- Quantum statistical mechanics
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.
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
integrableor 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 measureof "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 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 calculusfor unbounded operators.
One can easily show::
Note that if "S" is a
pure statecorresponding to the vector ψ, :
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:For such an "S", H("S") = log2 "n". The state "S" is called the maximally mixed state.
Recall that a
pure stateis 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 non-zero entry which is a 1.
Entropy can be used as a measure of
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 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 :
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 is
Under certain conditions, the Gibbs canonical ensemble maximizes the von Neumann entropy of the state subject to the energy conservation requirement.
Partition function (mathematics)
* J. von Neumann, "Mathematical Foundations of Quantum Mechanics", Princeton University Press, 1955.
* F. Reif, "Statistical and Thermal Physics", McGraw-Hill, 1965.
Wikimedia Foundation. 2010.
Look at other dictionaries:
quantum statistical mechanics — kvantinė statistinė mechanika statusas T sritis fizika atitikmenys: angl. quantum statistical mechanics vok. quantenstatistische Mechanik, f rus. квантовая статистическая механика, f pranc. mécanique quantostatistique, f … Fizikos terminų žodynas
Statistical mechanics — is the application of probability theory, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force.Statistical mechanics … Wikipedia
statistical mechanics — Physics, Chem. (used with a sing. v.) the science that deals with average properties of the molecules, atoms, or elementary particles in random motion in a system of many such particles and relates these properties to the thermodynamic and other… … Universalium
Microstate (statistical mechanics) — In statistical mechanics, a microstate is a specific microscopic configuration of a thermodynamic system that the system may occupy with a certain probability in the course of its thermal fluctuations. In contrast, the macrostate of a system… … Wikipedia
Partition function (statistical mechanics) — For other uses, see Partition function (disambiguation). Partition function describe the statistical properties of a system in thermodynamic equilibrium. It is a function of temperature and other parameters, such as the volume enclosing a gas.… … Wikipedia
Timeline of thermodynamics, statistical mechanics, and random processes — A timeline of events related to thermodynamics, statistical mechanics, and random processes. Ancient times *c. 3000 BC The ancients viewed heat as that related to fire. The ancient Egyptians viewed heat as related to origin mythologies. One… … Wikipedia
Correlation function (statistical mechanics) — For other uses, see Correlation function (disambiguation). In statistical mechanics, the correlation function is a measure of the order in a system, as characterized by a mathematical correlation function, and describes how microscopic variables… … Wikipedia
Statistical physics — is one of the fundamental theories of physics, and uses methods of statistics in solving physical problems. It can describe a wide variety of fields with an inherently stochastic nature. Examples include problems involving nuclear reactions, and… … Wikipedia
Quantum state — In quantum physics, a quantum state is a mathematical object that fully describes a quantum system. One typically imagines some experimental apparatus and procedure which prepares this quantum state; the mathematical object then reflects the… … Wikipedia
Quantum logic — In mathematical physics and quantum mechanics, quantum logic is a set of rules for reasoning about propositions which takes the principles of quantum theory into account. This research area and its name originated in the 1936 paper by Garrett… … Wikipedia