What for do we need operators in QM. Where is the complete state description of a quantum object?
You don't *need* operators--for example, Feynman's path integral approach to quantum mechanics only uses regular old numbers (no operators)... Of course, you may not like path integrals any better than operators since they are not really well-defined mathematically.
P.A.M. Dirac, “The Principles of Quantum Mechanics”, 4-th edition, Oxford, (1958).
The Hilbert spaces are perfectly defined mathematical structures.
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