Quantum Statistical Mechanics - Expectation

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. 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. EA 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 DA by

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

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,

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