Editing Newton's laws of motion (section)

===Quantum mechanics===
[[Quantum mechanics]] is a theory of physics originally developed in order to understand microscopic phenomena: behavior at the scale of molecules, atoms or subatomic particles. Generally and loosely speaking, the smaller a system is, the more an adequate mathematical model will require understanding quantum effects. The conceptual underpinning of quantum physics is [[Bell's theorem|very different from that of classical physics]]. Instead of thinking about quantities like position, momentum, and energy as properties that an object ''has'', one considers what result might ''appear'' when a [[Measurement in quantum mechanics|measurement]] of a chosen type is performed. Quantum mechanics allows the physicist to calculate the probability that a chosen measurement will elicit a particular result.<ref>{{cite journal|last=Mermin|first=N. David|author-link=N. David Mermin|year=1993|title=Hidden variables and the two theorems of John Bell|journal=[[Reviews of Modern Physics]]|volume=65|issue=3|pages=803&ndash;815|arxiv=1802.10119|bibcode=1993RvMP...65..803M|doi=10.1103/RevModPhys.65.803|s2cid=119546199 |quote=It is a fundamental quantum doctrine that a measurement does not, in general, reveal a pre-existing value of the measured property.}}</ref><ref>{{Cite journal|last1=Schaffer|first1=Kathryn|last2=Barreto Lemos|first2=Gabriela|date=24 May 2019|title=Obliterating Thingness: An Introduction to the "What" and the "So What" of Quantum Physics|journal=[[Foundations of Science]] |volume=26 |pages=7–26 |language=en|arxiv=1908.07936|doi=10.1007/s10699-019-09608-5|issn=1233-1821|s2cid=182656563}}</ref> The [[Expectation value (quantum mechanics)|expectation value]] for a measurement is the average of the possible results it might yield, weighted by their probabilities of occurrence.<ref>{{Cite journal|last1=Marshman|first1=Emily|last2=Singh|first2=Chandralekha|author-link2=Chandralekha Singh|date=2017-03-01|title=Investigating and improving student understanding of the probability distributions for measuring physical observables in quantum mechanics|journal=[[European Journal of Physics]]|volume=38|issue=2|pages=025705|doi=10.1088/1361-6404/aa57d1|bibcode=2017EJPh...38b5705M |s2cid=126311599 |issn=0143-0807|doi-access=free}}</ref>

The [[Ehrenfest theorem]] provides a connection between quantum expectation values and Newton's second law, a connection that is necessarily inexact, as quantum physics is fundamentally different from classical. In quantum physics, position and momentum are represented by mathematical entities known as [[Hermitian operator]]s, and the [[Born rule]] is used to calculate the expectation values of a position measurement or a momentum measurement. These expectation values will generally change over time; that is, depending on the time at which (for example) a position measurement is performed, the probabilities for its different possible outcomes will vary. The Ehrenfest theorem says, roughly speaking, that the equations describing how these expectation values change over time have a form reminiscent of Newton's second law. However, the more pronounced quantum effects are in a given situation, the more difficult it is to derive meaningful conclusions from this resemblance.{{refn|group=note|Details can be found in the textbooks by, e.g., Cohen-Tannoudji et al.<ref name="Cohen-Tannoudji">{{cite book|last1=Cohen-Tannoudji |first1=Claude |last2=Diu |first2=Bernard |last3=Laloë |first3=Franck |title=Quantum Mechanics |author-link1=Claude Cohen-Tannoudji |publisher=John Wiley & Sons |year=2005 |isbn=0-471-16433-X |translator-first1=Susan Reid |translator-last1=Hemley |translator-first2=Nicole |translator-last2=Ostrowsky |translator-first3=Dan |translator-last3=Ostrowsky}}</ref>{{Rp|242}} and Peres.<ref>{{cite book|last=Peres|first=Asher|author-link=Asher Peres |title=Quantum Theory: Concepts and Methods |title-link=Quantum Theory: Concepts and Methods|publisher=[[Kluwer]]|year=1993|isbn=0-7923-2549-4|oclc=28854083}}</ref>{{Rp|302}}}}