# Expectation value of an observable in quantum system

In quantum systems, sometimes if the wave function is not an eigen value of an operator then it is expanded in terms of bases states, then we apply the operator over the bases states and we get different eigen values for an observable with different probabilities corresponding to the projection coefficients of the bases states.

In a book, I read that after making a measurement we get an observation, and the probabilities of other observations shrink down to zero. The observation remains the same for howsoever times we make it after the measurement. Like if a coin is tossed, if we get head after first measurement it remains head, even if we make any number of observations. My question is that what is the need of expectation value of an observable (given by formula as summation of the product of probability associated with an observable and the eigen value of observable) as after measurement we get only one eigen value of the observable with 100% certainty. I think knowing only the possible values of the observable is sufficient. What is the significance of expectation value and then does the formula for expectation value given in books is for before the measurement?

• You perform measurements because you either (1) need to calibrate your measuring device or (2) do not know (and want to know) the value of a property for the system. There is no point otherwise in performing a measurement. Once you perform a measurement (that results say in an ensemble of expectation values for different possible values of the observable), you might want to compare the results to theoretical predictions. Once you have fully characterized the system, you might use it as a reference (as in case #1 above) to compare to other systems in future experiments. – Buck Thorn Apr 16 '20 at 15:29
• To whoever gave the -1. It's a brand new user. Rather than giving -1, why not edit the question? – user1271772 Apr 17 '20 at 1:53

## 1 Answer

Roughly, the significance is that before measurement the system can be in a superposition of states, the wavefunction of which can be represented by a linear combination of the individual eigenfunctions

$$\Psi = \sum_i c_i{\psi_i}$$

However the very act of observing the system reduces the wavefunction to a single eigenstate due to interaction with the external world

For most cases, the expectation value is just the quantum mechanical way of calculating properties. Remember, the wavefunction is not observable! Only properties like density, dipole moment, etc are and the expectation value allows us a way to go from wavefunctions to properties via operators.

These other answers may also be helpful