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Part of a series of articles about
Quantum mechanics
${\displaystyle i\hslash \frac{d}{dt}|\mathrm{\Psi }⟩=\hat{H}|\mathrm{\Psi }⟩}$ Schrödinger equation
Introduction Glossary History
Background Classical mechanics Old quantum theory Bra–ket notation Hamiltonian Interference
Fundamentals Complementarity Decoherence Entanglement Energy level Measurement Nonlocality Quantum number State Superposition Symmetry Tunnelling Uncertainty Wave function Collapse
Experiments Bell's inequality CHSH inequality Davisson–Germer Double-slit Elitzur–Vaidman Franck–Hertz Leggett inequality Leggett–Garg inequality Mach–Zehnder Popper Quantum eraser Delayed-choice Schrödinger's cat Stern–Gerlach Wheeler's delayed-choice
Formulations Overview Heisenberg Interaction Matrix Phase-space Schrödinger Sum-over-histories (path integral)
Equations Dirac Klein–Gordon Pauli Rydberg Schrödinger
Interpretations Bayesian Consciousness causes collapse Consistent histories Copenhagen de Broglie–Bohm Ensemble Hidden-variable Many-worlds Objective-collapse Quantum logic Superdeterminism Relational Transactional
Advanced topics Relativistic quantum mechanics Quantum field theory Quantum information science Quantum computing Quantum chaos EPR paradox Density matrix Scattering theory Quantum statistical mechanics Quantum machine learning
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Quantum statistical mechanics isstatistical mechanics applied toquantum mechanical systems. It relies on constructingdensity matrices that describe quantum systems inthermal equilibrium. Its applications include the study of collections ofidentical particles, which provides a theory that explains phenomena includingsuperconductivity andsuperfluidity.

In quantum mechanics, probabilities for the outcomes of experiments made upon a system are calculated from thequantum state describing that system. Each physical system is associated with avector space, or more specifically aHilbert space. Thedimension of the Hilbert space may be infinite, as it is for the space ofsquare-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 forspin degrees of freedom. A density operator, the mathematical representation of a quantum state, is apositive semi-definite,self-adjoint operator oftrace one acting on the Hilbert space of the system.^[1][2][3] A density operator that is a rank-1 projection is known as apure quantum state, and all quantum states that are not pure are designatedmixed.^[4] Pure states are also known aswavefunctions. Assigning a pure state to a quantum system implies certainty about the outcome of some measurement on that system. Thestate 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 aconvex set: Any mixed state can be written as aconvex combination of pure states, thoughnot in a unique way.^[5]

The prototypical example of a finite-dimensional Hilbert space is aqubit, a quantum system whose Hilbert space is 2-dimensional. An arbitrary state for a qubit can be written as a linear combination of thePauli matrices, which provide a basis for${\displaystyle 2\times 2}$ self-adjoint matrices:^[6]$${\displaystyle \rho =\frac{1}{2}\left(I+r_x{\sigma }_x+r_y{\sigma }_y+r_z{\sigma }_z\right),}$$where the real numbers${\displaystyle (r_x,r_y,r_z)}$ are the coordinates of a point within theunit ball and

In classical probability and statistics, theexpected (or expectation) value of arandom variable is themean 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 anobservable. Physically measurable quantities are represented mathematically byself-adjoint operators that act on the Hilbert space associated with a quantum system. The expectation value of an observable is theHilbert–Schmidt inner product of the operator representing that observable and the density operator:^[7]$${\displaystyle ⟨A⟩=\mathrm{tr}(A\rho ).}$$

Thevon Neumann entropy, named afterJohn von Neumann, quantifies the extent to which a state is mixed.^[8] It extends the concept ofGibbs entropy from classical statistical mechanics to quantum statistical mechanics, and it is the quantum counterpart of theShannon entropy from classicalinformation theory. For a quantum-mechanical system described by adensity matrixρ, the von Neumann entropy is^[9]$${\displaystyle S=-\mathrm{tr}(\rho \ln \rho ),}$$where${\displaystyle \mathrm{tr}}$ denotes thetrace and${\displaystyle \ln }$ denotes thematrix version of thenatural logarithm. If the density matrixρ is written in a basis of itseigenvectors${\displaystyle |1⟩,|2⟩,|3⟩,\dots }$ as$${\displaystyle \rho =\sum _j{\eta }_j|j⟩⟨j|,}$$then the von Neumann entropy is merely$${\displaystyle S=-\sum _j{\eta }_j\ln {\eta }_j.}$$In this form,S can be seen as the Shannon entropy of the eigenvalues, reinterpreted as probabilities.^[10]

The von Neumann entropy vanishes when${\displaystyle \rho }$ is a pure state. In the Bloch sphere picture, this occurs when the point${\displaystyle (r_x,r_y,r_z)}$ lies on the surface of the unit ball. The von Neumann entropy attains its maximum value when${\displaystyle \rho }$ is themaximally mixed state, which for the case of a qubit is given by${\displaystyle r_x=r_y=r_z=0}$.^[11]

The von Neumann entropy and quantities based upon it are widely used in the study ofquantum entanglement.^[12]

Consider an ensemble of systems described by a HamiltonianH with average energyE. IfH has pure-point spectrum and the eigenvalues${\displaystyle E_n}$ ofH go to +∞ sufficiently fast, e^{−r H} will be a non-negative trace-class operator for every positiver.

Thecanonical ensemble (or sometimesGibbs canonical ensemble) is described by the state^[13]$${\displaystyle \rho =\frac{{\mathrm{e}}^{-\beta H}}{\mathrm{Tr}({\mathrm{e}}^{-\beta H})},}$$where β is such that the ensemble average of energy satisfies$${\displaystyle \mathrm{Tr}(\rho H)=E}$$and$${\displaystyle \mathrm{Tr}({\mathrm{e}}^{-\beta H})=\sum _n{\mathrm{e}}^{-\beta E_n}=Z(\beta ).}$$

This is called thepartition function; it is the quantum mechanical version of thecanonical 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${\displaystyle E_m}$ is

$${\displaystyle \mathcal{P}(E_m)=\frac{{\mathrm{e}}^{-\beta E_m}}{\sum _n{\mathrm{e}}^{-\beta E_n}}.}$$

The Gibbs canonical ensemble maximizes the von Neumann entropy of the state subject to the condition that the average energy is fixed.^[14]

For open systems where the energy and numbers of particles may fluctuate, the system is described by thegrand canonical ensemble, described by the density matrix^[15]$${\displaystyle \rho =\frac{{\mathrm{e}}^{\beta (\sum _i{\mu }_i{N}_i-H)}}{\mathrm{Tr}\left({\mathrm{e}}^{\beta (\sum _i{\mu }_i{N}_i-H)}\right)}.}$$Here, theN₁,N₂, ... 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 differentN.

The grand partition function is^[16]$${\displaystyle \mathcal{Z}(\beta ,{\mu }_1,{\mu }_2,\cdots )=\mathrm{Tr}({\mathrm{e}}^{\beta (\sum _i{\mu }_i{N}_i-H)})}$$

Density matrices of this form maximize the entropy subject to the constraints that both the average energy and the average particle number are fixed.^[17]

In quantum mechanics,indistinguishable particles (also calledidentical orindiscernible particles) areparticles that cannot be distinguished from one another, even in principle. Species of identical particles include, but are not limited to,elementary particles (such aselectrons), compositesubatomic particles (such asatomic nuclei), as well asatoms andmolecules. 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 inquantum statistics. They were first discussed byWerner Heisenberg andPaul Dirac in 1926.^[18]

There are two main categories of identical particles:bosons, which are described by quantum states that are symmetric under exchanges, andfermions, which are described by antisymmetric states.^[19] Examples of bosons arephotons,gluons,phonons,helium-4 nuclei and allmesons. Examples of fermions areelectrons,neutrinos,quarks,protons,neutrons, andhelium-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.^[20] The theory of boson quantum statistics is the starting point for understandingsuperfluids,^[21] and quantum statistics are also necessary to explain the related phenomenon ofsuperconductivity.^[22]

• Quantum thermodynamics
• Thermal quantum field theory
• Stochastic thermodynamics
• Abstract Wiener space

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• Huang, Kerson (1987).Statistical Mechanics (2nd ed.). John Wiley & Sons.ISBN 0-471-81518-7.
• Nielsen, Michael A.;Chuang, Isaac L. (2010).Quantum Computation and Quantum Information (10th anniversary ed.). Cambridge: Cambridge Univ. Press.ISBN 978-0-521-63503-5.
• Peres, Asher (1993).Quantum Theory: Concepts and Methods. Kluwer.ISBN 0-7923-2549-4.
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• Rieffel, Eleanor; Polak, Wolfgang (2011).Quantum Computing: A Gentle Introduction. Scientific and engineering computation. Cambridge, Mass: MIT Press.ISBN 978-0-262-01506-6.
• Wilde, Mark M. (2017).Quantum Information Theory (2nd ed.). Cambridge University Press.arXiv:1106.1445.doi:10.1017/9781316809976.ISBN 9781316809976.
• Zwiebach, Barton (2022).Mastering Quantum Mechanics: Essentials, Theory, and Applications. MIT Press.ISBN 978-0-262-04613-8.

• Quantum mechanical entropy

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