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**operator the expectation value can be found by calculating the (relative) probabilities of each eigenvalue: square modulus of the projection of the state-vector along the corresponding eigenvector.**

*Hermitian*The normalization of these values give the absolute probabilities.

Alternatively it's possible to calculate directly the expectation value by the compact formula <A>=<psi|A|psi>.

I got stumbled considering what adjustment to take in case of an operator that is

**.**

*not Hermitian*In this case the eigenbasis is not orthonormal. But I feel that there should be some way to calculate the expectation value of the operator nonetheless.

Am I wrong? Or there's no meaningful way to define it in the non hermitian case?

If the procedure is more or less the same with some adaptation to make then:

Which projection to take? the components in the eigenbasis or the orthogonal projections of psi along the eigenvectors?

Is there an analog of the formula <A>=<psi|A|psi> that is correct in the non.orthonormal basis?

Any help appreciated

Thanks