quantum statistical mechanics - Expectation, Von Neumann entropy, Gibbs canonical ensemble

An extension of statistical mechanics in which quantum conditions on individual particles are taken into account, especially restrictions imposed by the uncertainty principle.

Expectation

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

assuming, of course, that the random variable is integrable or that the random variable is non-negative. 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 DA by

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

One can easily show:

Note that if S is a pure state corresponding 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

. 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. In fact T be the diagonal matrix

T is non-negative trace class and one can show T log2 T is not trace-class.

In analogy with classical entropy (notice the similarity in the definitions), H(S) measures the amount of randomness in the state S. 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.

Recall that a pure state is one of the form

for ψ a vector of norm 1.

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

Gibbs canonical ensemble

Consider an ensemble of systems described by a Hamiltonian H with average energy E.

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 Em is

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