Quantum statistical mechanics
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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 nonnegative, selfadjoint, traceclass 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.
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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 nonnegative. Similarly, let A be an observable of a quantum mechanical system. A is given by a densely defined selfadjoint operator on H. The spectral measure of A defined by
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 selfadjoint 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.
One can easily show:
Note that if S is a pure state corresponding to the vector ψ, then:
The trace of an operator A is written as follows:
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 log_{2} S is not necessarily traceclass. However, if S is a nonnegative selfadjoint 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 nonnegative trace class and one can show T log_{2} T is not traceclass.
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 finitedimensional, entropy is maximized for the states S which in diagonal form have the representation
For such an S, H(S) = log_{2} n. The state S is called the maximally mixed state.
Recall that a pure state is one of the form
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 nonzero 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 purepoint spectrum and the eigenvalues of H go to +∞ sufficiently fast, e^{−r H} will be a nonnegative traceclass 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
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 is
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
where the N_{1}, N_{2}, ... 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
References
 J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, 1955.
 F. Reif, Statistical and Thermal Physics, McGrawHill, 1965.