Why are the only possible values of an operator its eigenvalues?

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SUMMARY

The only possible values of an operator in quantum mechanics are its eigenvalues, as established in foundational texts such as Shankar's "Principles of Quantum Mechanics." This principle is rooted in the postulates of quantum mechanics, which dictate that observable quantities correspond to operators, and their measurable values are given by the eigenvalues of these operators. The discussion references multiple resources, including a Stack Exchange thread and quantum notes from the University of Washington, which provide deeper insights into this concept.

PREREQUISITES
  • Understanding of quantum mechanics principles
  • Familiarity with operators and eigenvalues in linear algebra
  • Knowledge of observable quantities in quantum systems
  • Basic grasp of postulates of quantum mechanics
NEXT STEPS
  • Study the postulates of quantum mechanics in detail
  • Explore linear algebra concepts related to eigenvalues and eigenvectors
  • Review Shankar's "Principles of Quantum Mechanics" for comprehensive insights
  • Investigate the implications of eigenvalues in quantum state measurements
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Students and professionals in physics, particularly those studying quantum mechanics, as well as educators looking to clarify the relationship between operators and eigenvalues in quantum systems.

randomafk
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Like the title says, why are the only possible values of an operator its eigenvalues?

reading shankar right now and I'm having difficulty understanding why this has to be the case, given some operator/variable Ω
 
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