Operators and Complete State Descriptions in Quantum Mechanics

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SUMMARY

Operators are essential in quantum mechanics (QM) for providing a complete state description of quantum objects. While Feynman's path integral approach utilizes regular numbers instead of operators, it lacks mathematical rigor compared to the operator framework. The discussion emphasizes the importance of Hilbert spaces as well-defined mathematical structures in QM, as highlighted in P.A.M. Dirac's "The Principles of Quantum Mechanics".

PREREQUISITES
  • Understanding of quantum mechanics fundamentals
  • Familiarity with Hilbert spaces
  • Knowledge of operators in quantum theory
  • Basic concepts of Feynman's path integral formulation
NEXT STEPS
  • Study the mathematical foundations of Hilbert spaces in quantum mechanics
  • Explore the role of operators in quantum mechanics
  • Research Feynman's path integral approach and its implications
  • Read P.A.M. Dirac's "The Principles of Quantum Mechanics" for deeper insights
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Students and professionals in physics, particularly those focusing on quantum mechanics, as well as researchers interested in the mathematical structures underlying quantum theory.

black_kitty
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What for do we need operators in QM. Where is the complete state description of a quantum object?
 
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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.
 
black_kitty said:
What for do we need operators in QM. Where is the complete state description of a quantum object?

P.A.M. Dirac, “The Principles of Quantum Mechanics”, 4-th edition, Oxford, (1958).

The Hilbert spaces are perfectly defined mathematical structures.

Regards, Dany.
 

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