Expectation value

QuestionsCategory: Quantum MechancisExpectation value
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Best Answer
Big Bang Boffin Staff answered 5 years ago

Although the notation used to denote the expectation value of some observable is the same as the one used for the mean(average), the expectation value is a little different than an ordinary average. Notation being: \left<p\right> for the expectation value of p.
You might have studied in Quantum Mechanics that the particles(systems) can be represented by wave functions and this wave function collapses when you measure something and you get a particular answer. But now since the wave function has collapsed to a particular state(one of its eigenstates), if you make the same measurement again, you will still get the same answer(which is the eigenvalue for that state). So no matter how many times you repeat the measurement you are going to get the same result because your wave function has collapsed to give a particular value, for the measurement of that quantity. 
This puts us in a pickle, as to how to measure average quantities for Quantum Mechanical systems, like average velocity, energies, etc.
The dilemma is overcome by doing the following.

Instead, of making multiple repeated measurements on the same system, we make a single measurement on a number of identical systems. Identical here means, that all the systems had the same wave functions.

When you make a measurement for energy on n-identical systems, what you expect to get is one of the eigenvalues of energy. So once you have made all the measurements, you can take their average(add them and divide by the number of systems), and this average is the expectation value.
Hope it helps!