Quantum mechanical observables of a system are represented by self - adjoint operators in a separable complex Hilbert space H. Now I understand a lot of operators employed in quantum mechanics are unbounded operators, in nutshell these operators cannot be defined for all vectors in H. For example according to "Stone - von Neumann", the canonical commutation relation [P,Q]=−iℏI has no solution for P and Q bounded ! My basic question is :
- If the state of our system ψ is for example not in the domain of P (because P is unbounded), i.e., if P⋅ψ does not, mathematically, make sense, what does this mean ? Does it mean we cannot extract any information about P when the system is in state ψ ?
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