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Why does the mathematical definition of a spectrum of an operator namely the set of complex numbers without the resolvent set, agreed with the real physical spectrum of an observable?

is this a postulate? and if it is, is there any motivation behind it?

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  • $\begingroup$ What do you mean by "Why does the mathematical definition of a spectrum agree with the real physical spectrum?". The Schroedinger equation is essentially an eigenvalue equation, therefore the energy is by definition the spectrum. Then why the Schroedinger equation is in fact an operator equation is another type of question. $\endgroup$ – gented Aug 28 '17 at 12:26
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It is indeed a postulate of quantum mechanics, as illuminated in Sakurai's quantum mechanics text, if you'd like a reference.

Specifically, to every observable we may associate a linear, Hermitian operator $\mathcal O$, whose spectrum of eigenvalues correspond to the spectrum of possible observations of said observable.

By the spectral theorem, we are guaranteed for the eigenvalues of $\mathcal O$ to be real, as required, since we expect measurements to yield real rather than complex values.

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Yes, in the standard (von Neumann-Dirac) axiomatization, it is said that whatever experimental values obtained when measuring the observable A, they are real numbers pertaining to the spectrum of the self-adjoint operator A describing the observable. The motivation: the Hilbert-space spectrum of a self-adjoint operator is a subset of $\mathbb{R}$. We expect to measure observables and find real numbers as results.

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