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**"Expectation value", a few questions**

I've read that in quantum mechanics we use the term "expectation value" for example for the energy of a system. Despite its name, the expectation value of the energy of for instance the quantum harmonic oscillator is not the most probable measured energy of the particle under such a potential. It is the mean value. And in the case of the harmonic oscillator, the mean value can be an energy that isn't allowed for the particle under the potential.

So what does it really -physically- mean? Is it just a mean? If so, why is it important? Why aren't we dealing with the "most probable" value(s), in quantum mechanics?

I don't understand why would the average of thousands of measures be important, especially when this average doesn't represent anything possible (like in the quantum harmonic oscillator).

And by the way, why is it called the "expectation value" rather than "mean value" or "average value"?