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In Quantum Mechanics, operators that represent physical observables are required to be Hermitian so that their eigenvalues are real and, hence, measurable.

However, the probability of observing an electron is obtained from a complex wave function by taking its modulus.

My question is why not allow non-hermitian operators and complex eigenvalues, with the prescription that the real measurable is the modulus of the complex eigenvalue?

john
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    When building a new theory, it is always helpful to first restrict the complexity of the theory by concentrating on nice subsets. It turns out, keeping ourselves to Hermitian operators can describe all the physics we want to cover, so that is what we teach. There have already been people who have discovered, say, that treating the QHO with non-Hermitian operators is slightly more mathematically convenient, but then the interpretation of the results is a lot more difficult. What is the goodness you are hoping to get out of this? – naturallyInconsistent Dec 19 '23 at 12:19
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    Does this answer your question? Why hermitian, after all? And links therein – Tobias Fünke Dec 19 '23 at 12:53
  • I think you have answered my question. If I understand your answer you are saying that Hermitian operators and real eigenvalues nicely describe all the physics we encounter. Basically all of the operators we encounter turn out to be Hermitian and everything works so why rock the boat? That's fine. This implies to me that being Hermitian is not an a priori requirement for an operator - it just turns out that they all are. – john Dec 19 '23 at 13:04
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    Not every operator is hermitian (and in particular not every operator we use in QM!) and as the linked question/answer discusses, there is no a priori need to use hermitian operators. But as suggested by the comment of @naturallyInconsistent, they are convenient and at least for the the standard formulation of QM, using projection valued measures, they suffice. – Tobias Fünke Dec 19 '23 at 13:15
  • https://physics.stackexchange.com/a/526270/28512 – alanf Dec 19 '23 at 16:46

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