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I thought the definition of the average value of $f(x)$ was $f* = \frac{1}{T}\int_0^T f(t)dt$

How do I get from this definition to the definition of the mean value of a function in terms of eigenvalues?

The mean value of a function is the sum of all the eigenvalues, each multiplied by a probability...

bobby
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  • are you talking about the expected value of an observable in quantum mechanics? see http://vergil.chemistry.gatech.edu/notes/quantrev/node15.html and http://en.wikipedia.org/wiki/Expectation_value_%28quantum_mechanics%29#Formalism_in_quantum_mechanics – symplectomorphic May 24 '14 at 22:52
  • There you've started by defining mean values via a probability density and ended up with them via eigenvalues by assuming the probability density is bilinear. In Landau's QM he starts by 'defining' mean values in terms of eigenvalues and ends up with a bilinear probability density - How does one even think of defining mean values in terms of eigenvalues without already knowing to define mean values in terms of a bilinear probability density, seems circular... – bobby May 25 '14 at 18:23

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