Quantum statistical mechanics

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

 \mathbb{E}(X) = \int_\mathbb{R} \lambda \, d \, \operatorname{D}_X(\lambda)




 \operatorname{D}_A(U) = \operatorname{Tr}(\operatorname{E}_A(U) S).


 \mathbb{E}(A) = \int_\mathbb{R} \lambda \, d \, \operatorname{D}_A(\lambda).



One can easily show:

 \mathbb{E}(A)  = \operatorname{Tr}(A S) = \operatorname{Tr}(S A).

Note that if S is a pure state corresponding to the vector ψ, then:

 \mathbb{E}(A) = \langle \psi | A | \psi \rangle.

The trace of an operator A is written as follows:

 \operatorname{Tr}(A) = \sum_{m} \langle m  |  A |  m \rangle  .

Von Neumann entropy[edit]

Main article: Von Neumann entropy


 \operatorname{H}(S) = -\operatorname{Tr}(S \log_2 S) .


 \begin{bmatrix} \lambda_1 & 0 & \cdots & 0 & \cdots \\ 0 & \lambda_2 & \cdots & 0 & \cdots\\ & & \cdots &  \\ 0 & 0 & \cdots &  \lambda_n & \cdots \\ & & \cdots & \cdots \end{bmatrix}

and we define

 \operatorname{H}(S) = - \sum_i \lambda_i \log_2 \lambda_i.

The convention is that  \; 0 \log_2 0 = 0, 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.


 T = \begin{bmatrix} \frac{1}{2 (\log_2  2)^2 }& 0 & \cdots & 0 & \cdots \\ 0 & \frac{1}{3 (\log_2  3)^2 } & \cdots & 0 & \cdots\\ & & \cdots &  \\ 0 & 0 & \cdots &  \frac{1}{n (\log_2  n)^2 } & \cdots \\ & & \cdots & \cdots \end{bmatrix}


Theorem. Entropy is a unitary invariant.


 \begin{bmatrix} \frac{1}{n} & 0 & \cdots & 0 \\ 0 & \frac{1}{n} & \dots & 0 \\ & & \cdots & \\ 0 & 0 & \cdots &  \frac{1}{n} \end{bmatrix}


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.


Entropy can be used as a measure of quantum entanglement.

Gibbs canonical ensemble[edit]

Main article: 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

 S= \frac{e^{- \beta H}}{\operatorname{Tr}(e^{- \beta H})}.

Where β is such that the ensemble average of energy satisfies

 \operatorname{Tr}(S H) = E


\operatorname{Tr}(e^{- \beta H}) = \sum_n e^{- \beta E_n} = Z(\beta)

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 E_m is

\frac{e^{- \beta E_m}}{\sum_n e^{- \beta E_n}}.


Grand canonical ensemble[edit]


 \rho= \frac{e^{- \beta (H + \mu_1 N_1 + \mu_2 N_2 + \cdots)}}{\operatorname{Tr}(e^{- \beta (H + \mu_1 N_1 + \mu_2 N_2 + \cdots)})}.


The grand partition function is

\mathcal Z(\beta, \mu_1, \mu_2, \cdots) = \operatorname{Tr}(e^{- \beta (H + \mu_1 N_1 + \mu_2 N_2 + \cdots)})


  • F. Reif, Statistical and Thermal Physics, McGraw-Hill, 1965.