View source for Physics:Quantum state space

{{Short description| Mathematical space representing physical quantum systems}}
In [[Physics:Physics|physics]], a '''quantum state space''' is an abstract [[Space|space]] in which different "positions" represent, not literal locations, but rather [[Physics:Quantum state|quantum state]]s of some [[Physics:Physical system|physical system]]. It is the quantum analog of the [[Physics:Phase space|phase space]] of [[Physics:Classical mechanics|classical mechanics]].

== Relative to Hilbert space ==
In [[Physics:Quantum mechanics|quantum mechanics]] a state space is a [[Hilbert space#Separable spaces|separable]] [[Complex number|complex]] [[Hilbert space]]. The [[Dimension (vector space)|dimension]] of this [[Hilbert space]] depends on the system we choose to describe.<ref>{{Cite book |last=McIntyre |first=David |title=Quantum Mechanics: A Paradigms Approach |publisher=Pearson |year=2012 |isbn=978-0321765796 |edition=1st}}</ref><ref>{{cite book|last1=Bengtsson |first1=Ingemar |last2=Życzkowski |first2=Karol |title=Geometry of Quantum States |edition=2nd |year=2017 |publisher=Cambridge University Press |isbn=978-1139207010}}</ref> The different states that could come out of any particular measurement form an [[Orthonormal basis|orthonormal basis]], so any  [[Physics:Quantum state|state vector]] in the state space can be written as a [[Linear combination|linear combination]] of these basis vectors. Having an nonzero component along multiple dimensions is called a [[Physics:Quantum superposition|superposition]].  In the [[Physics:Mathematical formulation of quantum mechanics|formalism of quantum mechanics]] these [[Physics:Quantum state|state vectors]] are often written using [[Biography:Paul Dirac|Dirac's]] compact [[Bra–ket notation|bra–ket notation]].<ref name=Schiff>{{Cite book |last=Schiff |first= Leonard  |title=Quantum mechanics |publisher= McGraw-Hill |year= 1949}}</ref>{{rp|165}}

== Examples ==

The [[Physics:Spin|spin]] state of a silver atom in the [[Physics:Stern–Gerlach experiment|Stern–Gerlach experiment]] can be represented in a two state space. The spin can be aligned with a measuring apparatus (arbitrarily called 'up') or oppositely ('down').<ref name=QM-minimum>{{Cite book |last=Susskind |first=Leonard |title=Quantum mechanics: the theoretical minimum; [what you need to know to start doing physics] |last2=Friedman |first2=Art |last3=Susskind |first3=Leonard |date=2014 |publisher=Basic Books |isbn=978-0-465-06290-4 |series=The theoretical minimum / Leonard Susskind and George Hrabovsky |location=New York, NY}}</ref>  In Dirac's notation these two states can be written as  <math> |u\rangle, |d\rangle </math>. The space of a two spin system has four states, <math> |uu\rangle, |ud\rangle, |du\rangle, |dd\rangle </math>.

The spin state is a discrete [[Chemistry:Degrees of freedom (physics and chemistry)|degree of freedom]]; quantum state spaces can have continuous degrees of freedom. For example, a particle in one space dimension has one degree of freedom ranging from <math>-\infty</math> to <math>\infty</math>. In Dirac notation, the states in this space might be written as <math>|q\rangle</math> or <math>|\psi\rangle</math>.<ref name=Messiah/>{{rp|302}}

== Relative to 3D space ==
Even in the early days of quantum mechanics, the state space (or configurations as they were called at first) was understood to be essential for understanding simple quantum-mechanical problems. In 1929, Nevill Mott showed that "tendency to picture the wave as existing in ordinary three dimensional space, whereas we are really dealing with wave functions in multispace" makes analysis of simple interaction problems more difficult.<ref>{{Cite journal |date=1929-12-02 |title=The wave mechanics of ∝-Ray tracks |url=https://royalsocietypublishing.org/doi/10.1098/rspa.1929.0205 |journal=Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character |language=en |volume=126 |issue=800 |pages=79–84 |doi=10.1098/rspa.1929.0205 |issn=0950-1207|doi-access=free }}</ref> Mott analyzes [[Physics:Alpha particle|<math>\alpha</math>-particle]] emission in a [[Physics:Cloud chamber|cloud chamber]]. The emission process is isotropic, a spherical wave in quantum mechanics, but the tracks observed are linear. 
[[File:Indeterminacy principle.gif|thumb|left|Spherical wave]]
[[File:AlphaTrackRutherfordScattering3.jpg|thumb|center|Alpha particle track in a cloud chamber]]
As Mott says, "it is a little difficult to picture how it is that an
outgoing spherical wave can produce a straight track; we think intuitively that it should ionise atoms at random throughout space". This issue became known at the [[Physics:Mott problem|Mott problem]]. Mott then derives the straight track by considering correlations between the positions of the source and two representative atoms, showing that consecutive ionization results from just that state in which all three positions are co-linear. <ref>{{Cite journal |last=Figari |first=Rodolfo |last2=Teta |first2=Alessandro |date=2013 |title=Emergence of classical trajectories in quantum systems: the cloud chamber problem in the analysis of Mott (1929) |url=http://link.springer.com/10.1007/s00407-012-0111-z |journal=Archive for History of Exact Sciences |language=en |volume=67 |issue=2 |pages=215–234 |doi=10.1007/s00407-012-0111-z |issn=0003-9519|arxiv=1209.2665 }}</ref>

== Relative to classical phase space ==
[[Physics:Classical mechanics|Classical mechanics]] for multiple objects describes their motion in terms of a list or [[Euclidean vector|vector]] of every object's coordinates and velocity. As the objects move, the values in the vector change; the set of all possible values is called a [[Physics:Phase space|phase space]].<ref>{{Cite book |last=Susskind |first=Leonard |title=The theoretical minimum: what you need to know to start doing physics |last2=Hrabovsky |first2=George |last3=Susskind |first3=Leonard |date=2014 |publisher=Basic Books |isbn=978-0-465-07568-3 |edition=Paperback 1. publ |series=The theoretical minimum / Leonard Susskind and George Hrabovsky |location=New York}}</ref>{{rp|88}} In quantum mechanics a state space is similar, however in the state space two vectors which are scalar multiples of each other represent the same state. Furthermore, the character of values in the quantum state differ from the classical values: in the quantum case the values can only be measured statistically (by repetition over many examples) and thus do not have well defined values at every instant of time. <ref name=Messiah>{{Cite book |last=Messiah |first=Albert |url=https://archive.org/details/quantummechanics0000mess/quantummechanics0000mess |title=Quantum Mechanics |date=1966 |publisher=North Holland, John Wiley & Sons |isbn=0486409244 |language=en}}</ref>{{rp|294}}

==See also==
* {{annotated link|Physics:Quantum mechanics}}
* {{annotated link|Physics:Quantum state}}
* {{annotated link|Physics:Configuration space}}

==References==
{{Reflist}}

==Further reading==
*{{cite book | author=Claude Cohen-Tannoudji |title=Quantum Mechanics | publisher=John Wiley & Sons. Inc. |year=1977|isbn=0-471-16433-X}}
*{{cite book | author=David J. Griffiths | title=Introduction to Quantum Mechanics | publisher=Prentice Hall |year=1995|isbn=0-13-124405-1}}
*{{cite book | author=David H. McIntyre |year=2012 |title=Quantum Mechanics: A Paradigms Approach |publisher=Pearson |isbn=978-0321765796}}

[[Category:Concepts in physics]]

{{Sourceattribution|Quantum state space}}