How to Calculate Uncertainty of an Operator with Known State

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    Operator Uncertainty
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Discussion Overview

The discussion focuses on calculating the uncertainty of an operator given a known quantum state. Participants explore different methods for determining this uncertainty, including the mathematical formulation and conceptual understanding of operators and their expectation values.

Discussion Character

  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • One participant presents a formula for calculating uncertainty, \(\Delta\Omega^2 = \langle\Psi|(\Omega - \langle\Omega\rangle)^2|\Psi\rangle\), and expresses confusion about the definition of \((\Omega - \langle\Omega\rangle)\), questioning if it is an operator minus a scalar.
  • Another participant clarifies that the scalar is multiplied by the identity operator, allowing for the subtraction to be valid.
  • There is mention of an alternative method to find \(\Delta\Omega^2\) by summing the products of probabilities of all states with their deviations from the expected value squared, although the first participant is seeking a method that does not require knowledge of all probabilities.
  • A later reply suggests that the initial formula indeed allows for breaking down the state into a weighted average of eigenstates, which facilitates the calculation of uncertainty without explicitly knowing all probabilities.

Areas of Agreement / Disagreement

Participants generally agree on the equivalence of the two methods for calculating uncertainty, but there is still some uncertainty regarding the interpretation of the operator and scalar subtraction.

Contextual Notes

Participants do not fully resolve the definitions and implications of the operator subtraction, nor do they clarify the assumptions behind the methods discussed.

Vaal
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Given the state and an operator I know the uncertainty of this operator can be calculated via

(see next post latex is being weird, sorry)
 
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\Delta\Omega2=<\Psi|(\Omega - <\Omega>)2|\Psi> (hope that is legible)

but I'm confused as to how the middle, (\Omega -<\Omega>) is defined. Isn't this an operator minus a scalar?

I know I can also find \Delta\Omega2 by summing the the products of the probabilities of all the states with the states deviation from the expected value squared, but I thought there was a way to do this without having to know all the probabilities. Thanks.
 
Vaal said:
\Delta\Omega2=<\Psi|(\Omega - <\Omega>)2|\Psi> (hope that is legible)

but I'm confused as to how the middle, (\Omega -<\Omega>) is defined. Isn't this an operator minus a scalar?
The scalar is multiplied by the identity operator. Then you can subtract them, and things work out like you'd expect.

Vaal said:
I know I can also find \Delta\Omega2 by summing the the products of the probabilities of all the states with the states deviation from the expected value squared, but I thought there was a way to do this without having to know all the probabilities. Thanks.

I think that's exactly what the above is doing. Whenever you have \langle\Psi|\Omega|\Psi\rangle, you can envision breaking down the state into a weighted average of eigenstates of the operator. Then you know that the operator's effect on each eigenstate will just be multiplying it by the eigenvalue, so that allows you to turn the calculation into a weighted average of eigenvalues. For the above expression, I believe you can do the same thing: break down \Psi into weighted eigenstates, then apply the operator to each eigenstate separately to get the eigenvalue, subtract the expectation value from it, square it, and then sum them all up according to the original weights on the states.
 
Yeah, I thought that might be that case but I wasn't sure. Thanks.

The two definitely are pretty much equivalent, I just wasn't quite seeing how so, thanks again.
 

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