# Quantum statistical mechanics

Quantum statistical mechanics is statistical mechanics applied to quantum mechanical systems. In quantum mechanics a statistical ensemble (probability distribution over possible quantum states) is described by a density operator S, which is a non-negative, self-adjoint, trace-class operator of trace 1 on the Hilbert space H describing the quantum system; this can be shown under various mathematical formalisms for quantum mechanics. One such formalism is provided by quantum logic.

## Expectation

From classical probability theory, we know that the expectation of a random variable X is defined by its distribution DX by

$\mathbb {E} (X)=\int _{\mathbb {R} }\lambda \,d\,\operatorname {D} _{X}(\lambda )$ assuming, of course, that the random variable is integrable or that the random variable is non-negative. Similarly, let A be an observable of a quantum mechanical system. A is given by a densely defined self-adjoint operator on H; the spectral measure of A defined by

$\operatorname {E} _{A}(U)=\int _{U}\lambda d\operatorname {E} (\lambda ),$ uniquely determines A and conversely, is uniquely determined by A. EA is a boolean homomorphism from the Borel subsets of R into the lattice Q of self-adjoint projections of H. In analogy with probability theory, given a state S, we introduce the distribution of A under S which is the probability measure defined on the Borel subsets of R by

$\operatorname {D} _{A}(U)=\operatorname {Tr} (\operatorname {E} _{A}(U)S).$ Similarly, the expected value of A is defined in terms of the probability distribution DA by

$\mathbb {E} (A)=\int _{\mathbb {R} }\lambda \,d\,\operatorname {D} _{A}(\lambda ).$ Note that this expectation is relative to the mixed state S which is used in the definition of DA.

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

One can easily show:

$\mathbb {E} (A)=\operatorname {Tr} (AS)=\operatorname {Tr} (SA).$ Note that if S is a pure state corresponding to the vector ψ, then:

$\mathbb {E} (A)=\langle \psi |A|\psi \rangle .$ The trace of an operator A is written as follows:

$\operatorname {Tr} (A)=\sum _{m}\langle m|A|m\rangle .$ ## Von Neumann entropy

Of particular significance for describing randomness of a state is the von Neumann entropy of S formally defined by

$\operatorname {H} (S)=-\operatorname {Tr} (S\log _{2}S)$ .

Actually, the operator S log2 S is not necessarily trace-class. However, if S is a non-negative self-adjoint operator not of trace class we define Tr(S) = +∞; also note that any density operator S can be diagonalized, that it can be represented in some orthonormal basis by a (possibly infinite) matrix of the form

${\begin{bmatrix}\lambda _{1}&0&\cdots &0&\cdots \\0&\lambda _{2}&\cdots &0&\cdots \\\vdots &\vdots &\ddots &\\0&0&&\lambda _{n}&\\\vdots &\vdots &&&\ddots \end{bmatrix}}$ and we define

$\operatorname {H} (S)=-\sum _{i}\lambda _{i}\log _{2}\lambda _{i}.$ The convention is that $\;0\log _{2}0=0$ , 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 of S.

Remark. It is indeed possible that H(S) = +∞ for some density operator S. In fact T be the diagonal matrix

$T={\begin{bmatrix}{\frac {1}{2(\log _{2}2)^{2}}}&0&\cdots &0&\cdots \\0&{\frac {1}{3(\log _{2}3)^{2}}}&\cdots &0&\cdots \\\vdots &\vdots &\ddots &\\0&0&&{\frac {1}{n(\log _{2}n)^{2}}}&\\\vdots &\vdots &&&\ddots \end{bmatrix}}$ T is non-negative trace class and one can show T log2 T is not trace-class.

Theorem. Entropy is a unitary invariant.

In analogy with classical entropy (notice the similarity in the definitions), H(S) measures the amount of randomness in the state S; the more dispersed the eigenvalues are, the larger the system entropy. For a system in which the space H is finite-dimensional, entropy is maximized for the states S which in diagonal form have the representation

${\begin{bmatrix}{\frac {1}{n}}&0&\cdots &0\\0&{\frac {1}{n}}&\dots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &{\frac {1}{n}}\end{bmatrix}}$ For such an S, H(S) = log2 n. The state S is called the maximally mixed state.

Recall that a pure state is one of the form

$S=|\psi \rangle \langle \psi |,$ for ψ a vector of norm 1.

Theorem. H(S) = 0 if and only if S is a pure state.

For S 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 of quantum entanglement.

## Gibbs canonical ensemble

Consider an ensemble of systems described by a Hamiltonian H with average energy E. If H has pure-point spectrum and the eigenvalues $E_{n}$ of H go to +∞ sufficiently fast, er H will be a non-negative trace-class operator for every positive r.

The Gibbs canonical ensemble is described by the state

$S={\frac {\mathrm {e} ^{-\beta H}}{\operatorname {Tr} (\mathrm {e} ^{-\beta H})}}.$ Where β is such that the ensemble average of energy satisfies

$\operatorname {Tr} (SH)=E$ and

$\operatorname {Tr} (\mathrm {e} ^{-\beta H})=\sum _{n}\mathrm {e} ^{-\beta E_{n}}=Z(\beta )$ This is called the 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 $E_{m}$ is

${\mathcal {P}}(E_{m})={\frac {\mathrm {e} ^{-\beta E_{m}}}{\sum _{n}\mathrm {e} ^{-\beta E_{n}}}}.$ Under certain conditions, the Gibbs canonical ensemble maximizes the von Neumann entropy of the state subject to the energy conservation requirement.[clarification needed]

## Grand canonical ensemble

For open systems where the energy and numbers of particles may fluctuate, the system is described by the grand canonical ensemble, described by the density matrix

$\rho ={\frac {\mathrm {e} ^{\beta (\sum _{i}\mu _{i}N_{i}-H)}}{\operatorname {Tr} \left(\mathrm {e} ^{-\beta (H+\sum _{i}\mu _{i}N_{i})}\right)}}.$ where the N1, 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

${\mathcal {Z}}(\beta ,\mu _{1},\mu _{2},\cdots )=\operatorname {Tr} (\mathrm {e} ^{\beta (\sum _{i}\mu _{i}N_{i}-H)})$ 