Expectation value of an observable in quantum system

Question

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?

Answer 1

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

https://physics.stackexchange.com/questions/67756/quantum-collapse https://philosophy.stackexchange.com/questions/48425/is-collapse-of-the-quantum-state-conceptually-misnamed https://physics.stackexchange.com/questions/151457/how-does-a-wavefunction-collapse