Friday, May 19, 2017

mathematical physics - Implications of unbounded operators in quantum mechanics


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]=iI 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 Pψ 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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classical mechanics - Moment of a force about a given axis (Torque) - Scalar or vectorial?

I am studying Statics and saw that: The moment of a force about a given axis (or Torque) is defined by the equation: $M_X = (\vec r \times \...