# “An operator is hermitian”. Implications?

Alastair Rae states that there are 4 postulates of Quantum Mechanics in his text on the subject matter. The first part of his second postulate can be stated as:

Every dynamical variable may be represented by a Hermitian operator whose eigenvalues represent the result of carrying out a measurement of the value of the dynamical variable...

My question is: just how much can we deduce from saying that the some of the operators involved in Quantum Mechanics are Hermitian?

When I use the word Hermitian, I am referring to the property that $A=A^+$ where $A$ is an operator and $A^+$ is the adjoint of $A$. Does an operator being Hermitian automatically imply that there will exist eigenfunctions and eigenvalues corresponding to this operator? Does being Hermitian imply that, should there exist eigenfunctions corresponding to this operator, they will form a complete and orthonormal set? That any wavefunction can be expanded in terms of these eigenfunction? Just how much is hidden in the word "Hermitian"?

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–  Emilio Pisanty Feb 17 '13 at 0:07

For an operator $A$ on a finite-dimensional Hilbert space $\mathcal H$, one can show that there exists an orthonormal basis for the Hilbert space consisting of eigenvectors of the operator $A$. Moreover, one can show that the eigenvalues corresponding to these eigenvectors are all real. This result is the finite-dimensional Spectral Theorem. The fact that eigenvalues of hermitian operators are real is of crucial importance in quantum mechanics since eigenvalues of observables are supposed to represent real-valued, physically measurable quantities.