Expectation value of a product of operators

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Discussion Overview

The discussion revolves around the expectation value of a product of operators, specifically examining the implications of one operator annihilating a state vector in the context of quantum mechanics. Participants explore the mathematical relationships between operators A and B and their effects on a state vector |a>.

Discussion Character

  • Technical explanation, Conceptual clarification, Debate/contested

Main Points Raised

  • One participant questions whether the condition B|a> = 0 implies that = 0, raising the issue of operator ordering.
  • Another participant confirms that = 0 under the given condition, explaining the operational sequence of applying B followed by A.
  • A third participant agrees with the initial assumption but adds a caveat regarding the non-commutativity of operators A and B, suggesting that may not equal 0 if the operators do not commute and if B is not hermitian.
  • A fourth participant discusses the conditions under which the Dirac notation is valid, emphasizing the self-adjoint nature of the operator product AB.

Areas of Agreement / Disagreement

Participants generally agree on the implication that = 0 when B|a> = 0, but there is disagreement regarding the implications for and the conditions under which the Dirac notation applies.

Contextual Notes

There are limitations regarding the assumptions about the operators A and B, particularly concerning their commutativity and hermitian properties, which remain unresolved in the discussion.

ryanwilk
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Just to check something:

If A and B are operators and B|a> = 0, does this imply that <a|AB|a> = 0 ?

Or can you not split up the operators like <a|A (B|a>) ?

Thanks.
 
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Yes. The meaning of AB|a> is to first operate with B, then with A - but the first operation produces a zero.
 
Yes, you are correct to assume <a|AB|a>=0; however, be sure to remember that, it's not necessarily true that <a|BA|a>=0 (if A and B don't commute, and if B is not hermitian, then this may not equal 0)
 
The Dirac notation makes sense, iff AB seen as an operator from D(B) to Ran(A) is self-adjoint. That's why you're permitted to convert the normal scalar product into the <|AB|> notation.
 

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