Understanding Operators & Observables: Averages & Variance

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SUMMARY

The average value and variance of an observable in quantum mechanics are directly dependent on the quantum state of the system. Specifically, when dealing with a time-independent Hamiltonian, operators that commute with the Hamiltonian will have constant average values. In contrast, operators that do not commute with the Hamiltonian will exhibit time-dependent averages. Understanding these relationships is crucial for analyzing the behavior of quantum systems.

PREREQUISITES
  • Quantum mechanics fundamentals
  • Understanding of Hamiltonian operators
  • Knowledge of commutation relations
  • Familiarity with observables in quantum systems
NEXT STEPS
  • Study the properties of time-independent Hamiltonians
  • Learn about commutation relations in quantum mechanics
  • Explore the implications of observables on quantum state evolution
  • Investigate the role of variance in quantum measurements
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Students and professionals in quantum mechanics, physicists analyzing quantum systems, and anyone interested in the mathematical foundations of quantum observables.

Feynmanfan
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I have trouble understanding operators and observables:

Does <A> the average value of an observable and the variance depend on the state of the system? What does it have to do with the Hamiltonian?

Thanks!
 
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Basically none.Yes,both the variance and the expectation value/average depend upon the quantum state.

Daniel.
 
This is too general question. With respect to Hamiltonian there are two types of operators- those which commute and those which does not commute with Hamiltonian.
If we have a time independent Hamiltonian, then those operators which commute with it will have their average value constant.
 

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