How to show that expectation value is always positive?

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SUMMARY

The discussion focuses on demonstrating that the expectation value in quantum mechanics is always non-negative. Participants emphasize the importance of showing that the operator A+A is self-adjoint, which is a prerequisite for establishing the non-negativity of the expectation value. A counter-example involving a Gaussian wavepacket centered at a negative position is mentioned to illustrate potential misconceptions. The conclusion drawn is that a rigorous approach to the properties of self-adjoint operators is essential for proving the positivity of expectation values.

PREREQUISITES
  • Understanding of quantum mechanics principles
  • Familiarity with self-adjoint operators
  • Knowledge of expectation values in quantum systems
  • Experience with Gaussian wavepackets
NEXT STEPS
  • Study the properties of self-adjoint operators in quantum mechanics
  • Learn about the mathematical formulation of expectation values
  • Explore examples of Gaussian wavepackets and their implications
  • Investigate the implications of non-negative operators in quantum theory
USEFUL FOR

Students of quantum mechanics, physicists exploring operator theory, and anyone interested in the mathematical foundations of expectation values in quantum systems.

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Homework Statement


In quantum mechanics, how to show that the expectation value is always positive?



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The Attempt at a Solution

 
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Can you be more specific? Because as you stated it, one can easily find a counter-example (e.g. <x> for a Gaussian wavepacket centered at some x0 < 0).
 
CompuChip said:
Can you be more specific? Because as you stated it, one can easily find a counter-example (e.g. <x> for a Gaussian wavepacket centered at some x0 < 0).
Questions just tell us to show that A+A is always self-adjoin and then show that expectation is non-negative.
 

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