"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"? 
Re: "Expectation value", a few questions
I don't believe the expectation value can be in the forbidden region. That should be prohibited by the Correspondence principle which says that expectation values evolve like their classical counterparts.
The term expectation value is very similar to the terms mean value or average value, but mean and average may make some people think of timeaveraging (e.g. like the RMS of the current in some AC system or something like). I think you call it expectation value because of the probabilistic nature of QM rather than confuse it with timeaveraging of a deterministic system. I don't know the exact historical reasons; however. The expectation value IS important because, like you said, it's the mean value if you took an average of a large ensemble. The "most probable" value(s) would be the modes of the distribution. Usually you care about both values, but with a slight emphasis on the average. 
Re: "Expectation value", a few questions
Actually the word "expectation value" is not specific to QM, it's just a term that they borrowed from general probability theory. If you consider a fair die, for example, then the expectation value is 3.5  which is impossible to get in a single throw :)
There is a bit of history on the origin of expectation values here By the way, Matterwave, wouldn't the potential [tex]V(x) = \begin{cases} 0 & 2a < x < a \\ \infty & \text{otherwise} \end{cases}[/tex] give <x> = 0 due to its symmetry? 
Re: "Expectation value", a few questions
It seems that you are right. Although, one could never actually detect the particle in the forbidden region, due to symmetry, it would appear that the expectation value must be there.

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0> + 1> The "expected" value of energy (measured from the ground state in units hbar omega) is (0+1)/2=1/2, but if you measure energy you will get either 0 or 1, never 1/2. 
Re: "Expectation value", a few questions
I think its expression is better for understanding. Expectation <Q>=<ψQψ>=Ʃpλ, which is exactly the same as that in statistics. To my perspective, since there is no such thing as probability wave in classical physics, expectation is the best approximation which can be reduced to classical mechanics.

Re: "Expectation value", a few questions
Admittedly, expectation is not necessarily expected. It's just arithmetical mean. Just like in statistics you can always get expectation like 3.5 persons per family, which is obviously not expected.
But when you come to a classical case, I think the variance of the distribution of certain observable of any macroscopic quantity is too small. When it is that small almost all the observables lies extremely close to the expectation. 
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Do you want me to apologize or what? 
Re: "Expectation value", a few questions
I think <expectation value> is just a misleading name given to what actually is the <average of an observable on an ensemble>.

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But actually I wanted to point out that the points are slightly different. My QM example just showed that the expectation value may be in the classically prohibited region, although x = 0 was still an "allowed" value (to be more precise: an eigenvalue of the position operator). If you measure the position, you could very well find x = 0. So even though according to Newton x = 0 is not allowed, it is a perfectly normal number if you use the QM model instead of the classical model of physics. Demystifier's example shows a system where the expectation value is not even an "allowed" value: whenever you measure the energy you will find 0 or 1 (in proper units :P). You can never find 1/2, not even in theory. The latter is more analogous to the example I gave of 3.5 dots on a die, or the statistical average of 1.4 children per family that was mentioned later. The first one is more analogous to having an expectation value of 180 children in a family, which is actually an "allowed" outcome according to our intuition about integer vs fractional numbers of people (it's just not feasible according to our "classical" model of society; which doesn't mean that if you use the Martian model instead of the earth model, that 180 cannot be a perfectly good outcome). 
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