The uncertainty operator and Heisenberg

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Discussion Overview

The discussion revolves around the Heisenberg uncertainty relation, specifically focusing on the mathematical treatment of uncertainty operators defined for Hermitian operators A and B. Participants explore the implications of subtracting expectation values from operators and the notation used in these expressions.

Discussion Character

  • Technical explanation, Debate/contested

Main Points Raised

  • One participant questions how it is valid to subtract an expectation value, which is a number, from an operator A.
  • Another participant argues that a number can also be considered an operator and provides a mathematical expression for variance to support this view.
  • A third participant notes the presence of an implicit identity operator multiplying the expectation value.
  • A later reply critiques the notation used, suggesting that it is misleading due to the omission of the unit operator when scaling by the expectation value.

Areas of Agreement / Disagreement

Participants express differing views on the validity of the notation and the treatment of expectation values in relation to operators, indicating that the discussion remains unresolved.

Contextual Notes

There are unresolved questions regarding the notation and the assumptions underlying the treatment of operators and expectation values.

dyn
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In deriving the Heisenberg uncertainty relation for 2 general Hermitian operators A and B , the uncertainty operators ΔA and ΔB are introduced defined by ΔA=A - (expectation value of A) and similarly for B.
My question is this - how can you subtract(or add) an expectation value , which is just a number to A which is an operator ?
 
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A number is also an operator. But you can also see it by the variance being <A.A>-<A>2, where <A.A> is a number and <A> is a number.

<(A-<A>)2>
= <A.A-2<A>A+<A>2>
= <A.A>-<2<A>A>+<<A>2>
= <A.A>-2<A><A>+<<A>2>
= <A.A>-<A>2
 
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There's an implicit identity operator multiplying ##\langle A \rangle##.
 
That's sloppy and indeed misleading notation. The unit operator on the states' space is not written when scaled by the expectation value.
 

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