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Quantum mechanics
${\displaystyle i\hslash \frac{d}{dt}|\mathrm{\Psi }⟩=\hat{H}|\mathrm{\Psi }⟩}$ Schrödinger equation
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Quantum statistical mechanics isstatistical mechanics applied toquantum mechanical systems. In quantum mechanics astatistical ensemble (probability distribution over possiblequantum states) is described by adensity operatorS, which is a non-negative,self-adjoint,trace-class operator of trace 1 on theHilbert spaceH describing the quantum system. This can be shown under variousmathematical formalisms for quantum mechanics.

From classical probability theory, we know that theexpectation of arandom variableX is defined by itsdistribution D_X by$${\displaystyle \mathbb{E}(X)={\int }_{\mathbb{R}}d\lambda {\mathrm{D}}_X(\lambda )}$$assuming, of course, that the random variable isintegrable or that the random variable is non-negative. Similarly, letA be anobservable of a quantum mechanical system.A is given by adensely definedself-adjoint operator onH. Thespectral measure ofA defined by

$${\displaystyle {\mathrm{E}}_A(U)={\int }_{U}d\lambda \mathrm{E}(\lambda ),}$$

uniquely determinesA and conversely, is uniquely determined byA. E_A is aBoolean homomorphism from theBorel subsets ofR into thelatticeQ of self-adjoint projections ofH. In analogy with probability theory, given a stateS, we introduce thedistribution ofA underS which is the probability measure defined on the Borel subsets ofR by$${\displaystyle {\mathrm{D}}_A(U)=\mathrm{Tr}({\mathrm{E}}_A(U)S).}$$Similarly, the expected value ofA is defined in terms of the probability distribution D_A by$${\displaystyle \mathbb{E}(A)={\int }_{\mathbb{R}}d\lambda {\mathrm{D}}_A(\lambda ).}$$Note that this expectation is relative to the mixed stateS which is used in the definition of D_A.

Remark. For technical reasons, one needs to consider separately the positive and negative parts ofA defined by theBorel functional calculus for unbounded operators.

One can easily show:$${\displaystyle \mathbb{E}(A)=\mathrm{Tr}(AS)=\mathrm{Tr}(SA).}$$Thetrace of an operatorA is written as follows:$${\displaystyle \mathrm{Tr}(A)=\sum _m⟨m|A|m⟩.}$$ Note that ifS is apure state corresponding to thevector${\displaystyle \psi }$, then:$${\displaystyle \mathbb{E}(A)=⟨\psi |A|\psi ⟩.}$$

Of particular significance for describing randomness of a state is the von Neumann entropy ofSformally defined by$${\displaystyle \mathrm{H}(S)=-\mathrm{Tr}(S{\log }_{2}S)}$$. Actually, the operatorS log₂S is not necessarily trace-class. However, ifS is a non-negative self-adjoint operator not of trace class we define Tr(S) = +∞. Also note that any density operatorS can be diagonalized, that it can be represented in some orthonormal basis by a (possibly infinite) matrix of the formand we define$${\displaystyle \mathrm{H}(S)=-\sum _i{\lambda }_i{\log }_2{\lambda }_i.}$$The convention is that${\displaystyle 0{\log }_{2}0=0}$, since an event with probability zero should not contribute to the entropy. This value is an extended real number (that is in [0, ∞]) and this is clearly a unitary invariant ofS.

Remark. It is indeed possible that H(S) = +∞ for some density operatorS. In factT be the diagonal matrixT is non-negative trace class and one can showT log₂T is not trace-class.

In analogy withclassical entropy (notice the similarity in the definitions), H(S) measures the amount of randomness in the stateS. The more dispersed the eigenvalues are, the larger the system entropy. For a system in which the spaceH is finite-dimensional, entropy is maximized for the statesS which in diagonal form have the representationFor such anS, H(S) = log₂n. The stateS is called the maximally mixed state.

Recall that apure state is one of the form$${\displaystyle S=|\psi ⟩⟨\psi |,}$$for ψ a vector of norm 1.

ForS is a pure state if and only if its diagonal form has exactly one non-zero entry which is a 1.

Entropy can be used as a measure ofquantum entanglement.

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.

TheGibbs canonical ensemble is described by the state$${\displaystyle S=\frac{{\mathrm{e}}^{-\beta H}}{\mathrm{Tr}({\mathrm{e}}^{-\beta H})}.}$$Where β is such that the ensemble average of energy satisfies$${\displaystyle \mathrm{Tr}(SH)=E}$$

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

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

Under certain conditions, the Gibbs canonical ensemble maximizes the von Neumann entropy of the state subject to the energy conservation requirement.^{[clarification needed]}

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$${\displaystyle \rho =\frac{{\mathrm{e}}^{\beta (\sum _i{\mu }_i{N}_i-H)}}{\mathrm{Tr}\left({\mathrm{e}}^{\beta (\sum _i{\mu }_i{N}_i-H)}\right)}.}$$where theN₁,N₂, ... are the particle number operators for the different species of particles that are exchanged with the reservoir. Note that this is a density matrix including many more states (of varying N) compared to the canonical ensemble.

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

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

• Quantum mechanical entropy

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