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,

Read more about this topic:  Quantum Statistical Mechanics

Famous quotes containing the word expectation:

    “No expectation fails there,
    No pleasing habit ends,
    No man grows old, no girl grows cold,
    But friends walk by friends.”
    —William Butler Yeats (1865–1939)

    “I have no expectation that any man will read history aright who thinks that what has been done in a remote age, by men whose names have resounded far, has any deeper sense than what he is doing to-day.”
    —Ralph Waldo Emerson (1803–1882)

    “The expectation that every neurotic phenomenon can be cured may, I suspect, be derived from the layman’s belief that the neuroses are something quite unnecessary which have no right whatever to exist. Whereas in fact they are severe, constitutionally fixed illnesses, which rarely restrict themselves to only a few attacks but persist as a rule over long periods throughout life.”
    —Sigmund Freud (1856–1939)