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
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