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I Can the expectation of an operator be imaginary?

  1. Nov 22, 2016 #1
    Assume ##\varPsi## is an arbitrary quantum state, and ##\hat{O}## is an arbitrary quantum operator, can the expectation $$\int\varPsi^{*}\hat{O}\varPsi$$ be imaginary?
     
  2. jcsd
  3. Nov 22, 2016 #2

    hilbert2

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    Not if the operator is hermitian. Otherwise it can be. For instance, think about the situation where ##\Psi## is normalized to unity and the operator is just a multiplication with ##i## (imaginary unit).
     
  4. Nov 22, 2016 #3
    Thank you very much! I have known the deduction.
    According to the definition of Hermitian operator denoted by ##\hat{O}## as follows, $$<f|\hat{O}g>=<\hat{O}f|g>,$$ and making ##f=g##, we have $$<f|\hat{O}f>=<\hat{O}f|f>=(<f|\hat{O}f>)^{*},$$ meaning ##<f|\hat{O}f>## is real, which resembles the equation $$a=a^{*},$$ meaning ##a## is real.
     
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