Solving an Interesting Problem with Two Hermitian Operators

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Discussion Overview

The discussion revolves around the properties of two Hermitian operators, O1 and O2, particularly focusing on the implications of measuring these operators in quantum mechanics. Participants explore the concept of uncertainty in the context of these operators, especially after measuring the energy associated with the Hamiltonian operator O1.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant presents a scenario involving two Hermitian operators, suggesting that measuring the energy leads to a state with zero uncertainty regarding the outcome of the second operator O2.
  • Another participant questions the definition of 'uncertainty' and asks for clarification on the role of the second operator in the context of the measurement.
  • A different participant argues that the uncertainty of O2 in state |1> cannot be zero, as |1> is not an eigenstate of O2.
  • One participant defines uncertainty mathematically, stating that the expectation value of O2 in state |1> is zero, while the expectation value of O2 squared is one, leading to an uncertainty of one.
  • Another participant agrees with the previous point, reinforcing that the uncertainty of O2 in state |1> is indeed one and questions the initial assumption of zero uncertainty.
  • One participant elaborates on the nature of measurements, explaining that if a measurement yields a single result, uncertainty is absent, but emphasizes that |1> and |2> are not eigenvectors of O2, suggesting a need for a different basis for analysis.
  • A later reply acknowledges the previous explanation and mentions the uncertainty relation for the operators and state |1>, indicating that the left-hand side of the relation is zero.

Areas of Agreement / Disagreement

Participants express disagreement regarding the uncertainty associated with operator O2 in state |1>. While some argue that the uncertainty is one, others initially suggest it could be zero, leading to an unresolved discussion on the correct interpretation of uncertainty in this context.

Contextual Notes

The discussion highlights the complexity of quantum measurements and the dependence on the definitions of eigenstates and operators. There are unresolved assumptions regarding the nature of the states involved and the implications of the measurements performed.

chafelix
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Interesting problem: Suppose one has 2 hermitian operators, O1,O2 with distinct eigenvalues.
Say the first is the Hamiltonian. We measure the energy, get a value, say E_1. So the system
is in |1>. Suppose now that the second operator has the property to turn state 1 into state |2> and operating on |2> gives |1>
Hence if after having measured the energy to be E_1, we operate on the state with the second op, we get |2>. No uncertainty about the outcome.
So should not the uncertainity be 0?
So O2|1>=|2>,<1|O2|1>=0
O2**2|1>=O2|2>=|1>. So <1|O2**2|1>=<1|1>=1
But uncertainity is (<1|O2**2|1>-<1|O2|1>**2)=1, not 0.
 
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Could you elaborate on how you define 'uncertainty' ?

I thought it was about two different non-commuting observables, but you describe the energy here only?

What is the other operator here?
 
Why do you think the uncertainty of O2 in state |1> should be 0? |1> is not an eigenstate of O2.
 
Uncertainity is sqrt(expectation value of operator**2)-square of expectation value)
So: Expectation value of operator O2 is <1|O2|1>=0
Expectation value of O2**2 is 1
 
Yes, and that means that the uncertainty of O2 in state |1> is 1. Why do you think it should be 0?
 
If any experiment can only give a single result, then there is no uncertainity.
In this case where we start with a prepared state |1>, a measurement of O2 can only give the result |2>. It has 100% |2> character and 0% |1> character.
In contrast if O2|1> were a|1>+b|2>, then the outcome would be uncertain, with respective probabilities to give |1> or |2> that are |a|**2 and |b|**2

The answer is that |1> and |2> are not eigenvectors of O2, so what one should do is use a different basis that is the basis of the O2 eigenvalues, Remember, the possible outcomes are only eigenvalues of the operator measured(here O2) and the state the system is left in is an eigensate of O2. If you do that , then the new basis is of the form
|P>=c|1>+d|2>, so that O2|P>=p|P>=>c|2>+d|1>=pd|2>+pc|1>=>
pd=c,pc=d=>p=+-1, so that a state |2> is not an eigenstate of O2, but a linear combination of its eigenstates (|1>+|2>)/sqrt(2) and (|1>-|2>)/sqrt(2)
 
That's right. (The second half of your post. I assume the first half is just what you were thinking before). The uncertainty relation for these operators and the state |1> is 0·1≥0. Well, I haven't actually calculated the right-hand side, but it would have to be 0, since the left-hand side is 0.
 

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