Editing Quantum statistical mechanics (section)

== Expectation ==
{{See also|Expectation value (quantum mechanics)|Density matrix#Measurement}}
In quantum mechanics a [[statistical ensemble (mathematical physics)|statistical ensemble]] ([[probability distribution]] over possible [[quantum state]]s) is described by a [[density matrix|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.

From classical probability theory, we know that the [[expected value|expectation]] of a [[random variable]] ''X'' is defined by its [[Probability distribution|distribution]] D<sub>''X''</sub> by
<math display="block"> \mathbb{E}(X) = \int_\mathbb{R}d \lambda \operatorname{D}_X(\lambda) </math>
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

<math display="block"> \operatorname{E}_A(U) = \int_U d\lambda \operatorname{E}(\lambda), </math>

uniquely determines ''A'' and conversely, is uniquely determined by ''A''.  E<sub>''A''</sub>  is a [[Boolean homomorphism]] from the [[Borel subset]]s of '''R''' into the [[Lattice (order)|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
<math display="block"> \operatorname{D}_A(U) = \operatorname{Tr}(\operatorname{E}_A(U) S). </math>
Similarly, the expected value of ''A'' is defined in terms of the probability distribution D<sub>''A''</sub> by
<math display="block"> \mathbb{E}(A) = \int_\mathbb{R} d\lambda \,  \operatorname{D}_A(\lambda).</math>
Note that this expectation is relative to the  mixed state ''S'' which is used in the definition of D<sub>''A''</sub>.

'''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:
<math display="block"> \mathbb{E}(A)  = \operatorname{Tr}(A S) = \operatorname{Tr}(S A). </math>
The [[trace of an operator]] ''A'' is written as follows:
<math display="block"> \operatorname{Tr}(A) = \sum_{m} \langle m  |  A |  m \rangle  . </math> 
Note that if ''S'' is a [[pure state]] corresponding to the [[Euclidean vector|vector]] <math>\psi</math>, then:
<math display="block"> \mathbb{E}(A) = \langle \psi | A | \psi \rangle. </math>