In Mathematical Quantum Physics
Let H be a complex separable Hilbert space, be a one-parametric group of unitary operators on H and be a statistical operator on H. Let A be an observable on H and let be the induced probability distribution of A with respect to ρ on the Borel σ-algebra on . Then the evolution of ρ induced by U is said to be bound with respect to A if, where .
Example: Let and let A be the position observable. Let have compact support and .
- If the state evolution of ρ "moves this wave package constantly to the right", e.g. if for all, then ρ is not a bound state with respect to the position.
- If does not change in time, i.e. for all, then is a bound state with respect to position.
- More generally: If the state evolution of ρ "just moves ρ inside a bounded domain", then ρ is also a bound state with respect to position.
Read more about this topic: Bound State
Famous quotes containing the words mathematical, quantum and/or physics:
“As we speak of poetical beauty, so ought we to speak of mathematical beauty and medical beauty. But we do not do so; and that reason is that we know well what is the object of mathematics, and that it consists in proofs, and what is the object of medicine, and that it consists in healing. But we do not know in what grace consists, which is the object of poetry.”
—Blaise Pascal (1623–1662)
“The receipt to make a speaker, and an applauded one too, is short and easy.—Take of common sense quantum sufficit, add a little application to the rules and orders of the House, throw obvious thoughts in a new light, and make up the whole with a large quantity of purity, correctness, and elegancy of style.”
—Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)
“The labor we delight in physics pain.”
—William Shakespeare (1564–1616)