imported>WikiHarold: over-write

2025-11-18T00:05:19Z

over-write

← Older revision	Revision as of 00:05, 18 November 2025
(No difference)

imported>WikiHarold: over-write

2025-11-18T00:05:19Z

over-write

New page

{{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}}

Physics:Quantum statistical mechanics - Revision history

imported>WikiHarold: over-write

imported>WikiHarold: over-write