measure after a very specific and predictable time a little flash on the screen. According to QM the wave function of the electron would spread out. Experiments show that the electron can be tracked in its trajectory. Quantum mechanic's differential equations are simply not cut out to be descriptive of this trajectory.

• This is not correct, QM is valid in the classical limit as @Rol pointed out below. So, in your given example, the wave function of the electrons would interfere destructive and we get the classical result.
– Moe
Jan 20 '17 at 18:45