Quantum statistical mechanics is the study of statistical ensembles of quantum mechanical systems. A statistical ensemble 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. In physics, a statistical ensemble is a very large set of similar systems, considered all at once. ... Fig. ... A density matrix is a self-adjoint (or Hermitian) positive-semidefinite matrix, (possibly infinite dimensional), of trace one, that describes the statistical state of a quantum system. ... In mathematics, a Hilbert space is a real or complex vector space with a positive definite sesquilinear form, that is complete under its norm. ... The mathematical formulation of quantum mechanics is the body of mathematical formalisms which permits a rigorous description of quantum mechanics. ... In mathematical physics and quantum mechanics, quantum logic can be regarded as a kind of propositional logic suitable for understanding the apparent anomalies regarding quantum measurement, most notably those concerning composition of measurement operations of complementary variables. ...

Expectation

From classical probability theory, we know that the expectation of a random variable X is completely determined by its distribution D_X by

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 Integrability is a mathematical concept used in different areas. ... In mathematics, projection-valued measures are used to express results in spectral theory. ...

uniquely determines A and conversely, is uniquely determined by A. E_A 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

Similarly, the expected value of A is defined in terms of the probability distribution D_A by

Note that this expectation is relative to the mixed state S which is used in the definition of D_A.

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. In functional analysis, the Borel functional calculus is a functional calculus (i. ...

One can easily show:

Note that if S is a pure state corresponding to the vector ψ, The term pure state refers to several related concepts in physics, particularly quantum mechanics and in functional analysis. ...

Von Neumann entropy

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

.

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

and we define

The convention is that , 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 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 Entropy of a Bernoulli trial as a function of success probability. ...

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 The term pure state refers to several related concepts in physics, particularly quantum mechanics and in functional analysis. ...

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. It has been suggested that Quantum coherence be merged into this article or section. ...

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, e^{-r H} will be a non-negative trace-class operator for every positive r.

The Gibbs canonical ensemble is described by the state

where β is such that the ensemble average of energy satisfies

,and

is the quantum mechanical version of the canonical partition function. The probability that a system chosen at random from the ensemble will be in a state corresponding to energy eigenvalue E_m is A canonical ensemble in statistical mechanics is an ensemble of dynamically similar systems, each of which can share its energy with a large heat reservoir, or heat bath. ...

Under certain conditions, the Gibbs canonical ensemble maximizes the von Neumann entropy of the state subject to the energy conservation requirement.

References

• J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, 1955.

• F. Reif, Statistical and Thermal Physics, McGraw-Hill, 1985.