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I understand that in order to calculate the average of a certain operator (observable), whatever that observable may be that we are interested in, we need to prepare many many many identical copies of the same state and apply the operator of interest to those identical state. By probability, we would then obtain several different values of that observable (these values are different eigenvalues of the operator). We would then take the total average of those many values to obtain the expectation value.

The expectation value of a particular observable is given by the integral:

$$<\hat A>=\int \Psi^*(x) \hat A \Psi(x) dx$$

where we sandwich the operator between the wavefunction ##\Psi(x)## and its complex conjugate wavefunction ##\Psi^*(x)##.

Does it matter what wavefunction ##\Psi(x)## we use to calculate this expectation value integral? Can the system be in any state/wavefunction ##\Psi(x)## when we calculate the expectation value? I don't think so since different functions ##\Psi(x)## would produce different values for the integral.

The wavefunction ##\Psi(x)## could be an eigenfunction of a certain operator, of the eigenfunction of other operator, etc.

In essence, what wavefunction do we use inside the integral?

thanks for any clarification.

Fog37

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# I Expectation of an operator (observable) how to calculate it

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