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)
“Sweet pliability of man’s spirit, that can at once surrender itself to illusions, which cheat expectation and sorrow of their weary moments!—long—long since had ye number’d out my days, had I not trod so great a part of them upon this enchanted ground.”
—Laurence Sterne (1713–1768)
“A youthful mind is seldom totally free from ambition; to curb that, is the first step to contentment, since to diminish expectation is to increase enjoyment.”
—Frances Burney (1752–1840)