# 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 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 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

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.

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:

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## Von Neumann entropy

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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 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* log_{2} *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*) = 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 non-zero entry which is a 1.

Entropy can be used as a measure of quantum entanglement.

## Gibbs canonical ensemble

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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

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.Template:Clarify

## Grand canonical ensemble

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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*, McGraw-Hill, 1965.