Force

=== Quantum mechanics ===
{{main|Quantum mechanics|Pauli exclusion principle}}
[[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 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>

In quantum mechanics, interactions are typically described in terms of energy rather than force. The [[Ehrenfest theorem]] provides a connection between quantum expectation values and the classical concept of force, a connection that is necessarily inexact, as quantum physics is fundamentally different from classical. In quantum physics, 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, with a force defined as the negative derivative of the potential energy. However, the more pronounced quantum effects are in a given situation, the more difficult it is to derive meaningful conclusions from this resemblance.<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 |page=242}}</ref><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 |page=302}}</ref>

Quantum mechanics also introduces two new constraints that interact with forces at the submicroscopic scale and which are especially important for atoms. Despite the strong attraction of the nucleus, the [[uncertainty principle]] limits the minimum extent of an electron probability distribution<ref name=Lieb-RMP>{{Cite journal |last=Lieb |first=Elliott H. |date=1976-10-01 |title=The stability of matter |url=https://link.aps.org/doi/10.1103/RevModPhys.48.553 |journal=Reviews of Modern Physics |language=en |volume=48 |issue=4 |pages=553–569 |doi=10.1103/RevModPhys.48.553 |issn=0034-6861 |quote=the fact that if one tries to compress a wave function ''anywhere'' then the kinetic energy will increase. This principle was provided by Sobolev (1938)...}}</ref> and the [[Pauli exclusion principle]] prevents electrons from sharing the same probability distribution.<ref name=Lieb-Bulletin>{{Cite journal |last=Lieb |first=Elliott H. |date=1990 |title=The stability of matter: from atoms to stars |url=https://www.ams.org/bull/1990-22-01/S0273-0979-1990-15831-8/ |journal=Bulletin of the American Mathematical Society |language=en |volume=22 |issue=1 |pages=1–49 |doi=10.1090/S0273-0979-1990-15831-8 |issn=0273-0979 |quote= bulk matter is stable, and has a volume proportional to the number of particles, because of the Pauli exclusion principle for fermions (Le., the electrons). Effectively the electrons behave like a fluid with energy density <math>\rho^{5/3}</math>, and this limits the compression caused by the attractive electrostatic forces. |doi-access=free }}</ref> This gives rise to an emergent pressure known as [[electron degeneracy pressure|degeneracy pressure]]. The dynamic equilibrium between the degeneracy pressure and the attractive electromagnetic force give atoms, molecules, liquids, and solids [[Stability of matter|stability]].<ref>{{cite book|last=Griffiths|title=Introduction to Quantum Mechanics, Second Edition|year=2005|publisher=[[Prentice Hall]]|location=London, UK|isbn=0131244051|pages=221–223}}</ref>