Physics:Quantum statistical mechanics - Revision history

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	<div style="flex:1; line-height:1.45; color:#006b45; column-count:2; column-gap:32px; column-rule:1px solid #b8d8c8;">		<div style="flex:1; line-height:1.45; color:#006b45; column-count:2; column-gap:32px; column-rule:1px solid #b8d8c8;">
	Quantum statistical mechanics is a Book I topic in the Quantum Collection. It extends statistical mechanics to systems whose microscopic states, observables, and probabilities are governed by quantum theory. Instead of classical phase-space distributions, it uses density matrices, partition functions, ensembles, occupation numbers, and quantum correlations. The subject explains blackbody radiation, Bose-Einstein and Fermi-Dirac statistics, heat capacity, magnetism, phase transitions, and the thermodynamic behavior of many-particle quantum systems.		Quantum statistical mechanics is a Book I topic in the Quantum Collection. It extends statistical mechanics to systems whose microscopic states, observables, and probabilities are governed by quantum theory. Instead of classical phase-space distributions, it uses density matrices, partition functions, ensembles, occupation numbers, and quantum correlations. The subject explains blackbody radiation, Bose-Einstein and Fermi-Dirac statistics, heat capacity, magnetism, phase transitions, and the thermodynamic behavior of many-particle quantum systems. It also clarifies how macroscopic equilibrium emerges from microscopic quantum states and constraints.
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	~~{{Thermodynamics sidebar\|expanded=branches}}~~		Quantum statistical mechanics is a Book I topic in the Quantum Collection. It extends statistical mechanics to systems whose microscopic states, observables, and probabilities are governed by quantum theory. Instead of classical phase-space distributions, it uses density matrices, partition functions, ensembles, occupation numbers, and quantum correlations. The subject explains blackbody radiation, Bose-Einstein and Fermi-Dirac statistics, heat capacity, magnetism, phase transitions, and the thermodynamic behavior of many-particle quantum systems.
	~~{{Quantum mechanics\|cTopic=Advanced topics}}~~

	~~'''~~Quantum statistical mechanics~~'''~~ is ~~[[Physics:Statistical mechanics\|~~statistical mechanics~~]] applied~~ to ~~[[Physics:Quantum mechanics\|~~quantum ~~mechanical systems]]~~. ~~It relies on constructing [[Density matrix\|~~density matrices~~]] that describe~~ quantum ~~systems in [[Physics:Thermal equilibrium\|thermal equilibrium]]~~. ~~Its applications include~~ the ~~study~~ of collections of [[Identical particles\|identical particles]], which provides a theory that explains phenomena including [[Physics:Superconductivity\|superconductivity]] and [[Physics:Superfluidity\|superfluidity]].
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	==Thermodynamic ensembles==		==Thermodynamic ensembles==
	=== Canonical ===		=== Canonical ===


	Consider an ensemble of systems described by a Hamiltonian ''H'' with average energy ''E''. If ''H'' has pure-point spectrum and the eigenvalues <math>E_n</math> of ''H'' go to +∞ sufficiently fast, e<sup>−''r H''</sup> will be a non-negative trace-class operator for every positive ''r''.		Consider an ensemble of systems described by a Hamiltonian ''H'' with average energy ''E''. If ''H'' has pure-point spectrum and the eigenvalues <math>E_n</math> of ''H'' go to +∞ sufficiently fast, e<sup>−''r H''</sup> will be a non-negative trace-class operator for every positive ''r''.
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	=== Grand canonical ===		=== Grand canonical ===


	For open systems where the energy and numbers of particles may fluctuate, the system is described by the [[Physics:Grand canonical ensemble\|grand canonical ensemble]], described by the density matrix{{sfn\|Kardar\|2007\|p=174}}		For open systems where the energy and numbers of particles may fluctuate, the system is described by the [[Physics:Grand canonical ensemble\|grand canonical ensemble]], described by the density matrix{{sfn\|Kardar\|2007\|p=174}}
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	There are two main categories of identical particles: [[Physics:Boson\|boson]]s, which are described by quantum states that are symmetric under exchanges, and [[Physics:Fermion\|fermion]]s, which are described by antisymmetric states.{{sfnm\|1a1=Huang \|1y=1987 \|1p=179 \|2a1=Kadanoff \|2a2=Baym \|2y=2018 \|2p=2 \|3a1=Kardar\|3y=2007\|3p=182}} Examples of bosons are [[Physics:Photon\|photon]]s, [[Physics:Gluon\|gluon]]s, [[Software:Phonon\|phonon]]s, [[Physics:Helium-4\|helium-4]] nuclei and all [[Software:Meson\|meson]]s. Examples of fermions are [[Physics:Electron\|electron]]s, [[Physics:Neutrino\|neutrino]]s, [[Company:Quark\|quark]]s, [[Software:Proton\|proton]]s, [[Physics:Neutron\|neutron]]s, and [[Physics:Helium-3\|helium-3]] nuclei.		There are two main categories of identical particles: [[Physics:Boson\|boson]]s, which are described by quantum states that are symmetric under exchanges, and [[Physics:Fermion\|fermion]]s, which are described by antisymmetric states.{{sfnm\|1a1=Huang \|1y=1987 \|1p=179 \|2a1=Kadanoff \|2a2=Baym \|2y=2018 \|2p=2 \|3a1=Kardar\|3y=2007\|3p=182}} Examples of bosons are [[Physics:Photon\|photon]]s, [[Physics:Gluon\|gluon]]s, [[Software:Phonon\|phonon]]s, [[Physics:Helium-4\|helium-4]] nuclei and all [[Software:Meson\|meson]]s. Examples of fermions are [[Physics:Electron\|electron]]s, [[Physics:Neutrino\|neutrino]]s, [[Company:Quark\|quark]]s, [[Software:Proton\|proton]]s, [[Physics:Neutron\|neutron]]s, and [[Physics:Helium-3\|helium-3]] nuclei.

	The fact that particles can be identical has important consequences in statistical mechanics, and identical particles exhibit markedly different statistical behavior from distinguishable particles.{{sfnm\|1a1=Huang\|1y=1987\|1pp=179–189 \|2a1=Kadanoff\|2y=2000\|2pp=187–192}} The theory of boson quantum statistics is the starting point for understanding [[Physics:Superfluidity\|superfluids]],{{sfn\|Kardar\|2007\|pp=200–202}} and quantum statistics are also necessary to explain the related phenomenon of [[Physics:Superconductivity\|superconductivity]].{{sfn\|Reichl\|2016\|pp=114–115,184}}		The fact that particles can be identical has important consequences in statistical mechanics, and identical particles exhibit markedly different statistical behavior from distinguishable particles.{{sfnm\|1a1=Huang\|1y=1987\|1pp=179–189 \|2a1=Kadanoff\|2y=2000\|2pp=187–192}} The theory of boson quantum statistics is the starting point for understanding [[Physics:Superfluidity\|superfluids]],{{sfn\|Kardar\|2007\|pp=200–202}} and quantum statistics are also necessary to explain the related phenomenon of [[Physics:Superconductivity\|superconductivity]].{{sfn\|Reichl\|2016\|pp=114–115,184}}

			=

	==See also==		== See also ==
	* [[Physics:~~Quantum thermodynamics~~\|~~Quantum thermodynamics]]~~		{{#invoke:PhysicsQC\|tocHeadingAndList\|Physics:Quantum basics/See also}}
	* [[Physics:~~Thermal quantum field theory\|Thermal quantum field theory]]~~
	* [[Stochastic thermodynamics]]
	* [[Abstract Wiener space]]

	==References==		=References==
	{{reflist}}		{{reflist}}

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	{{Quantum mechanics topics}}		{{Quantum mechanics topics}}


	[[Category:Quantum mechanical entropy]]		[[Category:Quantum mechanical entropy]]

	{{Sourceattribution\|Quantum statistical mechanics}}		{{Sourceattribution\|Quantum statistical mechanics}}

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	{{Short description\|Statistical mechanics of quantum-mechanical systems}}		{{Short description\|Statistical mechanics of quantum-mechanical systems}}

			{{Quantum book backlink\|Statistical mechanics and kinetic theory}}

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	{{Thermodynamics sidebar\|expanded=branches}}		{{Thermodynamics sidebar\|expanded=branches}}
	{{Quantum mechanics\|cTopic=Advanced topics}}		{{Quantum mechanics\|cTopic=Advanced topics}}

	'''Quantum statistical mechanics''' is [[Physics:Statistical mechanics\|statistical mechanics]] applied to [[Physics:Quantum mechanics\|quantum mechanical systems]]. It relies on constructing [[Density matrix\|density matrices]] that describe quantum systems in [[Physics:Thermal equilibrium\|thermal equilibrium]]. Its applications include the study of collections of [[Identical particles\|identical particles]], which provides a theory that explains phenomena including [[Physics:Superconductivity\|superconductivity]] and [[Physics:Superfluidity\|superfluidity]].		'''Quantum statistical mechanics''' is [[Physics:Statistical mechanics\|statistical mechanics]] applied to [[Physics:Quantum mechanics\|quantum mechanical systems]]. It relies on constructing [[Density matrix\|density matrices]] that describe quantum systems in [[Physics:Thermal equilibrium\|thermal equilibrium]]. Its applications include the study of collections of [[Identical particles\|identical particles]], which provides a theory that explains phenomena including [[Physics:Superconductivity\|superconductivity]] and [[Physics:Superfluidity\|superfluidity]].
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	== Density matrices, expectation values, and entropy ==		== Density matrices, expectation values, and entropy ==

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{{Short description|Statistical mechanics of quantum-mechanical systems}}

{{Thermodynamics sidebar|expanded=branches}}
{{Quantum mechanics|cTopic=Advanced topics}}

'''Quantum statistical mechanics''' is [[Physics:Statistical mechanics|statistical mechanics]] applied to [[Physics:Quantum mechanics|quantum mechanical systems]]. It relies on constructing [[Density matrix|density matrices]] that describe quantum systems in [[Physics:Thermal equilibrium|thermal equilibrium]]. Its applications include the study of collections of [[Identical particles|identical particles]], which provides a theory that explains phenomena including [[Physics:Superconductivity|superconductivity]] and [[Physics:Superfluidity|superfluidity]].

== Density matrices, expectation values, and entropy ==
{{main|Density matrix}}
In quantum mechanics, probabilities for the outcomes of experiments made upon a system are calculated from the [[Physics:Quantum state|quantum state]] describing that system. Each physical system is associated with a [[Vector space|vector space]], or more specifically a [[Hilbert space]]. The [[Dimension (vector space)|dimension]] of the Hilbert space may be infinite, as it is for the space of [[Square-integrable function|square-integrable function]]s on a line, which is used to define the quantum physics of a continuous degree of freedom. Alternatively, the Hilbert space may be finite-dimensional, as occurs for [[Physics:Spin|spin]] degrees of freedom. A density operator, the mathematical representation of a quantum state, is a [[Positive-definite matrix|positive semi-definite]], [[Self-adjoint operator|self-adjoint operator]] of trace one acting on the Hilbert space of the system.<ref name=fano1957>{{cite journal |doi=10.1103/RevModPhys.29.74 |title=Description of States in Quantum Mechanics by Density Matrix and Operator Techniques |journal=Reviews of Modern Physics |volume=29 |issue=1 |pages=74–93 |year=1957 |last1=Fano |first1=U. |bibcode=1957RvMP...29...74F }}</ref>{{sfn|Holevo|2001|pages=1,15}}<ref name=Hall2013pp419-440>{{cite book |doi=10.1007/978-1-4614-7116-5_19 |chapter=Systems and Subsystems, Multiple Particles |title=Quantum Theory for Mathematicians |volume=267 |pages=419–440 |series=[[Graduate Texts in Mathematics]] |year=2013 |last1=Hall |first1=Brian C. |isbn=978-1-4614-7115-8 |publisher=Springer}}</ref> A density operator that is a rank-1 projection is known as a ''pure'' quantum state, and all quantum states that are not pure are designated ''mixed''.{{sfn|Kardar|2007|p=172}} Pure states are also known as ''wavefunctions''. Assigning a pure state to a quantum system implies certainty about the outcome of some measurement on that system. The [[State space|state space]] of a quantum system is the set of all states, pure and mixed, that can be assigned to it. For any system, the state space is a [[Convex set|convex set]]: Any mixed state can be written as a [[Convex combination|convex combination]] of pure states, though not in a unique way.<ref>{{Cite journal|last=Kirkpatrick |first=K. A. |date=February 2006 |title=The Schrödinger-HJW Theorem |journal=Foundations of Physics Letters |volume=19 |issue=1 |pages=95–102 |doi=10.1007/s10702-006-1852-1 |issn=0894-9875 |arxiv=quant-ph/0305068|bibcode=2006FoPhL..19...95K }}</ref>

The prototypical example of a finite-dimensional Hilbert space is a [[Qubit|qubit]], a quantum system whose Hilbert space is 2-dimensional. An arbitrary state for a qubit can be written as a linear combination of the [[Pauli matrices]], which provide a basis for <math>2 \times 2</math> self-adjoint matrices:{{sfnm|1a1=Wilde|1y=2017|1p=126 |2a1=Zwiebach|2y=2022|2at=§22.2}}
<math display="block">\rho = \tfrac{1}{2}\left(I + r_x \sigma_x + r_y \sigma_y + r_z \sigma_z\right),</math>
where the real numbers <math>(r_x, r_y, r_z)</math> are the coordinates of a point within the [[Unit sphere|unit ball]] and
<math display="block">
\sigma_x =
\begin{pmatrix}
0&1\\
1&0
\end{pmatrix}, \quad
\sigma_y =
\begin{pmatrix}
0&-i\\
i&0
\end{pmatrix}, \quad
\sigma_z =
\begin{pmatrix}
1&0\\
0&-1
\end{pmatrix} .</math>

In classical probability and statistics, the [[Expected value|expected (or expectation) value]] of a [[Random variable|random variable]] is the [[Arithmetic mean|mean]] of the possible values that random variable can take, weighted by the respective probabilities of those outcomes. The corresponding concept in quantum physics is the expectation value of an [[Physics:Observable|observable]]. Physically measurable quantities are represented mathematically by [[Self-adjoint operator|self-adjoint operator]]s that act on the Hilbert space associated with a quantum system. The expectation value of an observable is the Hilbert–Schmidt inner product of the operator representing that observable and the density operator:{{sfnm|1a1=Holevo|1y=2001|1p=17|2a1=Peres|2y=1993|2pp=64,73 |3a1=Kardar|3y=2007|3p=172}}
<math display="block"> \langle A \rangle = \operatorname{tr}(A \rho).</math>

The [[Physics:Von Neumann entropy|von Neumann entropy]], named after [[Biography:John von Neumann|John von Neumann]], quantifies the extent to which a state is mixed.{{sfn|Holevo|2001|page=15}} It extends the concept of Gibbs entropy from classical statistical mechanics to quantum statistical mechanics, and it is the quantum counterpart of the Shannon entropy from classical [[Information theory|information theory]]. For a quantum-mechanical system described by a [[Density matrix|density matrix]] {{mvar|ρ}}, the von Neumann entropy is{{sfnm|1a1=Bengtsson|1a2=Życzkowski|1y=2017|1p=355 |2a1=Peres|2y=1993|2p=264}}
<math display="block"> S = - \operatorname{tr}(\rho \ln \rho),</math>
where <math>\operatorname{tr}</math> denotes the [[Trace (linear algebra)|trace]] and <math>\operatorname{ln}</math> denotes the matrix version of the [[Natural logarithm|natural logarithm]]. If the density matrix {{mvar|ρ}} is written in a basis of its eigenvectors <math>|1\rangle, |2\rangle, |3\rangle, \dots</math> as
<math display="block"> \rho = \sum_j \eta_j \left| j \right\rang \left\lang j \right| ,</math>
then the von Neumann entropy is merely
<math display="block"> S = -\sum_j \eta_j \ln \eta_j .</math>
In this form, ''S'' can be seen as the Shannon entropy of the eigenvalues, reinterpreted as probabilities.{{sfnm|1a1=Bengtsson|1a2=Życzkowski|1y=2017|1p=360 |2a1=Peres|2y=1993|2p=264}}

The von Neumann entropy vanishes when <math>\rho</math> is a pure state. In the Bloch sphere picture, this occurs when the point <math>(r_x, r_y, r_z)</math> lies on the surface of the unit ball. The von Neumann entropy attains its maximum value when <math>\rho</math> is the ''maximally mixed'' state, which for the case of a qubit is given by <math>r_x = r_y = r_z = 0</math>.{{sfnm|1a1=Rieffel|1a2=Polak|1y=2011|1pp=216–217 |2a1=Zwiebach|2y=2022|2at=§22.2}}

The von Neumann entropy and quantities based upon it are widely used in the study of [[Quantum entanglement|quantum entanglement]].{{sfn|Nielsen|Chuang|2010|p=700}}

==Thermodynamic ensembles==
=== Canonical ===

Consider an ensemble of systems described by a Hamiltonian ''H'' with average energy ''E''. If ''H'' has pure-point spectrum and the eigenvalues <math>E_n</math> of ''H'' go to +∞ sufficiently fast, e−''r H'' will be a non-negative trace-class operator for every positive ''r''.

The ''[[Physics:Canonical ensemble|canonical ensemble]]'' (or sometimes ''Gibbs canonical ensemble'') is described by the state{{sfnm|1a1=Huang|1y=1987|1p=177 |2a1=Peres|2y=1993|2p=266 |3a1=Kardar|3y=2007|3p=174}}
<math display="block"> \rho = \frac{\mathrm{e}^{- \beta H}}{\operatorname{Tr}(\mathrm{e}^{- \beta H})}, </math>
where β is such that the ensemble average of energy satisfies
<math display="block"> \operatorname{Tr}(\rho H) = E </math>
and
<math display="block">\operatorname{Tr}(\mathrm{e}^{- \beta H}) = \sum_n \mathrm{e}^{- \beta E_n} = Z(\beta). </math>

This is called the [[Partition function (mathematics)|partition function]]; it is the quantum mechanical version of the canonical partition function of classical statistical mechanics. The probability that a system chosen at random from the ensemble will be in a state corresponding to energy eigenvalue <math>E_m</math> is

<math display="block">\mathcal{P}(E_m) = \frac{\mathrm{e}^{- \beta E_m}}{\sum_n \mathrm{e}^{- \beta E_n}}.</math>

The Gibbs canonical ensemble maximizes the von Neumann entropy of the state subject to the condition that the average energy is fixed.{{sfn|Peres|1993|p=267}}

=== Grand canonical ===

For open systems where the energy and numbers of particles may fluctuate, the system is described by the [[Physics:Grand canonical ensemble|grand canonical ensemble]], described by the density matrix{{sfn|Kardar|2007|p=174}}
<math display="block"> \rho = \frac{\mathrm{e}^{\beta (\sum_i \mu_iN_i - H)}}{\operatorname{Tr}\left(\mathrm{e}^{ \beta ( \sum_i \mu_iN_i - H)}\right)}. </math>
Here, the ''N''1, ''N''2, ... are the particle number operators for the different species of particles that are exchanged with the reservoir. Unlike the canonical ensemble, this density matrix involves a sum over states with different ''N.''

The grand partition function is{{sfnm|1a1=Huang|1y=1987|1p=178 |2a1=Kadanoff|2a2=Baym|2y=2018|2pp=2–3 |3a1=Kardar|3y=2007|3p=174}}
<math display="block">\mathcal Z(\beta, \mu_1, \mu_2, \cdots) = \operatorname{Tr}(\mathrm{e}^{\beta (\sum_i \mu_iN_i - H)}) </math>

Density matrices of this form maximize the entropy subject to the constraints that both the average energy and the average particle number are fixed.{{sfn|Reichl|2016|pp=184–185}}

==Identical particles and quantum statistics==
{{see also|Physics:Bose–Einstein statistics|Fermi–Dirac statistics}}
In quantum mechanics, [[Indistinguishable particles|indistinguishable particles]] (also called ''identical'' or ''indiscernible particles'') are [[Physics:Particle|particle]]s that cannot be distinguished from one another, even in principle. Species of identical particles include, but are not limited to, [[Physics:Elementary particle|elementary particle]]s (such as [[Physics:Electron|electron]]s), composite [[Physics:Subatomic particle|subatomic particle]]s (such as atomic nuclei), as well as [[Physics:Atom|atom]]s and [[Physics:Molecule|molecule]]s. Although all known indistinguishable particles only exist at the quantum scale, there is no exhaustive list of all possible sorts of particles nor a clear-cut limit of applicability, as explored in [[Physics:Particle statistics#Quantum statistics|quantum statistics]]. They were first discussed by [[Biography:Werner Heisenberg|Werner Heisenberg]] and [[Biography:Paul Dirac|Paul Dirac]] in 1926.<ref>{{Cite journal |last=Gottfried |first=Kurt |date=2011 |title=P. A. M. Dirac and the discovery of quantum mechanics |url=https://pubs.aip.org/aapt/ajp/article-abstract/79/3/261/398648/P-A-M-Dirac-and-the-discovery-of-quantum-mechanics?redirectedFrom=fulltext |journal=American Journal of Physics |volume=79 |issue=3 |pages=2, 10 |arxiv=1006.4610 |doi=10.1119/1.3536639 |bibcode=2011AmJPh..79..261G |s2cid=18229595}}</ref>

There are two main categories of identical particles: [[Physics:Boson|boson]]s, which are described by quantum states that are symmetric under exchanges, and [[Physics:Fermion|fermion]]s, which are described by antisymmetric states.{{sfnm|1a1=Huang |1y=1987 |1p=179 |2a1=Kadanoff |2a2=Baym |2y=2018 |2p=2 |3a1=Kardar|3y=2007|3p=182}} Examples of bosons are [[Physics:Photon|photon]]s, [[Physics:Gluon|gluon]]s, [[Software:Phonon|phonon]]s, [[Physics:Helium-4|helium-4]] nuclei and all [[Software:Meson|meson]]s. Examples of fermions are [[Physics:Electron|electron]]s, [[Physics:Neutrino|neutrino]]s, [[Company:Quark|quark]]s, [[Software:Proton|proton]]s, [[Physics:Neutron|neutron]]s, and [[Physics:Helium-3|helium-3]] nuclei.

The fact that particles can be identical has important consequences in statistical mechanics, and identical particles exhibit markedly different statistical behavior from distinguishable particles.{{sfnm|1a1=Huang|1y=1987|1pp=179–189 |2a1=Kadanoff|2y=2000|2pp=187–192}} The theory of boson quantum statistics is the starting point for understanding [[Physics:Superfluidity|superfluids]],{{sfn|Kardar|2007|pp=200–202}} and quantum statistics are also necessary to explain the related phenomenon of [[Physics:Superconductivity|superconductivity]].{{sfn|Reichl|2016|pp=114–115,184}}

==See also==
* [[Physics:Quantum thermodynamics|Quantum thermodynamics]]
* [[Physics:Thermal quantum field theory|Thermal quantum field theory]]
* [[Stochastic thermodynamics]]
* [[Abstract Wiener space]]

==References==
{{reflist}}

* {{cite book|first1=Ingemar |last1=Bengtsson |first2=Karol |last2=Życzkowski |title=Geometry of Quantum States: An Introduction to Quantum Entanglement |title-link=Geometry of Quantum States |year=2017 |publisher=Cambridge University Press |edition=2nd |isbn=978-1-107-02625-4}}
* {{cite book|first=Alexander S. |last=Holevo |title=Statistical Structure of Quantum Theory |publisher=Springer |series=[[Physics:Lecture Notes in Physics|Lecture Notes in Physics. Monographs]] |year=2001 |isbn=3-540-42082-7}}
* {{cite book|first=Leo P. |last=Kadanoff |title=Statistical Physics: Statics, Dynamics and Renormalization |publisher=World Scientific |year=2000 |isbn=9810237588}}
* {{cite book|first1=Leo P. |last1=Kadanoff |first2=Gordon |last2=Baym |author-link2=Gordon Baym |title=Quantum Statistical Mechanics |publisher=CRC Press |year=2018 |orig-year=1989 |isbn= 978-0-201-41046-4 }}
* {{cite book|first=Mehran |last=Kardar |title=Statistical Physics of Particles |year=2007 |publisher=Cambridge University Press |isbn=978-0-521-87342-0 |title-link=Statistical Physics of Particles}}
* {{cite book|first=Kerson |last=Huang |title=Statistical Mechanics |edition=2nd |publisher=John Wiley & Sons |isbn=0-471-81518-7 |year=1987}}
* {{cite book |last1=Nielsen |first1=Michael A. |title=Quantum Computation and Quantum Information |title-link=Quantum Computation and Quantum Information |last2=Chuang |first2=Isaac L. |author-link2=Isaac Chuang |publisher=Cambridge Univ. Press |year=2010 |isbn=978-0-521-63503-5 |edition=10th anniversary|location=Cambridge}}
* {{cite book|first=Asher |last=Peres |title=Quantum Theory: Concepts and Methods |title-link=Quantum Theory: Concepts and Methods |publisher=Kluwer |year=1993 |isbn=0-7923-2549-4 }}
* {{cite book|last=Reichl |first=Linda E. |title=A Modern Course in Statistical Physics |year=2016 |publisher=Wiley |edition=4th |isbn=978-3-527-41349-2 |}}
* {{Cite book |last1=Rieffel |first1=Eleanor |title=Quantum Computing: A Gentle Introduction |title-link=Quantum Computing: A Gentle Introduction |last2=Polak |first2=Wolfgang |date=2011 |publisher=MIT Press |isbn=978-0-262-01506-6 |series=Scientific and engineering computation |location=Cambridge, Mass}}
* {{cite book|last=Wilde |first=Mark M. |title=Quantum Information Theory |edition=2nd |publisher=Cambridge University Press |year=2017 |doi=10.1017/9781316809976  |isbn=9781316809976 |arxiv=1106.1445}}
* {{cite book|first=Barton |last=Zwiebach |title=Mastering Quantum Mechanics: Essentials, Theory, and Applications |publisher=MIT Press |year=2022 |isbn=978-0-262-04613-8}}

== Further reading ==
{{refbegin}}
* Modern review for closed systems: {{Cite journal |last=Nandkishore |first=Rahul |last2=Huse |first2=David A. |date=2015-03-10 |title=Many-Body Localization and Thermalization in Quantum Statistical Mechanics |url=https://www.annualreviews.org/content/journals/10.1146/annurev-conmatphys-031214-014726 |journal=Annual Review of Condensed Matter Physics |language=en |volume=6 |pages=15–38 |doi=10.1146/annurev-conmatphys-031214-014726 |issn=1947-5454|arxiv=1404.0686 }}
* {{Cite book |last=Schieve |first=William C. |title=Quantum statistical mechanics |date=2009 |publisher=Cambridge University Press |isbn=978-0-521-84146-7 |location=Cambridge, UK}}
* Advanced graduate textbook {{Cite book |last=Bogoli︠u︡bov |first=N. N. |url=https://www.worldcat.org/title/526687587 |title=Introduction to quantum statistical mechanics |last2=Bogoli︠u︡bov |first2=N. N. |date=2010 |publisher=World Scientific |isbn=978-981-4295-19-2 |edition=2 |location=Hackensack, NJ |oclc=526687587}}
{{refend}}

{{Quantum mechanics topics}}

[[Category:Quantum mechanical entropy]]

{{Sourceattribution|Quantum statistical mechanics}}

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	== Density matrices, expectation values, and entropy ==		== Density matrices, expectation values, and entropy ==
	~~{{main\|Density matrix}}~~
	In quantum mechanics, probabilities for the outcomes of experiments made upon a system are calculated from the [[Physics:Quantum state\|quantum state]] describing that system. Each physical system is associated with a vector space, or more specifically a Hilbert space. The dimension of the Hilbert space may be infinite, as it is for the space of square-integrable functions on a line, which is used to define the quantum physics of a continuous degree of freedom. Alternatively, the Hilbert space may be finite-dimensional, as occurs for spin degrees of freedom. A density operator, the mathematical representation of a quantum state, is a positive semi-definite, self-adjoint operator of trace one acting on the Hilbert space of the system.<ref name=fano1957>{{cite journal \|doi=10.1103/RevModPhys.29.74 \|title=Description of States in Quantum Mechanics by Density Matrix and Operator Techniques \|journal=Reviews of Modern Physics \|volume=29 \|issue=1 \|pages=74–93 \|year=1957 \|last1=Fano \|first1=U. \|bibcode=1957RvMP...29...74F }}</ref>{{sfn\|Holevo\|2001\|pages=1,15}}<ref name=Hall2013pp419-440>{{cite book \|doi=10.1007/978-1-4614-7116-5_19 \|chapter=Systems and Subsystems, Multiple Particles \|title=Quantum Theory for Mathematicians \|volume=267 \|pages=419–440 \|series=Graduate Texts in Mathematics \|year=2013 \|last1=Hall \|first1=Brian C. \|isbn=978-1-4614-7115-8 \|publisher=Springer}}</ref> A density operator that is a rank-1 projection is known as a ''pure'' quantum state, and all quantum states that are not pure are designated ''mixed''.{{sfn\|Kardar\|2007\|p=172}} Pure states are also known as ''wavefunctions''. Assigning a pure state to a quantum system implies certainty about the outcome of some measurement on that system. The state space of a quantum system is the set of all states, pure and mixed, that can be assigned to it. For any system, the state space is a convex set: Any mixed state can be written as a convex combination of pure states, though not in a unique way.<ref>{{Cite journal\|last=Kirkpatrick \|first=K. A. \|date=February 2006 \|title=The Schrödinger-HJW Theorem \|journal=Foundations of Physics Letters \|volume=19 \|issue=1 \|pages=95–102 \|doi=10.1007/s10702-006-1852-1 \|issn=0894-9875 \|arxiv=quant-ph/0305068\|bibcode=2006FoPhL..19...95K }}</ref>		In quantum mechanics, probabilities for the outcomes of experiments made upon a system are calculated from the [[Physics:Quantum state\|quantum state]] describing that system. Each physical system is associated with a vector space, or more specifically a Hilbert space. The dimension of the Hilbert space may be infinite, as it is for the space of square-integrable functions on a line, which is used to define the quantum physics of a continuous degree of freedom. Alternatively, the Hilbert space may be finite-dimensional, as occurs for spin degrees of freedom. A density operator, the mathematical representation of a quantum state, is a positive semi-definite, self-adjoint operator of trace one acting on the Hilbert space of the system.<ref name=fano1957>{{cite journal \|doi=10.1103/RevModPhys.29.74 \|title=Description of States in Quantum Mechanics by Density Matrix and Operator Techniques \|journal=Reviews of Modern Physics \|volume=29 \|issue=1 \|pages=74–93 \|year=1957 \|last1=Fano \|first1=U. \|bibcode=1957RvMP...29...74F }}</ref>{{sfn\|Holevo\|2001\|pages=1,15}}<ref name=Hall2013pp419-440>{{cite book \|doi=10.1007/978-1-4614-7116-5_19 \|chapter=Systems and Subsystems, Multiple Particles \|title=Quantum Theory for Mathematicians \|volume=267 \|pages=419–440 \|series=Graduate Texts in Mathematics \|year=2013 \|last1=Hall \|first1=Brian C. \|isbn=978-1-4614-7115-8 \|publisher=Springer}}</ref> A density operator that is a rank-1 projection is known as a ''pure'' quantum state, and all quantum states that are not pure are designated ''mixed''.{{sfn\|Kardar\|2007\|p=172}} Pure states are also known as ''wavefunctions''. Assigning a pure state to a quantum system implies certainty about the outcome of some measurement on that system. The state space of a quantum system is the set of all states, pure and mixed, that can be assigned to it. For any system, the state space is a convex set: Any mixed state can be written as a convex combination of pure states, though not in a unique way.<ref>{{Cite journal\|last=Kirkpatrick \|first=K. A. \|date=February 2006 \|title=The Schrödinger-HJW Theorem \|journal=Foundations of Physics Letters \|volume=19 \|issue=1 \|pages=95–102 \|doi=10.1007/s10702-006-1852-1 \|issn=0894-9875 \|arxiv=quant-ph/0305068\|bibcode=2006FoPhL..19...95K }}</ref>

	The prototypical example of a finite-dimensional Hilbert space is a qubit, a quantum system whose Hilbert space is 2-dimensional. An arbitrary state for a qubit can be written as a linear combination of the Pauli matrices, which provide a basis for <math>2 \times 2</math> self-adjoint matrices:~~{{sfnm\|1a1=Wilde\|1y=2017\|1p=126 \|2a1=Zwiebach\|2y=2022\|2at=§22.2}}~~		The prototypical example of a finite-dimensional Hilbert space is a qubit, a quantum system whose Hilbert space is 2-dimensional. An arbitrary state for a qubit can be written as a linear combination of the Pauli matrices, which provide a basis for <math>2 \times 2</math> self-adjoint matrices:
	<math display="block">\rho = \tfrac{1}{2}\left(I + r_x \sigma_x + r_y \sigma_y + r_z \sigma_z\right),</math>		<math display="block">\rho = \tfrac{1}{2}\left(I + r_x \sigma_x + r_y \sigma_y + r_z \sigma_z\right),</math>
	where the real numbers <math>(r_x, r_y, r_z)</math> are the coordinates of a point within the unit ball and		where the real numbers <math>(r_x, r_y, r_z)</math> are the coordinates of a point within the unit ball and
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	\end{pmatrix} .</math>		\end{pmatrix} .</math>

	In classical probability and statistics, the expected (or expectation) value of a random variable is the mean of the possible values that random variable can take, weighted by the respective probabilities of those outcomes. The corresponding concept in quantum physics is the expectation value of an observable. Physically measurable quantities are represented mathematically by self-adjoint operators that act on the Hilbert space associated with a quantum system. The expectation value of an observable is the Hilbert–Schmidt inner product of the operator representing that observable and the density operator:~~{{sfnm\|1a1=Holevo\|1y=2001\|1p=17\|2a1=Peres\|2y=1993\|2pp=64,73 \|3a1=Kardar\|3y=2007\|3p=172}}~~		In classical probability and statistics, the expected (or expectation) value of a random variable is the mean of the possible values that random variable can take, weighted by the respective probabilities of those outcomes. The corresponding concept in quantum physics is the expectation value of an observable. Physically measurable quantities are represented mathematically by self-adjoint operators that act on the Hilbert space associated with a quantum system. The expectation value of an observable is the Hilbert–Schmidt inner product of the operator representing that observable and the density operator:
	<math display="block"> \langle A \rangle = \operatorname{tr}(A \rho).</math>		<math display="block"> \langle A \rangle = \operatorname{tr}(A \rho).</math>

	The von Neumann entropy, named after [[Biography:John von Neumann\|John von Neumann]], quantifies the extent to which a state is mixed.{{sfn\|Holevo\|2001\|page=15}} It extends the concept of Gibbs entropy from classical statistical mechanics to quantum statistical mechanics, and it is the quantum counterpart of the Shannon entropy from classical information theory. For a quantum-mechanical system described by a density matrix {{mvar\|ρ}}, the von Neumann entropy is~~{{sfnm\|1a1=Bengtsson\|1a2=Życzkowski\|1y=2017\|1p=355 \|2a1=Peres\|2y=1993\|2p=264}}~~		The von Neumann entropy, named after [[Biography:John von Neumann\|John von Neumann]], quantifies the extent to which a state is mixed.{{sfn\|Holevo\|2001\|page=15}} It extends the concept of Gibbs entropy from classical statistical mechanics to quantum statistical mechanics, and it is the quantum counterpart of the Shannon entropy from classical information theory. For a quantum-mechanical system described by a density matrix {{mvar\|ρ}}, the von Neumann entropy is
	<math display="block"> S = - \operatorname{tr}(\rho \ln \rho),</math>		<math display="block"> S = - \operatorname{tr}(\rho \ln \rho),</math>
	where <math>\operatorname{tr}</math> denotes the trace and <math>\operatorname{ln}</math> denotes the matrix version of the natural logarithm. If the density matrix {{mvar\|ρ}} is written in a basis of its eigenvectors <math>\|1\rangle, \|2\rangle, \|3\rangle, \dots</math> as		where <math>\operatorname{tr}</math> denotes the trace and <math>\operatorname{ln}</math> denotes the matrix version of the natural logarithm. If the density matrix {{mvar\|ρ}} is written in a basis of its eigenvectors <math>\|1\rangle, \|2\rangle, \|3\rangle, \dots</math> as
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	then the von Neumann entropy is merely		then the von Neumann entropy is merely
	<math display="block"> S = -\sum_j \eta_j \ln \eta_j .</math>		<math display="block"> S = -\sum_j \eta_j \ln \eta_j .</math>
	In this form, ''S'' can be seen as the Shannon entropy of the eigenvalues, reinterpreted as probabilities.~~{{sfnm\|1a1=Bengtsson\|1a2=Życzkowski\|1y=2017\|1p=360 \|2a1=Peres\|2y=1993\|2p=264}}~~		In this form, ''S'' can be seen as the Shannon entropy of the eigenvalues, reinterpreted as probabilities.

	The von Neumann entropy vanishes when <math>\rho</math> is a pure state. In the Bloch sphere picture, this occurs when the point <math>(r_x, r_y, r_z)</math> lies on the surface of the unit ball. The von Neumann entropy attains its maximum value when <math>\rho</math> is the ''maximally mixed'' state, which for the case of a qubit is given by <math>r_x = r_y = r_z = 0</math>.~~{{sfnm\|1a1=Rieffel\|1a2=Polak\|1y=2011\|1pp=216–217 \|2a1=Zwiebach\|2y=2022\|2at=§22.2}}~~		The von Neumann entropy vanishes when <math>\rho</math> is a pure state. In the Bloch sphere picture, this occurs when the point <math>(r_x, r_y, r_z)</math> lies on the surface of the unit ball. The von Neumann entropy attains its maximum value when <math>\rho</math> is the ''maximally mixed'' state, which for the case of a qubit is given by <math>r_x = r_y = r_z = 0</math>.

	The von Neumann entropy and quantities based upon it are widely used in the study of quantum entanglement.{{sfn\|Nielsen\|Chuang\|2010\|p=700}}		The von Neumann entropy and quantities based upon it are widely used in the study of quantum entanglement.{{sfn\|Nielsen\|Chuang\|2010\|p=700}}
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	Consider an ensemble of systems described by a Hamiltonian ''H'' with average energy ''E''. If ''H'' has pure-point spectrum and the eigenvalues <math>E_n</math> of ''H'' go to +∞ sufficiently fast, e<sup>−''r H''</sup> will be a non-negative trace-class operator for every positive ''r''.		Consider an ensemble of systems described by a Hamiltonian ''H'' with average energy ''E''. If ''H'' has pure-point spectrum and the eigenvalues <math>E_n</math> of ''H'' go to +∞ sufficiently fast, e<sup>−''r H''</sup> will be a non-negative trace-class operator for every positive ''r''.

	The ''canonical ensemble'' (or sometimes ''Gibbs canonical ensemble'') is described by the state~~{{sfnm\|1a1=Huang\|1y=1987\|1p=177 \|2a1=Peres\|2y=1993\|2p=266 \|3a1=Kardar\|3y=2007\|3p=174}}~~		The ''canonical ensemble'' (or sometimes ''Gibbs canonical ensemble'') is described by the state
	<math display="block"> \rho = \frac{\mathrm{e}^{- \beta H}}{\operatorname{Tr}(\mathrm{e}^{- \beta H})}, </math>		<math display="block"> \rho = \frac{\mathrm{e}^{- \beta H}}{\operatorname{Tr}(\mathrm{e}^{- \beta H})}, </math>
	where β is such that the ensemble average of energy satisfies		where β is such that the ensemble average of energy satisfies
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	Here, the ''N''<sub>1</sub>, ''N''<sub>2</sub>, ... are the particle number operators for the different species of particles that are exchanged with the reservoir. Unlike the canonical ensemble, this density matrix involves a sum over states with different ''N.''		Here, the ''N''<sub>1</sub>, ''N''<sub>2</sub>, ... are the particle number operators for the different species of particles that are exchanged with the reservoir. Unlike the canonical ensemble, this density matrix involves a sum over states with different ''N.''

	The grand partition function is~~{{sfnm\|1a1=Huang\|1y=1987\|1p=178 \|2a1=Kadanoff\|2a2=Baym\|2y=2018\|2pp=2–3 \|3a1=Kardar\|3y=2007\|3p=174}}~~		The grand partition function is
	<math display="block">\mathcal Z(\beta, \mu_1, \mu_2, \cdots) = \operatorname{Tr}(\mathrm{e}^{\beta (\sum_i \mu_iN_i - H)}) </math>		<math display="block">\mathcal Z(\beta, \mu_1, \mu_2, \cdots) = \operatorname{Tr}(\mathrm{e}^{\beta (\sum_i \mu_iN_i - H)}) </math>

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	==Identical particles and quantum statistics==		==Identical particles and quantum statistics==
	~~{{see also\|Physics:Bose–Einstein statistics\|Fermi–Dirac statistics}}~~
	In quantum mechanics, indistinguishable particles (also called ''identical'' or ''indiscernible particles'') are particles that cannot be distinguished from one another, even in principle. Species of identical particles include, but are not limited to, elementary particles (such as electrons), composite subatomic particles (such as atomic nuclei), as well as atoms and molecules. Although all known indistinguishable particles only exist at the quantum scale, there is no exhaustive list of all possible sorts of particles nor a clear-cut limit of applicability, as explored in quantum statistics. They were first discussed by [[Biography:Werner Heisenberg\|Werner Heisenberg]] and [[Biography:Paul Dirac\|Paul Dirac]] in 1926.<ref>{{Cite journal \|last=Gottfried \|first=Kurt \|date=2011 \|title=P. A. M. Dirac and the discovery of quantum mechanics \|url=https://pubs.aip.org/aapt/ajp/article-abstract/79/3/261/398648/P-A-M-Dirac-and-the-discovery-of-quantum-mechanics?redirectedFrom=fulltext \|journal=American Journal of Physics \|volume=79 \|issue=3 \|pages=2, 10 \|arxiv=1006.4610 \|doi=10.1119/1.3536639 \|bibcode=2011AmJPh..79..261G \|s2cid=18229595}}</ref>		In quantum mechanics, indistinguishable particles (also called ''identical'' or ''indiscernible particles'') are particles that cannot be distinguished from one another, even in principle. Species of identical particles include, but are not limited to, elementary particles (such as electrons), composite subatomic particles (such as atomic nuclei), as well as atoms and molecules. Although all known indistinguishable particles only exist at the quantum scale, there is no exhaustive list of all possible sorts of particles nor a clear-cut limit of applicability, as explored in quantum statistics. They were first discussed by [[Biography:Werner Heisenberg\|Werner Heisenberg]] and [[Biography:Paul Dirac\|Paul Dirac]] in 1926.<ref>{{Cite journal \|last=Gottfried \|first=Kurt \|date=2011 \|title=P. A. M. Dirac and the discovery of quantum mechanics \|url=https://pubs.aip.org/aapt/ajp/article-abstract/79/3/261/398648/P-A-M-Dirac-and-the-discovery-of-quantum-mechanics?redirectedFrom=fulltext \|journal=American Journal of Physics \|volume=79 \|issue=3 \|pages=2, 10 \|arxiv=1006.4610 \|doi=10.1119/1.3536639 \|bibcode=2011AmJPh..79..261G \|s2cid=18229595}}</ref>

	There are two main categories of identical particles: bosons, which are described by quantum states that are symmetric under exchanges, and fermions, which are described by antisymmetric states.~~{{sfnm\|1a1=Huang \|1y=1987 \|1p=179 \|2a1=Kadanoff \|2a2=Baym \|2y=2018 \|2p=2 \|3a1=Kardar\|3y=2007\|3p=182}}~~ Examples of bosons are photons, gluons, phonons, helium-4 nuclei and all mesons. Examples of fermions are electrons, neutrinos, quarks, protons, neutrons, and helium-3 nuclei.		There are two main categories of identical particles: bosons, which are described by quantum states that are symmetric under exchanges, and fermions, which are described by antisymmetric states. Examples of bosons are photons, gluons, phonons, helium-4 nuclei and all mesons. Examples of fermions are electrons, neutrinos, quarks, protons, neutrons, and helium-3 nuclei.

	The fact that particles can be identical has important consequences in statistical mechanics, and identical particles exhibit markedly different statistical behavior from distinguishable particles.~~{{sfnm\|1a1=Huang\|1y=1987\|1pp=179–189 \|2a1=Kadanoff\|2y=2000\|2pp=187–192}}~~ The theory of boson quantum statistics is the starting point for understanding superfluids,{{sfn\|Kardar\|2007\|pp=200–202}} and quantum statistics are also necessary to explain the related phenomenon of superconductivity.{{sfn\|Reichl\|2016\|pp=114–115,184}}		The fact that particles can be identical has important consequences in statistical mechanics, and identical particles exhibit markedly different statistical behavior from distinguishable particles. The theory of boson quantum statistics is the starting point for understanding superfluids,{{sfn\|Kardar\|2007\|pp=200–202}} and quantum statistics are also necessary to explain the related phenomenon of superconductivity.{{sfn\|Reichl\|2016\|pp=114–115,184}}

	=		=
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	* Advanced graduate textbook {{Cite book \|last=Bogoli︠u︡bov \|first=N. N. \|url=https://www.worldcat.org/title/526687587 \|title=Introduction to quantum statistical mechanics \|last2=Bogoli︠u︡bov \|first2=N. N. \|date=2010 \|publisher=World Scientific \|isbn=978-981-4295-19-2 \|edition=2 \|location=Hackensack, NJ \|oclc=526687587}}		* Advanced graduate textbook {{Cite book \|last=Bogoli︠u︡bov \|first=N. N. \|url=https://www.worldcat.org/title/526687587 \|title=Introduction to quantum statistical mechanics \|last2=Bogoli︠u︡bov \|first2=N. N. \|date=2010 \|publisher=World Scientific \|isbn=978-981-4295-19-2 \|edition=2 \|location=Hackensack, NJ \|oclc=526687587}}
	{{refend}}		{{refend}}
	~~[[Category:Quantum mechanical entropy]]~~

	{{Sourceattribution\|Quantum statistical mechanics}}		{{Sourceattribution\|Quantum statistical mechanics}}