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:
“The circumstances of human society are too complicated to be submitted to the rigour of mathematical calculation.”
—Marquis De Custine (1790–1857)
“A personality is an indefinite quantum of traits which is subject to constant flux, change, and growth from the birth of the individual in the world to his death. A character, on the other hand, is a fixed and definite quantum of traits which, though it may be interpreted with slight differences from age to age and actor to actor, is nevertheless in its essentials forever fixed.”
—Hubert C. Heffner (1901–1985)
“Mathematics should be mixed not only with physics but with ethics.”
—Henry David Thoreau (1817–1862)