Editing Quantum mechanics (section)

=== Classical mechanics ===

The rules of quantum mechanics assert that the state space of a system is a Hilbert space and that observables of the system are Hermitian operators acting on vectors in that space&nbsp;– although they do not tell us which Hilbert space or which operators. These can be chosen appropriately in order to obtain a quantitative description of a quantum system, a necessary step in making physical predictions. An important guide for making these choices is the [[correspondence principle]], a heuristic which states that the predictions of quantum mechanics reduce to those of [[classical mechanics]] in the regime of large [[quantum number]]s.<ref name="Tipler">{{cite book |last1=Tipler |first1=Paul |last2=Llewellyn |first2=Ralph |title=Modern Physics |edition=5th |year=2008 |publisher=W. H. Freeman and Company |isbn=978-0-7167-7550-8 |pages=160–161}}</ref> One can also start from an established classical model of a particular system, and then try to guess the underlying quantum model that would give rise to the classical model in the correspondence limit. This approach is known as [[Canonical quantization|quantization]].<ref name="Peres1993">{{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}}</ref>{{rp|299}}<ref>{{cite magazine |first=John C. |last=Baez |author-link=John C. Baez |url=https://nautil.us/the-math-that-takes-newton-into-the-quantum-world-237339/ |title=The Math That Takes Newton Into the Quantum World |magazine=[[Nautilus Quarterly]] |date=2019-02-26 |access-date=2024-03-23}}</ref>

When quantum mechanics was originally formulated, it was applied to models whose correspondence limit was [[theory of relativity|non-relativistic]] classical mechanics. For instance, the well-known model of the [[quantum harmonic oscillator]] uses an explicitly non-relativistic expression for the [[kinetic energy]] of the oscillator, and is thus a quantum version of the [[harmonic oscillator|classical harmonic oscillator]].<ref name="Zwiebach2022" />{{rp|234}}

Complications arise with [[chaotic systems]], which do not have good quantum numbers, and [[quantum chaos]] studies the relationship between classical and quantum descriptions in these systems.<ref name="Peres1993" />{{rp|353}}

[[Quantum decoherence]] is a mechanism through which quantum systems lose [[quantum coherence|coherence]], and thus become incapable of displaying many typically quantum effects: [[quantum superposition]]s become simply probabilistic mixtures, and quantum entanglement becomes simply classical correlations.<ref name="Zwiebach2022" />{{rp|687-730}} Quantum coherence is not typically evident at macroscopic scales, though at temperatures approaching [[absolute zero]] quantum behavior may manifest macroscopically.{{refn|group=note|See ''[[Macroscopic quantum phenomena]]'', ''[[Bose–Einstein condensate]]'', and ''[[Quantum machine]]''}}

Many macroscopic properties of a classical system are a direct consequence of the quantum behavior of its parts. For example, the stability of bulk matter (consisting of atoms and [[molecule]]s which would quickly collapse under electric forces alone), the rigidity of solids, and the mechanical, thermal, chemical, optical and magnetic properties of matter are all results of the interaction of [[electric charge]]s under the rules of quantum mechanics.<ref>{{cite web |url=http://academic.brooklyn.cuny.edu/physics/sobel/Nucphys/atomprop.html |title=Atomic Properties |publisher=Academic.brooklyn.cuny.edu |access-date=18 August 2012}}</ref>